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d-form sequences: families of sequences with low correlation values and large linear spans

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Andrew Klapper at University of Kentucky
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Large families of binary sequences with low correlation values and large linear span are critical for spread-spectrum communication systems. The author describes a method for constructing such families from families of homogeneous functions over finite fields, satisfying certain properties. He then uses this general method to construct specific families of sequences with optimal correlations and exponentially better linear span than No sequences (No. 1988)
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d
-Form Sequences: Families of Sequences with Low
Correlation Values and Large Linear Spans
Andrew Klapper
Department of Computer Science
915 Patterson Oce Tower
University of Kentucky
Lexington, KY 40506-0027
E-mail: klapper@ms.uky.edu
September 11, 1995
Abstract
Large families of binary sequences with low correlation values and large linear span are
critical for spread spectrum communication systems. In this paper we describ e a method for
constructing such families from families of homogeneous functions over nite elds, satisfying
certain properties. We then use this general method to construct specic families of sequences
with optimal correlations and exponentially better linear span than No sequences.
Index Terms
{Cross-correlation, autocorrelation, linear span, binary sequences, fam-
ilies of sequences, No sequences, spread spectrum, CDMA, nite elds, quadratic
forms.
I Introduction
The volume of communication trac over a variety of media has been steadily increasing over the
past few decades, and will continue to do so. This increase has led to a need for methods that allow
many users to share communication channels. Among proposed methods for such sharing, Co de
Division Multiple Access holds great promise, in part due to its ability to resist interference from
Project sponsored by the Natural Science and Engineering Research Council under Grant Number OGP0121648
and the National Security Agency under Grant Number MDA904-91-H-0012. The United States Government is
authorized to reproduce and distribute reprints notwithstanding any copyright notation hereon. Portions of this
work have b een presented at the 31st Annual Allerton Conference on Communications, Control, and Computing,
Urbana-Champaign, IL, Sept., 1993.
1
hostile agents [17]. In order for this promise to b e met, however, it is essential to nd large families
of easily generated binary sequences with high linear spans and low correlation function values.
The smaller the pairwise cross-correlations and the larger the family, the higher the capacity of
the system. Also, the higher the linear span, the harder it is for an adversary to jam or intercept
messages.
Unfortunately, there are a limited number of known instances of such sequences. The correlation
properties of sequences generated by various modied shift registers have b een studied, including
GMW sequences, [3], geometric sequences [8], cascaded GMW sequences [1, 9]. The results are
often closely related to results from coding theory, as in the case of
?
1 decimations of m-sequences
[18]. In a few cases families of sequences with go od correlation properties have been found, such
as Kasami sequences [4, 5], bent function sequences [10, 11, 15, 16], and No sequences [14]. Table
(1) summarizes the properties of some of these families.
In this paper we present a general method for constructing families of sequences with low
cross-correlation values from families of homogeneous functions (
d
-forms if the degree is
d
) with
certain properties. The families of sequences that arise include No sequences [14]. We then exhibit
a particular family of quadratic forms with the required property, and show that the linear spans
of the resulting sequences, trace norm (or TN) sequences, are larger (in some cases, asymptotically
exponentially larger) than the linear spans of No sequences. Moreover, we describe a method
for implementing TN sequences that shows that they are no more dicult to generate than No
sequences.
In his thesis No introduced a class of sequences called generalized No sequences [13]. These
sequences are formed by iteratively applying functions of the form \trace of a p ower of
x
" to
a No sequence with values in a nite eld. He derived the balance and cross-correlation and
autocorrelation values and their distributions for generalized No sequences. He described as open
problems the determination of the linear span and the implementation of generalized No sequences.
A sequence is a TN sequence if and only if it is both a
d
-form sequence with
d
= 2 and a gener-
alized No sequences. In this paper we resolve No's questions ab out linear span and implementation
for this case of generalized No sequences. We also count the numb er of distinct families of TN
sequences. No's results on the balance and correlation properties of generalized No sequences give
the balance and correlation prop erties of TN sequences. We have, however, included sketches of
the pro ofs of these results (Theorem I I I.2 and Proposition I I I.4) since they are examples of the ap-
plication of the general result on cross-correlations of
d
-form sequences, they are not yet available
in the archival literature, and they were discovered independently by the author.
In what follows, let
e
and
m
be positive integers, let
n
=
em
, and let
T r
n
m
be the trace function
from
GF
(2
n
) to
GF
(2
m
),
T r
n
m
(
x
) =
e
?
1
X
i
=0
x
2
mi
:
Also, let
N
n
m
be the norm function from
GF
(2
n
) to
GF
(2
m
),
N
n
m
(
x
) =
e
?
1
Y
i
=0
x
2
mi
=
x
(2
n
?
1)
=
(2
m
?
1)
:
2
Size of Maximum Maximum Range of
Family
n
Family Correlation Linear Span Imbalance
Gold 2
m
+ 1 2
n
+ 1 1 + 2
n
+1
2
2
n
[1
;
2
n
+1
2
+ 1]
Gold 4
m
+ 2 2
n
?
1 1 + 2
n
+2
2
2
n
[1
;
2
n
+2
2
+ 1]
Kasami 2
m
2
n
2
1 + 2
n
2
3
n
2
[1
;
2
n
2
+ 1]
(Small Set)
Kasami 4
m
+ 2 2
n
2
(2
n
+ 1) 1 + 2
n
+2
2
5
n
2
[1
;
2
n
+2
2
+ 1]
(Large Set)
Bent 4
m
2
n
2
1 + 2
n
2
n=
2
n=
4
2
n
4
1
No 2
m
2
n
2
1 + 2
n
2
m
(2
m
?
1) [1
;
2
n
2
+ 1]
TN 2
km
2
n
2
1 + 2
n
2
>
3
mk
(3
k
?
1)
m
?
2
[1
;
2
n
2
+ 1]
Table 1:
Comparison of Properties of Families of Sequences of Period
2
n
?
1
For simplicity we write
q
= 2
m
. By a
d
-form we mean a homogeneous function of degree
d
. That
is, a function
H
is a
d
-form, if for any
x
2
GF
(
q
e
) and
y
2
GF
(
q
), we have
H
(
yx
) =
y
d
H
(
x
)
:
For example, any
H
which is a homogeneous polynomial of degree
d
(thinking of
GF
(
q
e
) as an
e
-dimensional vector space over
GF
(
q
)) is a
d
-form. More generally, any sum of functions of the
form
H
(
x
) =
T r
km
m
(
N
n
km
(
x
r
))
;
where the sum of the coecients in the base
q
expansion of
r
(
q
e
?
1)
=
(
q
k
?
1) is congruent to
d
modulo
q
?
1, is a
d
-form. Every
d
-form is a sum of functions of this form.
We generate
d
-form sequences in three steps. Our construction is based on the nite elds
GF
(
q
) and
GF
(
q
e
). We start with a sequence of p owers of a primitive element in
GF
(
q
e
). To
this sequence we apply a
d
-form
H
mapping to
GF
(
q
). We then raise the result to some power.
Finally, we apply a trace function mapping to
GF
(2). The precise denition of
d
-form sequences
is as follows.
Denition I.1
Let
e
and
m
be positive integers, and let
q
= 2
m
. Let
r
and
d
be positive integers
such that
gcd(
r; q
?
1) = gcd(
d; q
?
1) = 1
. Let
be a primitive element in
GF
(
q
e
)
. Let
H
(
x
)
be
a
d
-form on
GF
(
q
e
)
over
GF
(
q
)
. Then the sequence
S
whose
i
th term is
s
i
=
T r
m
1
((
H
(
i
))
r
) (1)
is called a
d
-form sequence
.
3
II Cross-Correlations of
d
-form Sequences
Recall that the cross-correlation with shift
, of two sequences
S
= (
s
1
; s
2
;

) and
T
= (
t
1
; t
2
;

)
of perio d
N
, is dened by
S
;
T
(
) =
N
X
i
=1
(
?
1)
s
i
+
t
i
+
:
In our case
N
=
q
e
?
1. Our rst result is a complete description of the cross-correlations of
d
-form
sequences in terms of the zeros of
d
-forms.
Theorem I I.1
Let the integers
m
,
e
, and
r
, and primitive element
2
GF
(
q
e
)
be xed, and let
H
1
and
H
2
be
d
-forms on
GF
(
q
e
)
over
GF
(
q
)
dening
d
-form sequences
S
1
and
S
2
, as in equation
(1). For any shift
, let
z
=
jf
x
6
= 0
2
GF
(
q
e
) :
H
1
(
x
) +
H
2
(
x
) = 0
gj
. Then
S
1
;
S
2
(
) =
qz
?
(
q
e
?
1)
q
?
1
:
Proof:
The proof of this theorem is an extension of the pro of of the corresp onding theorem on
No sequences [14].
Let
L
= (
q
e
?
1)
=
(
q
?
1). For any
i
, 0
i < N
, we can write
i
=
i
1
L
+
i
2
;
0
i
1
< q
?
1
;
0
i
2
< L:
Since for any
x
2
GF
(
q
e
), we have
x
L
=
N
em
m
(
x
)
2
GF
(
q
), it follows that
H
j
(
x
i
) =
x
di
1
L
H
j
(
x
i
2
)
:
Thus the terms of the sequence
S
j
can be written
s
j
i
=
T r
m
1
(
dri
1
L
(
H
j
(
i
2
))
r
)
:
Letting
f
(
i
) = (
H
1
(
i
))
r
+ (
H
2
(
i
+
))
r
;
we have
s
1
i
+
s
2
i
+
=
T r
m
1
(
dri
1
L
f
(
i
2
))
:
(2)
Whenever
f
(
i
2
)
6
= 0, the sequence we get from equation (2) by letting
i
1
vary is an m-sequence
of p erio d
q
?
1. It follows from the balance properties of m-sequences [2] that for a xed
i
2
the
contribution of these terms to the cross-correlation is
?
1. On the other hand, when
f
(
i
2
) = 0,
every term is zero, so the contribution of these terms for a xed
i
2
is
q
?
1. If we let
z
be the
number of values of
i
2
, 0
i
2
< L
, for which
f
(
i
2
) = 0, then
S
1
;
S
2
(
) =
?
(
L
?
z
) + (
q
?
1)
z
=
qz
?
q
e
?
1
q
?
1
:
4
To complete the proof, we observe that
f
(
i
+
L
) =
drL
f
(
i
), so that
z
=
jf
i
:
f
(
i
) = 0
;
0
i < N
gj
q
?
1
:
This is precisely
z
=
(
q
?
1), since
i
ranges through all nonzero
x
as
i
ranges from 0 to
N
?
1, and
f
(
i
) = 0 if and only if
H
1
(
i
) =
H
2
(
i
+
).
2
In case
d
= 2 and
H
is a quadratic form (that is,
H
is a homogeneous polynomial of degree
two), the number of zeros of
H
is well understood. Recall that the
rank
of a quadratic form
H
is the smallest integer
t
such that there is a set of co ordinates in which
H
can b e represented
using only
t
variables. The number of solutions
x
to the equation
H
(
x
) = 0 (or, more generally,
H
(
x
) =
a
) is determined by the rank and, in the case of even rank, whether
H
is of one of two
types. A nice treatment of this analysis can b e found in Lidl and Neiderreiter's b o ok [12].
Corollary I I.2
Let
H
1
and
H
2
be quadratic forms dening
d
-form sequences
S
1
and
S
2
, and let
be any integer. If the rank of
H
1
(
x
) +
H
2
(
x
)
is
t
, then
S
1
;
S
2
(
) =
?
1
when
t
is odd, and
S
1
;
S
2
(
) =
q
e
?
t=
2
?
1
when
t
is even. In particular, if for every
the rank is
n
or is odd, then the cross-correlations of
S
1
and
S
2
are three valued with values in
f?
q
e=
2
?
1
;
?
1
; q
e=
2
?
1
g
.
Proof:
The corollary follows from the fact that a quadratic form with odd rank takes the value
zero
q
e
?
1
times, while a quadratic form with even rank
t
takes the value zero
q
e
?
1
(
q
?
1)
q
e
?
t=
2
?
1
times [12].
2
Corollary I I.3
If
F
is a family of quadratic forms on
GF
(
q
e
)
over
GF
(
q
)
such that for any
H
1
and
H
2
in
F
, and integer
, the rank of the quadratic form
H
1
(
x
) +
H
2
(
x
)
is
e
or odd,
then
F
denes a family of sequences with three valued cross-correlations with values in
f?
q
e=
2
?
1
;
?
1
; q
e=
2
?
1
g
.
III Trace Norm (TN) Sequences
In this section we describe a class of families of quadratic form sequences that achieve the cross-
correlations of Corollary II.3 and have large linear span. These are exactly the sequences that are
simultaneously
d
-form sequences and generalized No sequences [13]. The results in this section
were proved indep endently by No in his thesis. However, they do not app ear in the archival
literature and they are examples of the application of Theorem II.1, so we include the full details.
5
Denition I I I.1
Let
m
and
k
be positive integers, let
q
= 2
m
, and let
e
= 2
k
and
n
= 2
mk
. Let
r
be a positive integer such that
gcd(
r; q
?
1) = 1
. Let
be a primitive element in
GF
(
q
2
k
)
and
let
be an element of
GF
(
q
k
)
. Then the sequence
S
whose
i
th term is
s
i
=
T r
m
1
((
T r
mk
m
(
T r
2
mk
mk
(
2
i
) +
N
2
mk
mk
(
i
)))
r
)
is a
Trace Norm
(or
TN
) sequence.
Note that, if we let
T
=
q
k
+ 1, then
N
2
mk
mk
(
i
) =
T i
. This sequence is a 2-form sequence based
on
H
(
x
) =
T r
mk
m
(
T r
2
mk
mk
(
x
2
) +
N
2
mk
mk
(
x
))
:
We are interested in families of TN sequences with all parameters other than
xed.
Theorem I I I.2
Let
S
and
S
be two TN sequences, based on the same integers
m
,
k
, and
r
.
Then the cross-correlations of
S
and
S
are three valued, with values in
f?
q
k
?
1
;
?
1
; q
k
?
1
g
unless
S
=
S
and
= 0
.
Proof:
By Theorem II.1, it suces to determine the number of zeros
z
of the quadratic form
H
(
x
)
def
=
T r
mk
m
(
T r
2
mk
mk
(
x
2
) +
x
T
) +
T r
mk
m
(
T r
2
mk
mk
(
2
x
2
) +
T
x
T
)
=
T r
mk
m
(
T r
2
mk
mk
((1 +
2
)
x
2
) + (
+
T
)
x
T
)
:
For simplicity, we write
A
= (1 +
) and
B
= (
+
T
).
Let
G
(
x
) =
T r
2
mk
mk
(
A
2
x
2
) +
Bx
T
. Then
G
is a quadratic form on
GF
(
q
2
k
) over
GF
(
q
k
). Since
G
(
ax
) =
a
2
G
(
x
), and every element of a nite eld of characteristic two has a unique square ro ot,
G
takes on every nonzero value an equal numb er of times. If
w
=
jf
x
:
G
(
x
) = 0
gj
, then
G
takes
on any nonzero value precisely (
q
2
k
?
w
)
=
(
q
k
?
1) times. Moreover,
T r
mk
m
is a balanced function.
It follows that
z
=
wq
k
?
1
(
q
?
1) +
q
2
k
(
q
k
?
1
?
1)
q
k
?
1
?
1
.
The value of
w
can be computed by cases, dep ending on
,
, and
. For example, supp ose
6
=
T
and
6
= 1, i.e.,
B
6
= 0 and
A
6
= 0. Then
G
(
x
) = 0 if and only if there is a
2
GF
(
q
k
)
such that
T r
2
mk
mk
(
Ax
) =
and
Bx
T
=
2
:
Thus we must count the common zeros of a linear function and a quadratic form in two variables
over
GF
(
q
k
). These have been completely analyzed [7]. Observe that the quadratic form
Bx
T
is
zero only for
x
= 0, hence in the terminology of [7], is a Type II I quadratic form, and has rank
two. This allows us to apply Proposition 3.4 of [7] to show that
w
= 1 or 2
q
k
?
1. It follows that
S
;
S
(
) =
?
q
k
?
1 or
q
k
?
1.
2
The imbalance
I
(
S
) of a binary sequence
S
is the number of zeros in
S
minus the number of
ones. The proof of the Theorem II I.2 can be used to nd the imbalance of a TN sequence.
6
Proposition II I.3
The imbalance of a TN sequence is
?
1
,
?
q
k
?
1
, or
q
k
?
1
.
Proof:
If
= 0, then we have a GMW sequence, which is well known to have imbalance
?
1. If
6
= 0, then we can count as in Case 3 of the proof of Theorem I I I.2 with
A
= 1 and
B
=
. The
proposition follows.
2
There are six possible distributions of values of the cross-correlation of two TN sequences.
Proposition II I.4
Let
S
and
S
be two TN sequences based on the same eld parameters
m
and
k
and exponent
r
. The distribution of values of the cross-correlation of
S
and
S
is given by one
of the following cases. The rst three cases correspond to autocorrelations.
1.
q
2
k
?
1
occurs once and
?
1
occurs
q
2
k
?
2
times.
2.
q
2
k
?
1
occurs once,
?
1
occurs
q
k
times,
q
k
?
1
occurs
q
2
k
=
2
times, and
?
q
k
?
1
occurs
q
2
k
=
2
?
q
k
?
2
times.
3.
q
2
k
?
1
occurs once,
?
1
occurs
q
k
times,
q
k
?
1
occurs
q
2
k
=
2
?
2
times, and
?
q
k
?
1
occurs
q
2
k
=
2
?
q
k
times.
4.
q
k
?
1
occurs
(
q
2
k
+
q
k
)
=
2
?
1
times and
?
q
k
?
1
occurs
(
q
2
k
?
q
k
)
=
2
times.
5.
q
k
?
1
occurs
(
q
2
k
+
q
k
)
=
2
times and
?
q
k
?
1
occurs
(
q
2
k
?
q
k
)
=
2
?
1
times.
6.
?
1
occurs
q
k
+ 1
times,
q
k
?
1
occurs
(
q
2
k
+
q
k
)
=
2
times, and
?
q
k
?
1
occurs
(
q
2
k
?
3
q
k
)
=
2
?
2
times.
7.
?
1
occurs
q
k
+ 1
times,
q
k
?
1
occurs
(
q
2
k
+
q
k
)
=
2
?
2
times, and
?
q
k
?
1
occurs
(
q
2
k
?
3
q
k
)
=
2
times.
8.
?
1
occurs
q
k
+ 1
times,
q
k
?
1
occurs
(
q
2
k
?
q
k
)
=
2
?
1
times, and
?
q
k
?
1
occurs
(
q
2
k
?
q
k
)
=
2
?
1
times.
Proof:
If
=
= 0, then we have the autocorrelation of a GMW sequence. Otherwise, let
C
(respectively,
D
; resp ectively,
E
) be the number of occurences of
S
;
S
(
) =
?
1 (respectively,
S
;
S
(
) =
?
q
k
?
1; respectively,
S
;
S
(
) =
q
k
?
1). If
6
=
= 0, then in the proof of Theorem
III.2 we have
B
6
= 0, and the only values that appear are
?
q
k
?
1 and
q
k
?
1. That is,
C
= 0.
Otherwise the cross-correlation is
?
1 for
q
k
+ 1 shifts, so
C
=
q
k
+ 1. In general, we have
X
S
;
S
(
) =
I
(
S
)
I
(
S
)
;
where
I
(
S
) and
I
(
S
) are the imbalances of
S
and
S
, respectively.
Thus we have an equation
?
C
+ (
?
q
k
?
1)
D
+ (
q
k
?
1)
E
=
I
(
S
)
I
(
S
)
:
Moreover,
C
+
D
+
E
=
q
2
k
?
1
:
Considering the various values of
,
,
, and
C
, we can solve these equations to get the various
possibilities in the statement of the proposition.
2
7
IV Linear Span of TN Sequences
In this section we show that the linear span of a TN sequence is at least that of a GMW sequence
with the same period, and that for
q
large enough and
k
= 2, it exceeds the linear span of a No
sequence with the same perio d. The development is similar to that in the case of No sequences,
with some additional complication due to the additional trace function.
Key [6] showed that if we express the
i
th term of a sequence
S
as a polynomial in
i
, then
the linear span
S
is the number of monomials in the polynomial. That is, we must count the
monomials in the polynomial
s
(
x
) =
T r
m
1
((
T r
mk
m
(
T r
2
mk
mk
(
x
2
) +
x
T
))
r
)
=
T r
m
1
((
T r
mk
m
(
x
2
+
x
2
q
k
+
x
T
))
r
)
=
T r
m
1
((
T r
mk
m
(
x
2
(1 +
y
+
y
2
)))
r
)
;
where
y
=
x
q
k
?
1
. Expanding the trace functions, we see that
s
(
x
) =
m
?
1
X
j
=0
(
k
?
1
X
i
=0
x
2
j
+1
q
i
(1 +
y
+
y
2
)
2
j
q
i
)
r
:
Lemma IV.1
Distinct terms from the outer sum have distinct degree monomials.
Proof:
First reduce all exp onents mo d
q
k
?
1 (which divides
q
2
k
?
1, so this reduction is compatible
with
x
q
2
k
=
x
). Then
y
becomes 1, so the degrees of the monomials resulting from the
j
th term
are the degrees of the monomials in
(
k
?
1
X
i
=0
x
2
j
+1
q
i
)
r
= (
k
?
1
X
i
=0
x
q
i
)
r
2
j
+1
; j
= 0
;

; m
?
1
:
Thus it suces to compare the degrees of the monomials in the terms
(
k
?
1
X
i
=0
x
q
i
)
r
and (
k
?
1
X
i
=0
x
q
i
)
r
2
j
:
These polynomials have monomials with degrees
A
=
k
?
1
X
i
=0
a
i
q
i
;
with
k
?
1
X
i
=0
a
i
=
r < q
(3)
and
B
=
k
?
1
X
i
=0
b
i
2
j
q
i
;
with
k
?
1
X
i
=0
b
i
=
r < q ;
(4)
respectively. In particular, 0
a
i
; b
i
< q
, so the representations in equations (3) and (4) are the
(unique) base
q
representations.
8
Let
b
i
=
c
i
+ 2
m
?
j
d
i
;
0
c
i
<
2
m
?
j
;
0
d
i
<
2
j
:
Then
B
=
k
?
1
X
i
=0
c
i
2
j
q
i
+
d
i
q
i
+1
k
?
1
X
i
=0
(
d
i
?
1
+
c
i
2
j
)
q
i
(mod
q
k
?
1)
is the (unique) base
q
representation mo dulo
q
k
?
1 (where arithmetic on the subscripts is modulo
k
).
Suppose
A
=
B
. Then for every
i
,
a
i
=
d
i
?
1
+
c
i
2
j
. Consequently,
r
=
k
?
1
X
i
=0
(
d
i
?
1
+
c
i
2
j
) =
k
?
1
X
i
=0
(
c
i
+
d
i
2
m
?
j
)
:
Letting
C
=
P
i
c
i
and
D
=
P
i
d
i
, this reduces to
r
=
D
+ 2
j
C
=
C
+ 2
m
?
j
D;
so
(2
j
?
1)
C
= (2
m
?
j
?
1)
D:
Let
t
be the greatest common divisor of
m
and
j
. Then the greatest common divisor of 2
j
?
1 and
2
m
?
j
?
1 is 2
t
?
1, so there is an integer
E
such that
C
=
2
m
?
j
?
1
2
t
?
1
E
and
D
=
2
j
?
1
2
t
?
1
E:
It follows that
r
=
E
2
j
?
1
2
t
?
1
+ 2
j
2
m
?
j
?
1
2
t
?
1
!
=
E
2
m
?
1
2
t
?
1
:
However, the greatest common divisor of
r
and 2
m
?
1 is one, so we must have
t
=
m
, i.e.,
m
divides
j
, which is impossible.
2
This lemma implies that
S
=
m
8
<
:
monomials in
k
?
1
X
i
=0
x
2
q
i
(1 +
y
+
y
2
)
q
i
!
r
9
=
;
:
(5)
9
Let
r
=
m
?
1
X
j
=0
r
j
2
:
Then for any
z
,
k
?
1
X
i
=0
z
q
i
!
r
=
Y
r
j
6
=0
k
?
1
X
i
=0
z
2
j
q
i
!
=
X
(
i
j
)
Y
r
j
6
=0
z
2
j
q
i
j
=
X
(
i
j
)
z
P
r
j
6
=0
2
j
q
i
j
;
where the outer sums are over all vectors of
i
j
, indexed by those
j
such that
r
j
6
= 0, and with
0
i
j
< k
. Consider a term with exp onent
P
r
j
6
=0
2
j
q
i
j
. For each
i
= 0
;

; k
?
1, let
a
i
be the
number whose binary representation has a 1 in the
j
th bit if and only if
i
j
=
i
. Then
X
r
j
6
=0
2
j
q
i
j
=
k
?
1
X
i
=0
a
i
q
i
;
and this last is the base
q
expansion of this exponent. For a given bit
j
, at most one
a
i
has a 1 in
bit
j
, so the bitwise \AND" of any two dierent
a
i
s is zero. If
r
has a 1 in bit
j
, then some
a
i
has
a 1 in bit
j
, so the bitwise \OR" of all the
a
i
equals
r
. That is, the bits of
r
are distributed among
the
a
i
, maintaining their relative bit positions. In particular,
a
i
r < q
. For example, writing
r
and the
a
i
s in base 2, if
k
= 2 and
r
= 11110, then we can have
a
0
= 11110 and
a
1
= 00000, or
a
0
= 11010 and
a
1
= 00100, or
a
0
= 01010 and
a
1
= 10100, etc.
It follows that the
r
th power in equation (5) can be expanded as
X
a
k
Y
i
=1
(
x
2
q
i
(1 +
y
+
y
2
)
q
i
)
a
i
=
X
a
x
2
P
a
i
q
i
(1 +
y
+
y
2
)
P
a
i
q
i
;
where the sum is over all
a
= (
a
0
;

; a
k
?
1
) such that 0
a
i
, the bitwise \AND" of any two
dierent
a
i
s is zero, and the bitwise \OR" of all the
a
i
equals
r
. Reducing exponents mo dulo
q
k
?
1, we get monomials of degree 2
P
i
a
i
q
i
. Each such expression is twice the unique base
q
representation of an integer, hence these monomials are pairwise distinct. Thus we have the
following prop osition.
Proposition IV.2
The linear span of the TN sequence
S
is given by
S
=
m
X
a
jf
monomials in
(1 +
y
+
y
2
)
P
a
i
q
i
gj
;
where the sum is over al l
a
= (
a
0
;

; a
k
?
1
)
such that
0
a
i
, the bitwise \AND" of any two
a
i
s is
zero, and the bitwise \OR" of al l the
a
i
equals
r
.
10
The cardinalities in the sums in this proposition have been evaluated by No and Kumar [14]
in describing the linear span of No sequences as follows. Supp ose
t < q
k
=
2 is a positive integer
with
R
runs of ones, of lengths
L
1
;

; L
R
and
6
= 0 in its base two expansion
1
. Let
=
?
1 if the
quadratic
y
2
+
y
+ 1 is reducible over
GF
(
q
k
) (that is, if
T r
mk
1
(1
=
) = 0) and
= 1 otherwise. For
a xed primitive element
2
GF
(
q
2
k
), there is a unique (principal) root
=
b
of the quadratic
y
2
+
y
+ 1 such that 0
b <
(
q
2
k
?
1)
=
2. When
=
?
1, we have
b
=
c
(
q
k
+ 1), and when
= 1,
we have
b
=
c
(
q
k
?
1). Thus 0
c
q
k
=
2. Let
g
= gcd(
c; q
k
+
). Then the number of nonzero
terms in (
y
2
+
y
+ 1)
t
is
R
Y
j
=1
2
L
j
+1
?
1
?
2
$
(2
L
j
?
1)
g
q
k
+
%!
:
When
= 0, the number of nonzero terms is 2
wt(
t
)
, where wt(
t
) =
P
j
L
j
is the number of ones in
the binary expansion of
t
.
Combining this with Proposition IV.2, we have
Theorem IV.3
Let
S
be a TN sequence based on elds
GF
(
q
)
and
GF
(
q
k
)
, exponent
r
, and
coecient
2
GF
(
q
k
)
. When
6
= 0
, let
=
?
1
if the quadratic
y
2
+
y
+ 1
is reducible over
GF
(
q
k
)
and
= 1
otherwise. Let
c
be such that
=
c
(
q
k
+1)
when
=
?
1
and
=
c
(
q
k
?
1)
when
= 1
is a root of
y
2
+
y
+ 1
, with
0
c
q
k
=
2
. Let
g
= gcd(
c; q
k
+
)
.
For each
a
= (
a
0
;

; a
k
?
1
)
such that
0
a
i
, the bitwise \AND" of any two dierent
a
i
s is
zero, and the bitwise \OR" of all the
a
i
equals
r
, let
R
a
be the number of runs of ones in
P
a
i
q
i
,
and let
L
a;
1
;

; L
a;R
a
be the lengths of the runs. Then the linear span of
S
is given by
S
=
m
X
a
R
a
Y
j
=1
2
L
a;j
+1
?
1
?
2
$
(2
L
a;j
?
1)
g
q
k
+
%!
:
If
= 0
, then the linear span of
S
is given by
S
=
m
X
a
2
wt(
r
)
:
1
It should be noted that the necessary assumption that
t < q
k
=
2 was not explicitly stated by No and Kumar. It
poses no diculty for their analysis, however, since any exponent over
GF
(
q
k
) which is relatively prime to
q
k
?
1
can be assumed to be less than
q
k
?
1. Such an exponent thus has at least one zero in the rst
mk
bits of its base
two expansion. Since No sequences are invariant under cyclic shift of the exp onent
r
, we can assume the exponent
has a zero as its high bit. In the more general case of TN sequences, we can also cyclically shift
r
so its high bit is
zero, and therefore the high bit of each
P
i
a
i
q
i
is zero.
11
We next want to determine how the linear span can be maximized. First observe that if we
choose
so that
gcd(
c; q
k
+
)
<
q
k
+
2
m
?
1
?
1
q
k
+
2
L
a;j
?
1
;
then all terms involving the oor will disappear, and the linear span will be
S
=
m
X
a
R
a
Y
j
=1
2
L
a;j
+1
?
1
:
This can be done indep endently of the choice of
r
(say by cho osing
c
relatively prime to
q
k
+
),
so the maximum value of the linear span o ccurs with such a choice.
Next observe that the eect on an
a
of increasing the number of ones in the binary expansion
of
r
is either to increase the length of a run of ones in
P
i
a
i
q
i
by one, or to merge a run of length
L
and a run of length
K
into a run of length
L
+
K
+ 1. In either case, the contribution to the
linear span is increased, since
(2
L
+1
?
1)
<
(2
L
+2
?
1) and (2
L
+1
?
1)(2
K
+1
?
1)
<
(2
K
+
L
+2
?
1)
:
Thus the linear span is maximized by maximizing wt(
r
). We have
r < q
?
1, so wt(
r
) is maximized
at
m
?
1, that is, when
r
has one zero and
m
?
1 ones in its binary expansion. Since the sequence,
and hence the linear span, is indep endent of cyclic shifts of
r
, the maximal linear span occurs when
r
is a string of ones followed by a single zero. That is
r
= 2
m
?
1
?
1.
We next estimate the linear span for these
r
by recursively estimating the linear span for the
exponent
r
i
which consists of
i
ones followed by
m
?
i
zeros. If
i >
0, we can produce
r
i
from
r
i
?
1
by replacing the rst zero with a one. Each
a
0
for
r
i
?
1
gives rise to
k
a
's for
r
i
, dep ending on which
a
i
receives the new bit. For
k
?
1 of them, we are adding a run of length 1, hence multiplying the
contribution to the linear span by 3. For the remaining
a
, we are increasing the length of a run
by 1, and this run has length at most
i
?
1. Thus the contribution of this term is
=
2
t
+1
?
1
2
t
?
1
times the contribution of the original term, for some
t
such that 2
t < i
. Therefore 2
<
7
=
3.
Also, when
r
has a single one in its binary expansion, the linear span is exactly 3
k
. It follows that
the linear span satises
3
mk
(3
k
?
1)
m
?
2
<
S
< m
(3
k
)
2
(3
k
?
2
3
)
m
?
3
:
For large enough
n
= 2
mk
, the linear span is maximized by taking
k
= 2, and can b e made at
least
3
n
2
5
n=
4
?
2
:
12
We have shown experimentally that the base of 5 in this expression in fact gets close to 5
:
24 as
n
increases, and we can come closer to this value theoretically by improving the estimates for
r
i
for
small
i
. By comparing the values achieved, we nd that for
n
40 the linear span is maximized
by taking
k
= 2. For smaller
n
, experimental data show that the linear span is maximized by
taking
k
= 1, i.e., by No sequences. The linear span in this case is at most
n
2
(2
n=
2
?
1)
<
n
2
4
n=
4
:
The maximum linear span of TN sequences with
k
= 2 and p erio d 2
n
?
1 grows with
n
at a rate
of
O
(
n
5
n=
4
). This is exp onentially larger than the rate of growth of the maximum linear span of
No sequences with the same p erio d, that growth rate b eing
O
(
n
4
n=
4
).
Theorem IV.4
The maximum possible linear span for a TN sequence is achieved by taking
k
= 2
and
r
= 2
n=
4
?
1
?
1
when
n
40
, and by taking
k
= 1
and
r
= 2
n=
2
?
1
?
1
when
n <
40
.
In general, we see that the linear span of TN sequences can be the same as or greater than that
of No sequences while p erforming the exponentiation in a smaller eld, with a smaller exponent
r
. This leads to more ecient implementation of generators of the sequences, as discussed in the
next section.
We would also like to know how many sequences in a family have this maximum linear span.
In fact, this happ ens for most sequences when
r
= 2
m
?
1
?
1. As has been shown above, for the
linear span to be maximal we must have
gcd(
c;
2
mk
+
)
<
2
mk
+
2
m
?
1
+
and 0
c
2
km
?
1
:
(6)
Thus we need to know how many such choices of
c
and
arise from parameters
2
GF
(
q
k
).
Lemma IV.5
For any
2 f
1
g
and
c
such that
0
c
q
k
=
2
;
there is a (unique)
2
GF
(
q
k
)
which gives rise to
; c
.
Proof:
First let
=
?
1, so
=
c
(
q
k
+1)
:
Then
=
N
2
km
km
(
c
)
2
GF
(
q
k
). Therefore,
= (
2
+ 1)
=
2
GF
(
q
k
) and
y
2
+
y
+ 1 is a reducible
polynomial with root
.
Next let
= 1, so
=
c
(
q
k
?
1)
:
Then
2
GF
(
q
2
k
), so it satises a quadratic equation
y
2
+
y
+
for some
;
2
GF
(
q
k
). By
Galois theory we have
=
N
2
mk
mk
(
) =
q
k
+1
= 1. Furthermore, if this equation were reducible,
we would have
2
GF
(
q
k
), so
q
k
=
. That is,
c
(
q
k
?
1)
2
= 1. This implies that
q
k
+ 1 divides
c
,
which is false.
2
13
Thus it suces to count the number of
c
and
satisfying equation (6). We rst observe that
c
= 0 corresp onds to
= 0. Thus if we let
c
range from 1 to
q
k
+
?
1, we will have counted each
exactly twice (
y
2
+ 1 is the only quadratic equation with constant term 1 that has a repeated
root). It follows that for each
, the number of
is
1
2
jf
c
: gcd(
c;
2
mk
+
)
2
mk
+
2
m
?
1
?
1
gj
=
1
2
X
t
j
2
mk
+
2
m
?
1
<t
(
t
)
;
where
(
t
) is Euler's function. The total,
1
2
0
B
B
B
@
X
t
j
2
mk
+1
2
m
?
1
<t
(
t
) +
X
t
j
2
mk
?
1
2
m
?
1
<t
(
t
)
1
C
C
C
A
;
is greater than (
(2
mk
?
1) +
(2
mk
+ 1))
=
2, and is very nearly 2
mk
?
1. That is, the great ma jority
of sequences in this family have the maximum linear span.
V Implementation of TN Sequence Generators
Consider the TN sequence
S
with
s
i
=
T r
m
1
((
T r
mk
m
(
T r
2
mk
mk
(
2
i
) +
iT
))
r
)
=
T r
m
1
((
T r
2
mk
m
(
2
i
) +
T r
mk
m
(
iT
))
r
)
and let
n
= 2
mk
. Since
is a primitive element in
GF
(
q
2
k
),
2
is also a primitive element in
GF
(
q
2
k
) and
def
=
T
is a primitive element in
GF
(
q
k
). It follows that the sequence
T r
2
mk
m
(
2
i
)
is an m-sequence of elements in
GF
(
q
), and can be generated by a linear feedback shift register
(LFSR) of length
n=m
over
GF
(
q
). That is, the elements of the register are elements of
GF
(
q
),
and the feedback function is a linear function in
n=m
variables. Such a register requires only
n
bits
of storage. The arithmetic required is at most
n=m
multiplications by constants in
GF
(
q
) (the
coecients of the minimum polynomial of
2
, some of which may b e zero), and at most
n=m
?
1
additions in
GF
(
q
). The arithmetic can be minimized if
is chosen to minimize the number of
nonzero coecients in its minimal p olynomial.
Similarly, the sequence
T r
mk
m
(
iT
) =
T r
mk
m
(
i
)
is an m-sequence over
GF
(
q
) which can be generated by a LFSR of length
n=
(2
m
) over
GF
(
q
).
This requires
n=
2 bits, and at most
n=
(2
m
) multiplications and
n=
(2
m
)
?
1 additions in
GF
(
q
),
which can be minimized by choosing
to minimize the number of nonzero co ecients in the
14
minimal polynomial of
. Thus the total amount of
GF
(
q
) arithmetic required to implement a TN
sequence is minimized by choosing
so the total numb er of nonzero coecients in the minimal
polynomials of
and
is minimized.
One extra addition is required to combine the outputs of the two LFSRs. The result is then
raised to the
r
th power, and the trace to
GF
(2) computed. However, in representing elements of
GF
(
q
) as
m
-bit vectors, we can cho ose a basis so that the trace of an element is always given by
projection onto a xed comp onent, say the rst. Thus we only need to compute a single bit of the
r
th power.
The dierent choices of
correspond to dierent initial loadings of the second LFSR. Thus an
entire family of TN sequences can be implemented by a single hardware circuit. Changing to a
new sequence is possible by simply resetting the initial loading of the second LFSR.
VI The Number of Distinct Families of TN Sequences
It is useful to know how many distinct families of TN sequences (not necessarily with maximum
linear span) there are with the parameters
n
= 2
`
, and thus the p erio d, xed. In this section we
keep
n
xed and let the factorization
`
=
mk
vary. We show that each choice of parameters
m
,
k
,
r
(up to multiplication by a power of 2), and
(up to raising to an exponent which is a power of
two) gives rise to a distinct family of sequences, in the sense that no sequence in one family is a
cyclic shift of a sequence in another family. For any xed even integer
n
we write
S
(
m; ; r
) =
f
S
:
s
i
=
T r
m
1
((
T r
mk
m
(
T r
2
mk
mk
(
2
i
) +
iT
))
r
) and
2
GF
(2
k
)
g
;
where
m
divides
n=
2,
k
=
n=
(2
m
),
is a primitive element of
GF
(2
n
), and
r
is relatively prime
to 2
m
?
1.
Proposition VI.1
Let
n
= 2
`
,
N
= 2
n
?
1
,
m
1
, and
m
2
be divisors of
`
. Let
r
1
and
r
2
be integers
such that
1
r
i
<
2
m
i
?
1
and
r
i
is relatively prime to
2
m
i
?
1
. Let
1
and
2
be primitive elements
in
GF
(2
n
)
. Then
S
(
m
1
;
1
; r
1
)
and
S
(
m
2
;
2
; r
2
)
are distinct unless either
1.
m
1
=
m
2
, and for some integers
u
and
v
,
0
u<n
, and
0
v < m
1
,
2
=
2
u
1
, and
r
1
= 2
v
r
2
, or
2.
r
1
and
r
2
are powers of
2
and for some integer
u
,
0
u < n
and
2
=
2
u
1
.
In each of these cases
S
(
m
1
;
1
; r
1
) =
S
(
m
2
;
2
; r
2
)
:
Proof:
Suppose that we have a pair of sequences in
S
(
m
1
;
1
; r
1
) and
S
(
m
2
;
2
; r
2
) resp ectively,
such that one is a cyclic shift of the other. Since
n
(and hence
`
) is xed, there are integers
a
,
b
,
c
, and
d
with
abcd
=
`
and
b
relatively prime to
c
, such that
m
1
=
ab
and
m
2
=
ac
. If
r
1
and
r
2
are powers of two, they can b e factored out of the trace functions. This is the second case of the
proposition, and the sequences can b e written with
m
1
=
m
2
. The following lemma shows that
when
r
1
and
r
2
are not both powers of two, the degrees
m
1
and
m
2
must be equal.
15
Lemma VI.2
Assume
r
1
and
r
2
are not both powers of two. Then
m
1
=
m
2
.
Proof of lemma:
For some
1
;
2
2
GF
(2
`
), and
2
GF
(2
n
) we have
T r
ab
1
((
T r
`
ab
(
T r
n
`
(
2
i
1
) +
1
iT
1
))
r
1
) =
T r
ac
1
((
T r
`
ac
(
T r
n
`
(
2
i
2
) +
2
iT
2
))
r
2
) (7)
for every
i
. For some
e
we have
2
=
e
1
, so equation (7) holds if and only if for every
x
2
GF
(2
n
),
T r
ab
1
((
T r
`
ab
(
T r
n
`
(
x
2
) +
1
x
T
))
r
1
) =
T r
ac
1
((
T r
`
ac
(
T r
n
`
(
x
2
e
) +
2
x
T e
))
r
2
)
:
(8)
Let
2
GF
(2
n
) satisfy
T r
n
`
(
2
) +
1
T
6
= 0 (such a
always exists) and restrict equation (8)
to
x
of the form
x
=
y
,
y
2
GF
(2
`
). Then
T r
ab
1
((
T r
`
ab
(
1
y
2
))
r
1
) =
T r
ac
1
((
T r
`
ac
(
2
y
2
e
))
r
2
)
;
(9)
where
1
=
T r
n
`
(
2
) +
1
T
and
2
=
T r
n
`
(
2
e
) +
2
T e
. Note that
2
6
= 0. As in the derivation
of the linear span of a GMW sequence, it can b e shown that the number of nonzero terms in the
expansion of the left hand side is
ab
(
cd
)
wt(
r
1
)
;
while the numb er of nonzero terms on the right hand side is
ac
(
bd
)
wt(
r
2
)
:
It follows that
c
wt(
r
1
)
?
1
d
wt(
r
1
)
=
b
wt(
r
2
)
?
1
d
wt(
r
2
)
:
Without loss of generality, we may assume that wt(
r
1)
wt(
r
2
). It follows from the fact that
b
and
c
are relatively prime that either
b
= 1, or wt(
r
2
) = 1. In the latter case we also must have
wt(
r
1
) = 1, that is,
r
1
and
r
2
are powers of two, which we have assumed is not the case. Therefore
b
= 1 and
c
wt(
r
1
)
?
1
=
d
wt(
r
2
)
?
wt(
r
1
)
:
(10)
Now if we further restrict
y
to be in
GF
(2
ac
) in equation (9), we get
T r
a
1
((
T r
ac
a
(
1
y
2
))
r
1
) =
T r
ac
1
(
2
y
2
er
2
)
;
for nonzero
1
and
2
. The number of nonzero terms on the left hand side is
a
c
wt(
r
1
)
while the number of nonzero terms on the right hand side is
ac
. It follows that either
c
= 1, in
which case
m
1
=
m
2
, or wt (
r
1
) = 1. In the latter case, from equation (10) and the fact that
r
1
and
r
2
are not both powers of two, we see that
d
= 1. This implies that
T r
`
1
(
T r
n
`
(
x
2
) +
1
x
T
) =
T r
`
1
((
T r
n
`
(
x
2
e
) +
2
x
T e
)
r
2
)
:
By considering the linear spans of the sequences corresponding to these two functions, we see that
this is possible only if wt(
r
2
) = 1. This proves the lemma.
2
The completion of the proof of the proposition is essentially the same as the proof of the
corresponding result of No and Kumar (Lemma 2 of [14]). The details are left to the reader.
2
16
n
Period
N
TN
6 63 12
8 255 32
10 1 023 360
12 4 095 1 008
14 16 383 13 608
16 65 535 34 816
18 262 143 381 024
20 1 048 575 1 560 000
22 4 194 303 21 125 632
24 16 777 215 41 748 480
Table 2:
Number of Distinct Families of TN Sequences of Period
2
n
?
1
Theorem VI.3
Let
n
= 2
`
. The number
N
TN
of distinct families of TN sequences of period
2
n
?
1
is given by
N
TN
=
(2
n
?
1)
n
0
@
X
m
j
`
(2
m
?
1)
m
?
1
!
+ 1
1
A
;
where
(
)
is Euler's phi function.
Proof:
If
1
and
2
are primitive elements in
GF
(2
n
), then they are equivalent for purp oses of
generating families of TN sequences if they are in the same Galois coset, i.e., if
2
=
2
j
1
for some
j
. If
m
j
`
is chosen, and 0
r
1
; r
2
<
2
m
?
1, then we say
r
1
is equivalent to
r
2
, written
r
1
m
r
2
,
if
r
2
2
j
r
1
mod 2
m
?
1 for some
j
. By Proposition VI.1, a family of TN sequences is uniquely
determined by the following: a choice of Galois coset of primitive elements of
GF
(2
n
); a choice of
divisor
m
of
`
; and a choice of
m
equivalence class
r
, with
r
6
m
1. In addition, there is a family
of TN sequences for each choice of Galois equivalence class of primitive elements and
r
= 1.
The number of Galois equivalence classes of primitive elements is
(2
n
?
1)
=n:
For a given
m
,
the number of
m
equivalence classes is
(2
m
?
1)
=m;
which proves the theorem.
2
The values of
N
TN
for the rst few
n
are summarized in Table 2. Note that the No sequences
correspond to the choice
m
=
`
. Thus, if we restrict the sum in Theorem VI.3 to
m
=
`
, we obtain
the number of No sequences, as computed previously [14].
VII Complete Families of Degree
d d
-Form Sequences
In this section we consider families of
d
-form sequences in which the elds
GF
(
q
) and
GF
(
q
e
),
exponent
r
, the degree
d
, and the primitive element
are xed, but the
d
-form
H
(
x
) is free to vary
through all nonzero polynomials that are homogeneous of degree
d
on
GF
(
q
e
) over
GF
(
q
). We
call such a family a
complete family of degree
d d
-form sequences
. Here we restrict our attention
to the case when
e
= 2. Note that for
d
= 2, this gives a family of No sequences. In general the
17
size
N
of such a family (identifying sequences that are cyclic shifts of each other) is
N
=
8
>
>
<
>
>
:
q
d
+1
?
1
q
2
?
1
if
d
is odd,
q
d
+1
?
q
q
2
?
1
+ 1 if
d
is even,
q
d
?
1
:
Furthermore, cross-correlations in a complete family of degree
d d
-form sequences are bounded
as follows.
Theorem VI I.1
If
F
is a complete family of degree
d d
-form sequences, then every cross-correlation
of any two sequences from
F
is contained in the set
f?
q
?
1
;
?
1
; q
?
1
;
2
q
?
1
;

;
(
d
?
1)
q
?
1
g
.
In particular, the cross-correlations are at most
d
+ 1
valued.
Proof:
As we have shown, it suces to compute the number
z
of nontrivial zeros of any
d
-form
H
(
x; y
) in two variables over
GF
(
q
). We show by induction that
z
is in the set
f
i
(
q
?
1) : 0
i
d
g
,
and the theorem follows from Theorem II.1. We can take
d
= 2 as base case, since any quadratic
polynomial in two variables over
GF
(
q
) has 0, 1, or 2 roots.
For the induction case, observe that if we apply a change of coordinates
x
7!
ax
+
by; y
7!
cx
+
dy;
then
H
(
x; y
) will be put in the form
H
(
x; y
) =
xG
(
x; y
) +
H
(
b; d
)
y
d
where
G
(
x; y
) is a
d
?
1-form. If
H
has no nontrivial ro ots, we are done. Otherwise, we can nd
such a change of coordinates for which
H
(
b; d
) = 0. In other words, we can assume
H
(
x; y
) =
xG
(
x; y
)
:
It follows that
H
(
x; y
) = 0 if either
x
= 0 or
G
(
x; y
) = 0. There are exactly
q
?
1 nontrivial
pairs (
x; y
) with
x
= 0. By the homogeneity of
G
, they are either all roots of
G
, or none are. Thus
the number of roots of
H
is either equal to the number of roots of
G
, or is
q
?
1 greater than the
number of roots of
G
. The theorem follows by induction on
G
.
2
If
d
= 3, the size of such a family is about the same as a family of Gold sequences. The
maximum correlations are the same as for Gold sequences with even
n
. The linear spans of
d
-form
sequences, however, are generally much larger.
If
d
= 4, the size of such a family is ab out the same as a large set of Kasami sequences. The
maximum correlations are about one and a half times those of a large set of Kasami sequences.
Again, the linear spans of
d
-form sequences are generally much larger.
For
d
5, the maximum correlations increase linearly in
d
, while the size of the family increases
exponentially in
d
.
18
VIII Acknowledgements
The author thanks the referees of this pap er for pointing out the previous results on generalized
No sequences in No's thesis and for making a number of suggestions that improved the manuscript.
References
[1] M. Antweiller and L. Bomer, \Complex sequences over
GF
(
p
m
) with a two-level autocorre-
lation function and a large linear span,"
IEEE Trans. on Info. Theory
, vol. 38, pp. 120-130,
Jan. 1992.
[2] S. Golomb,
Shift Register Sequences,
Aegean Park Press: Laguna Hills, CA, 1982.
[3] B. Gordon, W. H. Mills, and L. R. Welch, \Some new dierence sets,"
Canad. J. Math.
vol. 14
pp. 614-625, 1962.
[4] T. Kasami, \Weight distribution formula for some classes of cyclic co des," Co ordinated Science
Laboratory, University of Illinois, Urbana, Tech. Rep. R-285 (AD632574), 1966.
[5] T. Kasami, \Weight distribution of Bose-Chaudhuri-Hocquenghem codes," in
Combinatorial
Mathematics and its Applications.
Chapel Hill, NC: University of North Carolina Press, 1969.
[6] E. L. Key, \An Analysis of the structure and complexity of nonlinear binary sequence gener-
ators,"
IEEE Trans. Info. Theory,
vol. IT-22 no. 6, pp. 732-736, Nov. 1976.
[7] A. Klapper, \Cross-correlations of geometric sequences in characteristic two,"
Designs, Codes,
and Cryptography
, vol. 3, pp. 347-377, 1993.
[8] A. Klapp er, A.H. Chan, and M. Goresky, \Cross-correlations of linearly and quadratically
related geometric sequences and GMW Sequences,"
Discrete Applied Mathematics,
vol. 46,
pp. 1-20, 1993.
[9] A. Klapper, A.H. Chan, and M. Goresky, \Cascaded GMW Sequences,"
IEEE Trans. on
Info. Theory,
vol. IT-39, pp. 177-183, 1993.
[10] P. V. Kumar and R. A. Scholtz, \Bounds on the linear span of bent sequences,"
IEEE
Trans. Info. Theory,
vol. IT-29, pp. 854-862, Nov. 1983.
[11] A. Lempl and M. Cohn, \Maximal families of bent sequences,"
IEEE Trans. Info. Theory,
vol. IT-28, pp. 865-868, Nov. 1982.
[12] R. Lidl and H. Niederreiter
Finite Fields
in
Encyclopedia of Mathematics Vol. 20,
Cambridge
University Press: Cambridge, 1983.
19
[13] J. No,
A new family of binary pseudorandom sequences having optimal periodic correlation
properties and large linear span,
Doctoral Dissertation, University of Southern California,
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[14] J. No and P. V. Kumar, \A new family of binary pseudorandom sequences having optimal
periodic correlation prop erties and large linear span,"
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pp. 371-379, 1989.
[15] J. D. Olsen, R. A. Scholtz, and L. R. Welch, \Bent-function sequences,"
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[16] O. Rothaus, \On bent functions,"
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20
Table 1:
Size of Maximum Maximum Range of
Family
n
Family Correlation Linear Span Imbalance
Gold 2
m
+ 1 2
n
+ 1 1 + 2
n
+1
2
2
n
[1
;
2
n
+1
2
+ 1]
Gold 4
m
+ 2 2
n
?
1 1 + 2
n
+2
2
2
n
[1
;
2
n
+2
2
+ 1]
Kasami 2
m
2
n
2
1 + 2
n
2
3
n
2
[1
;
2
n
2
+ 1]
(Small Set)
Kasami 4
m
+ 2 2
n
2
(2
n
+ 1) 1 + 2
n
+2
2
5
n
2
[1
;
2
n
+2
2
+ 1]
(Large Set)
Bent 4
m
2
n
2
1 + 2
n
2
n=
2
n=
4
2
n
4
1
No 2
m
2
n
2
1 + 2
n
2
m
(2
m
?
1) [1
;
2
n
2
+ 1]
TN 2
km
2
n
2
1 + 2
n
2
>
3
mk
(3
k
?
1)
m
?
2
[1
;
2
n
2
+ 1]
Table 1:
Comparison of Properties of Families of Sequences of Period
2
n
?
1
21
Table 2:
n
Period
N
TN
6 63 12
8 255 32
10 1 023 360
12 4 095 1 008
14 16 383 13 608
16 65 535 34 816
18 262 143 381 024
20 1 048 575 1 560 000
22 4 194 303 21 125 632
24 16 777 215 41 748 480
Table 2:
Number of Distinct Families of TN Sequences of Period
2
n
?
1
22
Captions of Tables:
1.
Comparison of Properties of Families of Sequences of Period
2
n
?
1
2.
Number of Distinct Families of TN Sequences of Period
2
n
?
1
23
Footnote 1:
It should be noted that the necessary assumption that
t < q
k
=
2 was not explicitly stated by
No and Kumar. It p oses no diculty for their analysis, however, since any exponent over
GF
(
q
k
)
which is relatively prime to
q
k
?
1 can be assumed to b e less than
q
k
?
1. Such an exp onent
thus has at least one zero in the rst
mk
bits of its base two expansion. Since No sequences are
invariant under cyclic shift of the exp onent
r
, we can assume the exp onent has a zero as its high
bit. In the more general case of TN sequences, we can also cyclically shift
r
so its high bit is zero,
and therefore the high bit of each
P
i
a
i
q
i
is zero.
24
Biography
Andrew M. Klapper
was born in White Plains, New York, in 1952. He received the A.B. degree
in mathematics from New York University, New York, NY, in 1974, the M.S. degree in applied
mathematics from SUNY at Binghamton, Binghamton, NY, in 1975, the M.S. degree in math-
ematics from Stanford University, Stanford, CA, in 1976, and the Ph.D. degree in mathematics
from Brown University, Providence, RI, in 1982. His thesis, in the area of arithmetic geometry,
concerned the existence of canonical subgroups in formal grouplaws.
From 1981 to 1984 he was a post doctoral fellow and visiting professor in the Department
of Mathematics and Computer Science at Clark University, Worcester, MA. From 1984 to 1991
he was an assistant professor in the College of Computer Science at Northeastern University,
Boston, MA. From 1991 to 1993 he was an assistant professor in the Computer Science Department
at the University of Manitoba in Winnipeg, Manitoba, Canada. Currently he is an assistant
professor in the Computer Science Department at the University of Kentucky. His past research has
included work on algebraic geometry over
p
-adic integer rings, computational geometry, mo delling
distributed systems, structural complexity theory, and cryptography. His current interests include
statistical properties of pseudo-random sequences, stream ciphersm, public key cryptosystems, and
morris dancing.
25
... As you know, d-form functions give a rich source of functions with difference balanced property, which were first defined in [26]. ...
... = (33,24,24,24,25,1,2,3,26,5,6,7,27,9,10,11,28,13,14,15,29,17,18,19,30,21,22,23,31,2,3,0,32,6,7,4,25,10,11,8,26,14,15,12,27,18,19,16,28,22,23,20,29,3,0,1,30,7,4,5,31,11,8,9,32,15,12,13,25,19,16,17,26,23,20,21,27,0,1,2,28,4,5,6,29,8,9,10,30,12,13,14,31,16,17,18,32,20,21,22). ...
... = (33,24,24,24,25,1,2,3,26,5,6,7,27,9,10,11,28,13,14,15,29,17,18,19,30,21,22,23,31,2,3,0,32,6,7,4,25,10,11,8,26,14,15,12,27,18,19,16,28,22,23,20,29,3,0,1,30,7,4,5,31,11,8,9,32,15,12,13,25,19,16,17,26,23,20,21,27,0,1,2,28,4,5,6,29,8,9,10,30,12,13,14,31,16,17,18,32,20,21,22). ...
Three classes of partitioned difference families and their optimal constant composition codes
Article
Full-text available
  • Jan 2019
  • ADV MATH COMMUN
Cyclotomy, firstly introduced by Gauss, is an important topic in Mathematics since it has a number of applications in number theory, combinatorics, coding theory and cryptography. Depending on \begin{document}$ v $\end{document} prime or composite, cyclotomy on a residue class ring \begin{document}$ {\mathbb{Z}}_{v} $\end{document} can be divided into classical cyclotomy or generalized cyclotomy. Inspired by a foregoing work of Zeng et al. [40], we introduce a generalized cyclotomy of order \begin{document}$ e $\end{document} on the ring \begin{document}$ {\rm GF}(q_1)\times {\rm GF}(q_2)\times \cdots \times {\rm GF}(q_k) $\end{document}, where \begin{document}$ q_i $\end{document} and \begin{document}$ q_j $\end{document} (\begin{document}$ i\neq j $\end{document}) may not be co-prime, which includes classical cyclotomy as a special case. Here, \begin{document}$ q_1 $\end{document}, \begin{document}$ q_2 $\end{document}, \begin{document}$ \cdots $\end{document}, \begin{document}$ q_k $\end{document} are powers of primes with an integer \begin{document}$ e|(q_i-1) $\end{document} for any \begin{document}$ 1\leq i\leq k $\end{document}. Then we obtain some basic properties of the corresponding generalized cyclotomic numbers. Furthermore, we propose three classes of partitioned difference families by means of the generalized cyclotomy above and \begin{document}$ d $\end{document}-form functions with difference balanced property. Afterwards, three families of optimal constant composition codes from these partitioned difference families are obtained, and their parameters are also summarized.
View
... Kasami sequences can be constructed using m-sequences and their decimations [12]. Besides, there are some known families of sequences of length 2 n − 1 with good correlation properties, such as bent function sequences [25], No sequences [23], Trace Norm sequences [14]. In 2011, Zhou and Tang generalized the modified Gold sequences and obtained binary sequences with length 2 n −1 for n = 2m+1, size 2 ℓn +· · ·+2 n −1, and correlation 2 m+ℓ +1 for each 1 ℓ m [34]. ...
... Trace Norm [14] 2 n − 1, n even ...
Binary sequences with a low correlation via cyclotomic function fields with odd characteristics
Preprint
Full-text available
  • Oct 2022
Sequences with a low correlation have very important applications in communications, cryptography, and compressed sensing. In the literature, many efforts have been made to construct good sequences with various lengths where binary sequences attracts great attention. As a result, various constructions of good binary sequences have been proposed. However, most of the known constructions made use of the multiplicative cyclic group structure of finite field $\mathbb{F}_{p^n}$ for a prime $p$ and a positive integer $n$. In fact, all $p^n+1$ rational places including the place at infinity of the rational function field over $\mathbb{F}_{p^n}$ form a cyclic structure under an automorphism of order $p^n+1$. In this paper, we make use of this cyclic structure to provide an explicit construction of binary sequences with a low correlation of length $p^n+1$ via cyclotomic function fields over $\mathbb{F}_{p^n}$ for any odd prime $p$. Each family of binary sequences has size $p^n-2$ and its correlation is upper bounded by $4+\lfloor 2\cdot p^{n/2}\rfloor$. To the best of our knowledge, this is the first construction of binary sequences with a low correlation of length $p^n+1$ for odd prime $p$. Moreover, our sequences can be constructed explicitly and have competitive parameters.
View
... Kasami sequences can be constructed using m-sequences and their decimations [11]. Besides, there are some known families of sequences of length 2 n − 1 with good correlation properties, such as bent function sequences [24], No sequences [22], Trace Norm sequences [13]. In 2011, Zhou and Tang generalized the modified Gold sequences and obtained binary sequences with length 2 n − 1 for n = 2m + 1, size 2 n + · · · + 2 n − 1, and correlation 2 m+ + 1 for each 1 m [33]. ...
... For the case n ≡ 2(mod 4), compared with the Gold sequences of length 2 n − 1, our sequences have a smaller correlation and a larger length although the family size of our sequences is slightly smaller. For the case n ≡ 0(mod 4), all families of sequences given in [11], [13], [22], [24] have size 2 n/2 which is much less than the size 2 n − 1 of our family of sequences, although the correlation of their sequences is one half of our correlation. Hence, it turns out that our sequences have competitive parameters for even n. ...
Binary Sequences With a Low Correlation via Cyclotomic Function Fields
Article
Full-text available
  • May 2022
Due to wide applications of binary sequences with a low correlation to communications, various constructions of such sequences have been proposed in the literature. Many efforts have been made to construct good binary sequences with various lengths. However, most of the known constructions make use of the multiplicative cyclic group structure of finite field $\mathbb {F}_{2^{n}}$ for a positive integer $n$ . It is often overlooked in this community that all $2^{n}+1$ rational places (including “the place at infinity”) of the rational function field over $\mathbb {F}_{2^{n}}$ form a cyclic structure under an automorphism of order $2^{n}+1$ . In this paper, we make use of this cyclic structure to provide an explicit construction of binary sequences with a low correlation of length $2^{n}+1$ via cyclotomic function fields over $\mathbb {F}_{2^{n}}$ . Each family of our sequences has size $2^{n}-1$ and its correlation is upper bounded by $\lfloor 2^{(n+2)/2}\rfloor $ . To the best of our knowledge, this is the first construction of binary sequences with a low correlation of length $2^{n}+1$ . Moreover, our sequences can be constructed explicitly and have competitive parameters. In particular, compared with the Gold sequences of length $2^{n}-1$ for even $n$ , our sequences have a smaller correlation and a larger length although the family size of our sequences is slightly smaller.
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... Ces codes sont aussi des codes binaires utilisés en cryptographie. Leur génération est quasisimilaire à celle des codes de No à l'exception de l'utilisation de la fonction Norme [43]. Les caractéristiques de cette famille sont une cardinalité de √ + 1 = 2 /2 séquence de longueur avec des niveaux de corrélations (hors pic principal) bornés par √ + 1 + 1. En conclusion, malgré les problèmes relatifs au contrôle du spectre, ces trois familles semblent être de bonnes candidates pour une application MSPSR active. ...
... Les caractéristiques de cette famille sont une cardinalité de √ + 1 = 2 /2 séquence de longueur avec des niveaux de corrélations (hors pic principal) bornés par √ + 1 + 1. En conclusion, malgré les problèmes relatifs au contrôle du spectre, ces trois familles semblent être de bonnes candidates pour une application MSPSR active.  TN [43]  BBAG (Bruit Blanc Additif Gaussien) Toutefois, dans la partie suivante nous allons seulement caractériser les signaux OFDM ayant donnés les meilleurs résultats c'est-à-dire ceux modulés par du QPSK ou des m-séquences. [64]. ...
Formes d’ondes MSPSR, traitements et performances associés
Thesis
  • Dec 2017
Aujourd’hui, les systèmes MSPSR (Multi-Static Primary Surveillance Radar) passifs se sont installés de manière durable dans le paysage de la surveillance aérienne [1]. L’intérêt que suscitent ces nouveaux systèmes provient du fait qu’en comparaison aux radars mono-statiques utilisés actuellement, les systèmes MSPSR reposent sur une distribution spatiale d’émetteurs et de récepteurs offrant des avantages en termes de fiabilité (redondance), de coûts (absence de joints tournants et émetteurs moins puissants) et de performances (diversité spatiale). Toutefois, le défaut majeur du MSPSR passif réside en l’absence de formes d’ondes dédiées due à l’exploitation d’émetteurs d’opportunités tels que les émetteurs de radio FM (Frequency Modulation) et/ou de DVB-T (Digital Video Broadcasting-Terrestrial) [2]. Afin de pallier à ce défaut, il est envisagé d’utiliser des émetteurs dédiés permettant l’emploi de formes d’ondes optimisées pour une application radar, on parle alors de MSPSR actif. Cette thèse se place dans ce cadre et a pour objectif d’étudier et de définir la ou les formes d’ondes ainsi que les traitements associés permettant d’atteindre de meilleurs performances : une meilleure flexibilité sur la disposition du système (positionnement des émetteurs libres), une continuité de service (non dépendance d’un système tiers) et de meilleurs performances radars (e.g. en terme de précision des mesures, détections, …). Dans ce but, cette thèse étudie : - Les critères de sélection des codes : comportement des fonctions d’ambiguïtés, PAPR (Peak to Average Power Ratio), efficacité spectrale, etc... ; - Les formes d’ondes utilisées en télécommunication (scrambling code, OFDM) afin d’identifier leur possible réemploi pour une application radar ; - L’utilisation d’algorithmes cycliques pour générer des familles de séquences adaptées à notre problème ; - Une approche basée sur une descente de gradient afin de générer des familles de codes de manière plus efficiente ; - Et l’évaluation des performances de ces différents algorithmes à travers l’établissement d’une borne supérieure sur le niveau maximum des lobes secondaires et à travers le dépouillement des données enregistrées suite à des campagnes d’essais
View
... Kasami sequences can be constructed using m-sequences and their decimations [9]. Besides, there are some known families of sequences of length 2 n −1 with good correlation properties, such as bent function sequences [20], No sequences [18], TN sequences [10]. In 2011, Zhou and Tang generalized the modified Gold sequences and obtained binary sequences with length 2 n − 1, n = 2m + 1, size 2 ℓn + · · · + 2 n − 1, correlation 2 m+ℓ + 1 [28]. ...
Binary sequences with low correlation via cyclotomic function fields
Preprint
Full-text available
  • Jul 2021
Due to wide applications of binary sequences with low correlation to communications, various constructions of such sequences have been proposed in literature. However, most of the known constructions via finite fields make use of the multiplicative cyclic group of $\F_{2^n}$. It is often overlooked in this community that all $2^n+1$ rational places (including "place at infinity") of the rational function field over $\F_{2^n}$ form a cyclic structure under an automorphism of order $2^n+1$. In this paper, we make use of this cyclic structure to provide an explicit construction of families of binary sequences of length $2^n+1$ via the finite field $\F_{2^n}$. Each family of sequences has size $2^n-1$ and its correlation is upper bounded by $\lfloor 2^{(n+2)/2}\rfloor$. Our sequences can be constructed explicitly and have competitive parameters. In particular, compared with the Gold sequences of length $2^n-1$ for even $n$, we have larger length and smaller correlation although the family size of our sequences is slightly smaller.
View
... The δ-homogeneous function from F q n to F q is introduced by Klapper [26], which is defined as H(xy) = y δ H(x) for any x ∈ F q n and y ∈ F q . Then, Kim et al. constructed an RDS from a δ-homogeneous function on F * q n [27]. ...
Construction of New Fractional Repetition Codes from Relative Difference Sets with λ=1
Article
Full-text available
  • Oct 2017
  • Entropy
Fractional repetition (FR) codes are a class of distributed storage codes that replicate and distribute information data over several nodes for easy repair, as well as efficient reconstruction. In this paper, we propose three new constructions of FR codes based on relative difference sets (RDSs) with λ = 1 . Specifically, we propose new ( q 2 - 1 , q , q ) FR codes using cyclic RDS with parameters ( q + 1 , q - 1 , q , 1 ) constructed from q-ary m-sequences of period q 2 - 1 for a prime power q, ( p 2 , p , p ) FR codes using non-cyclic RDS with parameters ( p , p , p , 1 ) for an odd prime p or p = 4 and ( 4 l , 2 l , 2 l ) FR codes using non-cyclic RDS with parameters ( 2 l , 2 l , 2 l , 1 ) constructed from the Galois ring for a positive integer l. They are differentiated from the existing FR codes with respect to the constructable code parameters. It turns out that the proposed FR codes are (near) optimal for some parameters in terms of the FR capacity bound. Especially, ( 8 , 3 , 3 ) and ( 9 , 3 , 3 ) FR codes are optimal, that is, they meet the FR capacity bound for all k. To support various code parameters, we modify the proposed ( q 2 - 1 , q , q ) FR codes using decimation by a factor of the code length q 2 - 1 , which also gives us new good FR codes.
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Binary Sequences With a Low Correlation via Cyclotomic Function Fields of Odd Characteristic
Article
  • Jan 2023
Sequences with a low correlation have very important applications in communications, cryptography, and compressed sensing. In the literature, many efforts have been made to construct good sequences with various lengths, where binary sequences attract great attention. As a result, various constructions of good binary sequences have been proposed. However, most of the known constructions made use of the multiplicative cyclic group structure of finite field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> p<sup>n</sup> </sub> for a prime p and a positive integer n . In fact, all p<sup>n</sup> + 1 rational places including the place at infinity of the rational function field over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> p<sup>n</sup> </sub> can form a cyclic structure under an automorphism of order p<sup>n</sup> + 1. In this paper, we make use of this cyclic structure to provide an explicit construction of binary sequences with a low correlation of length p<sup>n</sup> + 1 via cyclotomic function fields over Fpn for any odd prime p . Each family of binary sequences has size p<sup>n</sup> -2 and its correlation is upper bounded by 4+⎣2 · p<sup>n/2</sup> ⎦. To the best of our knowledge, this is the first construction of binary sequences with a low correlation of length p<sup>n</sup> +1 for odd prime p . Moreover, our sequences can be constructed explicitly and have competitive parameters.
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Construction of Binary Sequences With Low Correlation via Multiplicative Quadratic Character Over Finite Fields of Odd Characteristics
Article
  • Feb 2021
In literature, there are several methods to construct Gold sequences. One of the constructions is via the trace function from extension field of $\mathbb {F}_{2}$ . Estimation of correlation of this construction is based on number of rational points of elliptic curves. In this article, we generalize this construction from finite fields of even characteristic to odd characteristics by using multiplicative quadratic character. Again, estimation of correlation of this construction is based on number of rational points of elliptic curves. Thus, we obtain binary sequences which have more flexibility on length while still possessing low correlation property. Moreover, some of the sequences are optimally balanced.
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New Optimal Difference Systems of Sets from Ideal Sequences and Perfect Ternary Sequences
Article
  • May 2020
Difference systems of sets (DSSs) introduced by Levenstein are combinatorial structures used to construct comma-free codes for synchronization. In this letter, two classes of optimal DSSs are presented. One class is obtained based on q-ary ideal sequences with d-form property and difference-balanced property. The other class of optimal and perfect DSSs is derived from perfect ternary sequences given by Ipatov in 1995. Compared with known constructions (Zhou, Tang, Optimal and perfect difference systems of sets from q-ary sequences with difference-balanced property, Des. Codes Cryptography, 57(2), 215-223, 2010), the proposed DSSs lead to comma-free codes with nonzero code rate.
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Cross-correlations of geometric sequences in characteristic two
Article
Full-text available
  • Oct 1993
Cross-correlation functions are determined for a large class of geometric sequences based on m-sequences in characteristic two. These sequences are shown to have low cross-correlation values in certain cases. They are also shown to have significantly higher linear complexities than previously studied geometric sequences. These results show that geometric sequences are candidates for use in spread-spectrum communications systems in which cryptographic security is a factor.
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Some New Difference Sets
Article
  • Jan 1962
  • CAN J MATH
A difference set is a set D = {d 1 , d 2 , … , d k ] of k distinct residues modulo v such that each non-zero residue occurs the same number of times among the k(k — 1) differences d i — d j , i ≠ j. If λ is the number of times each difference occurs, then (1) When we wish to emphasize the particular values of v, k , and λ involved we will call such a set a (v, k , λ) difference set. Another (v, k , λ) difference set E = {e 1 , e 2 , … e k } is said to be equivalent to the original one if there exist a and t such that (t, v) = 1 and E = {a + td 1 , … , a + td k } . If t = 1 we will call the set E a slide of the set D . If D = E , then t is called a multiplier of D .
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On “bent” functions
Article
  • May 1976
Let P(x) be a function from GF(2n) to GF(2). P(x) is called “bent” if all Fourier coefficients of (−1)P(x) are ±1. The polynomial degree of a bent function P(x) is studied, as are the properties of the Fourier transform of (−1)P(x), and a connection with Hadamard matrices.
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New bounds on cyclic codes from algebraic curves
Conference Paper
  • Nov 1988
Starting from a deep link between the words of cyclic codes and plane algebraic curves over finite fields we use bounds on the number of rational points of these curves to obtain general bounds for the weights of cyclic codes.
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Maximal Families of Bent Sequences
Article
  • Nov 1982
In a recent paper Olsen, Scholtz, and Welch (OSW) describe the use of bent functions to construct families of sequences with asymptotically optimal correlation properties. We show that the sequences produced by the OSW construction possess the claimed correlation properties if and only if the underlying bent functions are pairwise orthogonal. Consequently, the cardinality of a family of OSW sequences of length 2^{n}-1 cannot exceed 2^{n/2} . We also describe methods of selecting a maximal set of orthogonal bent functions.
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Bent-function sequences
Article
  • Dec 1982
In this paper we construct a new family of nonlinear binary signal sets which achieve Welch's lower bound on simultaneous cross correlation and autocorrelation magnitudes. Given a parameter n with n=0 pmod{4} , the period of the sequences is 2^{n}-1 , the number of sequences in the set is 2^{n/2} , and the cross/auto correlation function has three values with magnitudes leq 2^{n/2}+1 . The equivalent linear span of the codes is bound above by sum_{i=1}^{n/4}left(stackrel{n}{i} right) . These new signal sets have the same size and correlation properties as the small set of Kasami codes, but they have important advantages for use in spread spectrum multiple access communications systems. First, the sequences are "balances," which represents only a slight advantage. Second, the sequence generators are easy to randomly initialize into any assigned code and hence can be rapidly "hopped" from sequence to sequence for code division multiple access operation. Most importantly, the codes are nonlinear in that the order of the linear difference equation satisfied by the sequence can be orders of magnitude larger than the number of memory elements in the generator that produced it. This high equivalent linear span assures that the code sequence cannot be readily analyzed by a sophisticated enemy and then used to neutralize the advantages of the spread spectrum processing.
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Cross-Correlations of Linearly and Quadratically Related Geometric Sequences and GMW Sequences.
Article
  • Sep 1993
  • DISCRETE APPL MATH
In this paper we study the cross-correlation function values of geometric sequences obtained from q-ary m-sequences whose underlying m-sequences are linearly or quadratically related. These values are determined by counting the points of intersection of pairs of hyperplanes or of hyperplanes and quadric hypersurfaces of a finite geometry. The results are applied to obtain the cross-correlations of m-sequences and GMW sequences with different primitive polynomials.
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