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arXiv:2203.11567v1 [cs.IT] 22 Mar 2022
The b-symbol weight distribution of irreducible cyclic
codes and related consequences
Hongwei Zhu ∗Minjia Shi†
Abstract
The b-symbol read channel is motivated by the limitations of the reading process
in high density data storage systems. The corresponding new metric is a generaliza-
tion of the Hamming metric known as the b-symbol weight metric and has become
an important object in coding theory. In this paper, the general b-symbol weight
enumerator formula for irreducible cyclic codes is presented by using the Gaussian
period and a new invariant #U(b, j, N1). The related b-symbol weight hierarchies
{d1(C), d2(C),...,dK(C)}(K= dim(C)) are given for some cases. The shortened
codes which are optimal from some classes of irreducible cyclic codes are given,
where the shorten set Tis the complementary set of b-symbol support of some
codeword with the minimal b-symbol weight.
Keywords: b-symbol weight, irreducible cyclic code, Gaussian period, shortened code
MSC(2010): 94 B15, 94 B25, 05 E30
1 Introduction
The theory of error-control codes aims to recover the original information units when
some bound is given on their corruption. These corruption bounds can be defined at
the code-block level, such as a certain number of errors in Hamming metric codes, or
at the individual-symbol level, such as symbol-transition restrictions in asymmetric or
unidirectional error-correcting codes. The alphabet on which the information unit is
defined may change throughout the coding problem, like in soft-decoding, but it is still
typically the same unit that is tracked and analyzed. In 2011, Cassuto and Blaum [3,
4] proposed a new coding framework for channels whose outputs are overlapping pairs
of symbols. Such channels are motivated by storage applications in which the spatial
resolution of the reader may be insufficient to isolate adjacent symbols. Codes are still
defined by an alphabet, as usual. The goal is to protect against a certain number of
pairwise errors, not a certain number of symbol errors. A pair-error is defined as a pair-
read in which one or more of the symbols is read in error. Due to physical limitations,
individual symbols cannot be read off the channel. Therefore, each channel read contains
∗School of Mathematical Sciences, Anhui University, Hefei, China. E-mail: zhwgood66@163.com
†School of Mathematical Sciences, Anhui University, Hefei, China. E-mail: smjwcl.good@163.com
contributions from two adjacent symbols. The constructions of symbol-pair codes are
studied in a series of papers [5–7, 9, 11, 14, 19, 21, 24, 25, 35, 37]. Later, Yaakobi et al. [38]
generalized the symbol-pair read channel to the b-symbol read channel. The contributions
to the b-symbol codes can be found in [12, 31, 40–43] and the references therein.
The definition of b-symbol metric will be introduced in detail in Section II of this
paper. It is not hard to see that the b-symbol metric is a natural generalization of
Hamming metric. Another generalization of Hamming metric is the b-th generalized
Hamming metric, which has appeared as early as in 1970s [17, 20] and has become an
important research topic in coding theory after the famous paper [36] in 1991, where
Wei gave a series of wonderful consequences on the b-th generalized Hamming metric and
indicates that it completely characterizes the performance of a linear code when it is used
on the wire-tap channel of type II. For more details on the b-th generalized Hamming
metric, we refer the readers to [36].
Is there any connection between the two types of generalizations? Shi et al. [32]
considered this question and showed many interesting results, especially if Cis constayclic.
Let Cdenote a linear code with dimension k. We use db(C) to denote the minimum b-th
generalized Hamming distance of C. When b= 1, d1(C) is the minimal Hamming distance
of C. The set
{db(C)|1≤b≤k}
is called the weight hierarchy of C. To distinguish it from the later definition, let us call
it the generalized weight hierarchy in the sequel.
For a code C,db(C) denotes the minimum b-symbol distance of C. When b= 1,
d1(C) is also the minimum Hamming distance of C. The b-symbol metric is also called
symbol-pair metric if b= 2. The set
{db(C)|1≤b≤n}
is called the b-symbol weight hierarchy of C. Note that Ccould be an unrestricted code
under the b-symbol metric. If Cis a cyclic code (or a constacyclic code), then the b-symbol
weight hierarchy of Chas the following property:
d1(C)< d2(C)<···< dk−1(C)< dk(C) = dk+1 (C) = ···=dn(C) = n. (1)
The generalized Hamming weight hierarchy of Chas a similar property to (1), and C
could be a linear code but not cyclic.
Theorem 1. [32] If Cis a cyclic code with length nand dimension K, then db(C) = n
for K≤b≤n. Moreover, if Cis a cyclic code, then db(C) = db(C)if b= 1 or b= dim(C).
It is worth mentioning that there are other interesting connections between the two
metrics. Liu and Pan [22, 30] considered the superposition of two metrics. They call this
superposition a generalized b-weight (we prefer to call it a generalized b-symbol weight).
For more details, we refer the readers to [22, 30].
The following result shows that db(C) is a lower bound of db(C).
Theorem 2. [32] If Cis a cyclic code, then db(C)≥db(C).
2
The b-symbol weight distribution of irreducible cyclic codes and related consequences
It is very meaningful to determine the b-symbol weight hierarchy of cyclic codes, since
it provides a nice upper bound on their generalized weight hierarchy. Moreover, there is
another application for determining the b-symbol weight hierarchy of cyclic codes. If the
b-symbol weight hierarchy of cyclic codes is known, we can obtain a shortened code by
shortening some coordinates associated with the codeword with the minimum b-symbol
weight. We will discuss this in detail in Section V of this paper.
Besides the weight hierarchy of cyclic codes, the Hamming weight structure of cyclic
codes is also a hot topic in coding theory. The Hamming weight structure of irreducible
cyclic codes has been a research topic since the first works of McEliece and others [10,27,28]
due to their connection to Gaussian sums and L-functions, and its intrinsic complexity.
As we all know, it is very difficult to determine the Hamming weight distribution of
irreducible cyclic codes. Predictably, determining the b-symbol weight distribution of an
irreducible cyclic code is even more difficult. To the best of our knowledge, there are only
a few papers dealing with the b-symbol weight distribution of some cyclic codes:
•Sun et al. [35] considered the symbol-pair distance distribution of a class of repeated-
root cyclic codes;
•Ma and Luo [25] considered the symbol-pair weight distribution of MDS codes and
Simplex codes;
•Shi et al. [31] gave some bounds on the b-symbol minimum distance of cyclic codes
by a geometric approach;
•Zhu et al. considered the complete b-symbol weight distribution of a class of irre-
ducible cyclic codes [41] and the b-symbol weight hierarchy of a class of reducible
cyclic codes called Kasami codes [42].
This paper is a further study of the paper [41]. We give a formula for computing the
b-symbol weight of a codeword of an irreducible cyclic code by using the Gaussian period
and a new invariant #U(b, j, N1). The definitions of the Gaussian period and #U(b, j, N1)
will be defined in Section II and Section III, respectively. The formula is a generalization
of the formula for computing the Hamming weight of a codeword of an irreducible cyclic
code given in [13]. We consider the b-symbol weight hierarchy of some irreducible cyclic
codes. In particular, the two types of weight hierarchies of the same irreducible cyclic
code are equal under some restrictions. Some optimal shortened codes are obtained by
shortening some special coordinates, where these special coordinates are related to the
codeword with the minimum b-symbol distance.
The paper is organized as follows. In Section II, we introduce various notations, defini-
tions, and basic facts. Then, in Section III, we present a general formula for the b-symbol
weight distribution of irreducible cyclic codes and some specific cases. In Section IV, we
compute the b-symbol weight hierarchy of some classes of irreducible cyclic codes and
compare these results with the known results on the generalized hierarchy of irreducible
cyclic codes. In Section V, we present an application of the b-symbol weight hierarchy
of cyclic codes in the shortening technique and construct some new shortened codes with
nice parameters. Section VI concludes this paper.
3
2 Preliminaries
Throughout this paper we assume and fix the following:
•Let q=ps,Q=qm, where pis a prime number, s, m are positive integers.
•Let ndenote the length of the code, where n|Q−1 and gcd(n, q) = 1. Let k0be
the multiplicative order of qmodulo nand k0|m.
•Let N=Q−1
n,αbe a primitive element of FQand θ=αN.
•Let TQ/q denote the trace function from FQto Fq.
•Let supp(x) denote the support of the vector x.
2.1 The b-symbol metric
Let bbe a positive integer with 1 ≤b≤n. For any x∈Fn
q, the Hamming weight wH(x)
is defined as the number of nonzero coordinates in x. Let πb(x) denote the vector
πb(x) = ((x0, . . . , xb−1),(x1,...,xb),···,(xn−1,...,xb+n−2)) ∈Fb
qn,
where the indices are taken modulo n. The b-symbol weight of xis defined as
wb(x) = wH(πb(x)).
Example 3. Let x= (0,0, a, 0,0,0, b, 0,0,0,0, c, 0, a)∈F14
q,where a, b, c ∈F∗
q.Then the
b-symbol weight of xare the following.
(i) w1(x) = wH(x) = 4;
(ii) w2(x) = wH(π2(x)) = wH(0,0),(0, a),(a, 0),(0,0),(0,0),(0, b),(b, 0),(0,0),(0,0),
(0,0),(0, c),(c, 0),(0, a),(a, 0)= 8;
(iii) w3(x) = wH(π3(x)) = wH(0,0, a),(0, a, 0),(a, 0,0),(0,0,0),(0,0, b),(0, b, 0),
(b, 0,0),(0,0,0),(0,0,0),(0,0, c),(0, c, 0),(c, 0, a),(0, a, 0),(a, 0,0)= 11;
(iv) w4(x) = wH(π4(x)) = wH(0,0, a, 0),(0, a, 0,0),(a, 0,0,0),(0,0,0, b),(0,0, b, 0),(0, b,
0,0),(b, 0,0,0),(0,0,0,0),(0,0,0, c),(0,0, c, 0),(0, c, 0, a),(c, 0, a, 0),(0, a, 0,0),(a, 0,
0, a)= 13;
(iv) wb(x) = 14 if b≥5.
For any x,y∈Fn
q, the b-symbol distance between xand yis defined as
db(x,y) = wb(x−y).
When b= 1, w1(x) = wH(x) and d1(x,y) = dH(x,y).For convenience, we adopt w1(x)
and d1(x,y) to represent the Hamming weight of xand the Hamming distance between
4
The b-symbol weight distribution of irreducible cyclic codes and related consequences
xand y, respectively. Let Ebe a subset of Fn
q. The minimum b-symbol distance of db(E)
is defined as
db(E) = min{db(x,y)|x,y∈Eand x6=y}.
A linear [n, K, db(C)] code Cover Fqis a K-dimensional subspace of Fn
qwith minimum
b-symbol distance db(C). Let Ab
idenote the number of codewords with b-symbol weight i
in a code of length n. The b-symbol weight enumerator of Cis defined by
1 + Ab
1T+Ab
1T2+···+Ab
nTn.
In fact, the b-symbol weight of a nonzero vector will never less than bby the definition of
b-symbol metric. Therefore, the b-symbol weight enumerator of Cis better to write as
1 + Ab
bTb+···+Ab
nTn.
2.2 Cyclic codes
Let τ(x0, x1,...,xn−1) denote the vector (xn−1, x0,...,xn−2) obtained from (x0, x1,...,xn−1)
by the cyclic shift of the coordinates i7→ i+ 1 mod n. A linear [n, k] code Cover Fqis
called cyclic if c∈ C implies τ(c)∈ C.Let gcd(n, q) = 1.The set
C(Q, N) = c(β) = (TQ/q (β), TQ/q (βθ)),...,TQ/q (βθn−1)|β∈FQ(2)
is called an irreducible cyclic code over Fqwith parameters [n, k0]. It is worth mentioning
that the celebrated Golay code is an irreducible cyclic code and was used on the Mariner
Jupiter-Saturn Mission. An irreducible cyclic code is said to be semi-primitive if n=Q−1
N
where N > 2 divides qj+ 1 for some j≥1.
2.3 Group character, Gaussian sum, Gaussian periods
An additive character of Fqis a nonzero function χfrom Fqto the set of complex numbers
such that χ(x+y) = χ(x)χ(y) for any (x, y)∈F2
q.For each b∈Fq, the function
χb(c) = e2π√−1Tq/p (bc)/p,for all c∈Fq
defines an additive character of Fq. χ1is called the canonical additive character of Fq.
A multiplicative character of Fqis a nonzero ψfrom F∗
qto the set of complex numbers
such that ψ(xy) = ψ(x)ψ(y) for all pairs (x, y)∈F∗
q×F∗
q. Let gbe a fixed primitive
element of Fq. For each j∈ {1,2,...,q−1}, the function ψjwith
ψj(gk) = e2π√−1jk/(q−1),for k∈ {0,1,2,...,q−1}
defines a multiplicative character with order q−1
gcd(q−1,j)of Fq.
Let ψbe a multiplicative character with order kwhere k|(q−1) and χan additive
character of Fq. Then the Gaussian sum G(ψ, χ) of order kis defined by
G(ψ, χ) = X
c∈F∗
q
ψ(c)χ(c).
5
For convenience, let G(ψ) denote G(ψ, χ1) in the sequel.
Let C(k,Q)
i=αihαkifor i∈ {0,1,2,...,k−1}, where hαkidenotes the subgroup of F∗
Q
generated by αk. The cosets C(k,Q)
iare called the cyclotomic classes of order kin FQ.
The Guassian periods are defined by
η(k,Q)
i=X
x∈C(k,Q)
i
χ1(x), i ∈ {0,1,...,k−1}.
By the discrete Fourier transform, Gaussian periods and Gaussian sum have the following
relationship,
η(k,Q)
i=−1 + Pk−1
j=1 ξ−ij
kG(ψj)
k,
where ξk=e2π√−1/k and ψis a primitive multiplicative character of order kover F∗
Q.
Lemma 4. Let symbols be the same as before. Then we have
(1) [34] Pk−1
i=0 η(k,Q)
i=−1.
(2) [34] Pk−1
i=0 η(k,Q)
i2=Qθj−Q−1
kfor all j={0,1,...,k−1}, where
θj=
1if Q−1
kis even and j= 0
1,if Q−1
kis odd and j=k
2
0,otherwise,
and equivalently θj= 1 if and only if −1∈C(k,Q)
j.
(3) η(k,Q)
0, η(k,Q)
1,...,η(k,Q)
k−1can not be all the same if and only if k≥2.
(4) Let f(x) = Pk−1
i=0 η(k,Q)
ixiand ωdenote a primitive k-th root of unity. The circulant
matrix
A=
η(k,Q)
0η(k,Q)
1··· η(k,Q)
k−1
η(k,Q)
k−1η(k,Q)
0··· η(k,Q)
k−2
.
.
..
.
.....
.
.
η(k,Q)
1η(k,Q)
2··· η(k,Q)
0
k×k
(3)
is invertible if and only if f(ωi)6= 0 for all i∈ {0,1,...,k−1}.
Proof. The first two properties of Gaussian periods are from [34]. Assume that
η(k,Q)
0=η(k,Q)
1=···=η(k,Q)
k−1=λ. (4)
From assertions (1) and (2) of this Lemma, we have
kλ =−1,
kλ2=Q−Q−1
k.
6
The b-symbol weight distribution of irreducible cyclic codes and related consequences
Then the Eq. (4) holds if and only if k= 1.
Let Pdenote the permutation matrix
P=
010··· 0
001··· 0
.
.
..
.
..
.
..
.
..
.
.
000··· 1
100··· 0
.
Then Pk=Ikand A=η(k,Q)
0Ik+η(k,Q)
1P+···+η(k,Q)
k−1Pk−1,where Ikdenotes the unit
matrix. Therefore, the circulant matrix Ais invertible if and only if f(x) is coprime to
xk−1.This completes the proof.
Since the values of the Gaussian sums in general are very hard to compute, the values
of η(k,Q)
iare also hard to compute. Some known results on η(k,Q)
iare the following [2,29].
Lemma 5. The known results on the Gaussian periods are the following.
1. If k= 2, then
η(2,Q)
0=(−1+(−1)sm−1Q1
2
2,if p≡1 (mod 4)
−1+(−1)sm−1(√−1)smQ1
2
2,if p≡3 (mod 4)
and η(2,Q)
1=−1−η(2,Q)
0.
2. If k= 3 and p≡2(mod 3), then
η(3,Q)
0=−1−(√−1)sm2Q1
2
3, η(3,Q)
1=η(3,Q)
2=−1 + (√−1)smQ1
2
3.(5)
3. If k= 4 and p≡3(mod 4), then
η(4,Q)
0=−1−(√−1)sm3Q1
2
4, η(4,Q)
1=η(4,Q)
2=η(4,Q)
3=−1 + (√−1)smQ1
2
4.(6)
4. (Semi-primitive case)If k > 2and there exists a positive integer jsuch that pj≡
−1(mod k), and the jis the least such. Let Q=p2jγ for some integer γ.
•When γ, p and pj+1
kare all odd, then
η(k,Q)
k
2
=(k−1)Q1
2−1
k, η(k,Q)
i=−Q1
2+ 1
kfor i6=k
2.(7)
•In all other cases,
η(k,Q)
0=(−1)γ+1(k−1)Q1
2−1
k, η(k,Q)
i=(−1)γQ1
2−1
kfor i6= 0.(8)
7
3 The complete b-symbol weight enumerators
Ding and Yang [13] proved that the determination of the Hamming weight distribu-
tion of an irreducible cyclic codes is equivalent to that of the Gaussian periods of order
gcd Q−1
q−1, N. McEliece [28] gave another proof by Gaussian sums.
Lemma 6. [13, 28] Let c(β)be a codeword of the irreducible cyclic code C(Q, N)as in
(2). If 06=β∈C(gcd(Q−1
q−1,N),Q)
i, then the Hamming weight of c(β)is
w1(c(β)) = (q−1)(Q−1)
qN −
(q−1) gcd Q−1
q−1, Nη(gcd(Q−1
q−1,N),Q)
i
qN .
Very recently, Shi et al. [32] studied the relationship between the b-th generalized
Hamming weight metric and b-symbol weight metric. A very interesting expression on
the b-symbol weight of a vector cis given in that paper, and we present it in the following.
It is very important for giving the expression of the b-symbol weight of a codeword in
C(Q, N).
Lemma 7. [32] Let c∈Fn
qand denote by Vb(c)the codewords generated by all linear
combinations of cand its first b−1cyclic shifts. Then
wb(c) = 1
qb−1(q−1) X
c′∈Vb(c)
w1(c′).
Combining the two lemmas above, we obtain the following result, which is the key
observation of this paper. For convenience, let N1= gcd Q−1
q−1, Nin the sequel.
Theorem 8. Let c(β)be a codeword of the irreducible cyclic code C(Q, N )as in (2).
Let αbe a primitive element of FQ,θ=αNand (u1,...,ub)∈Fb
q\ {0}. Assume that
1≤b≤k0−1and the nonzero elements Pb
i=1 uiθi−1belongs to the cyclotomic class
C(N1,Q)
k(u1,...,ub). If 06=β∈C(N1,Q)
i, then the b-symbol weight of c(β)is
wb(c(β)) = (qb−1)(Q−1)
qbN−N1
qbNX
(u1,...,ub)∈Fb
q\{0}
η(N1,Q)
i+k(u1,...,ub),
where the indices are taken modulo N1.
Proof. For any nonzero element β∈FQ, according to the definition of Vb(c(β)) in Lemma
7 and the property of the trace function, we have
Vb(c(β)) = (b
X
j=1
ujτ(c(β))
(u1,...,ub)∈Fb
q)
=(b
X
j=1
ujc(βθj−1)
(u1,...,ub)∈Fb
q)
=(c(β
b
X
j=1
ujθj−1)
(u1,...,ub)∈Fb
q\ {0})∪ {0}.
8
The b-symbol weight distribution of irreducible cyclic codes and related consequences
Since β∈C(N1,Q)
iand Pb
j=1 ujθj−1∈C(N1,Q)
k(u1,...,ub), we have
β
b
X
j=1
ujθj−1∈C(N1,Q)
i+k(u1,...,ub),
where the indices are taken modulo N1. Combining Lemma 6 and Lemma 7, the b-symbol
weight of c(β) equals
wb(c(β)) = 1
qb−1(q−1)
X
c′∈Vb(c(β))\{0}
w1(c′) + w1(0)
=(qb−1)(Q−1)
qbN−X
(u1,...,ub)∈Fb
q\{0}
N1η(N1,Q)
i+k(u1,...,ub)
qbN,
where the indices are taken modulo N1. This completes the proof.
Definition 9. Define U(b, i, N1) be the set
U(b, i, N1) = ((u1, . . . , ub)
b
X
i=1
uiθi−1∈C(N1,Q)
iand (u1,...,ub)∈Fb
q\ {0}),
where θ=αN.
Lemma 10. We have the following properties on U(b, i, N1).
(1) Fb
q={0} ∪ SN1−1
i=0 U(b, i, N1).
(2) PN1−1
i=0 #U(b, i, N1) = qb−1, where #U(b, j, N1)denotes the size of U(b, j, N1).
(3) U(1,0, N1) = F∗
qand U(1, j, N1) = ∅for all j∈ {1,2,...,N1−1}.
(4) #U(k0, i, N1) = qk0−1
N1for all i∈ {0,1,...,N1−1},where k0is the multiplicative
order of qmodulo n.
(5) #U(b, 0, N1)≥b(q−1). Moreover, #U(b, i, N1)≤Q−1
N1and #Ub, 0,Q−1
(q−1)b=
b(q−1) if b≤k0.
(6) F∗
q=U(1,0, N1)⊂U(2,0, N2)⊂ ·· · ⊂ U(k0,0, N1) = C(N1,qk0)
0and ∅=U(1, j, N1)⊂
U(2, j, N2)⊂ · ·· ⊂ U(k0, j, N1) = C(N1,qk0)
jfor all j∈ {1,2,...,N1−1}.
Proof. The first two statements are trivial. When b= 1, u1has to be a nonzero elements
of Fq. Since F∗
q=hαQ−1
q−1iand N1= gcd Q−1
q−1, N, we obtain u1∈C(N1,Q)
0,k(u1)= 0,
U(1,0, N1) = F∗
qand U(1, i, N1) = ∅for i6= 0.
9
When b=k0,the set nPk0
i=1 uiθi−1(u1,...,uk0)∈Fk0
q\ {0}o=F∗
qk0since k0is the
multiplicative order of qmodulo n. Then #U(k0, i, N1) equals the size of the following
set
nαα∈C(N1,Q)
i∩F∗
qk0o=α
α∈C(N1,qk0)
i.
Therefore, #U(k0, i, N1) = qk0−1
N1for all i∈ {0,1,...,N1−1}.
For any i∈ {1,2,...,b}, we have uiθi−1∈C(N1,Q)
0if ui6= 0. Then
(0,...,0, ui,0,...,0) ∈U(b, 0, N1).
Therefore, #U(b, 0, N1)≥b(q−1).If b≤k0, then #U(b, i, N1)≤C(N1,Q)
i=Q−1
N1.
Moreover,
b(q−1) ≤#Ub, 0,Q−1
(q−1)b≤Q−1
Q−1
(q−1)b
=b(q−1).
Therefore, #Ub, 0,Q−1
(q−1)b=b(q−1).
Combining the parts (3), (4) and (5) of this lemma, we obtain the last desired result.
Example 11. The numerical examples in Table 1 are computed by Magma. In these
examples, we let b
Q−1
q−1, 1 ≤b≤k0and N1=Q−1
q−1. The value of #U(b, 0, N1) computed
by Magma is consistent with the part (5) of Lemma 10.
Table 1: Numeral examples of Lemma 10
Q q b N N1#U(b, 0, N1)
242 3 5 5 3
262 3 21 21 3
282 5 51 51 5
210 2 3 341 341 3
464 3 455 455 9
464 5 273 273 15
484 5 4369 4369 15
343 2 20 20 4
363 2 182 182 4
383 2 1640 1640 4
383 4 820 820 8
383 5 656 656 10
Therefore, the determination of the b-symbol weight distribution of an irreducible
cyclic code is equivalent to the values of #U(b, i, N1) and η(N1,Q)
i. The following result is
a generalization of Lemma 6.
10
The b-symbol weight distribution of irreducible cyclic codes and related consequences
Corollary 12. Let 1≤b≤k0and let c(β)be a codeword of the irreducible cyclic code
C(Q, N)as in (2). Let αbe a primitive element of FQ,θ=αN,N1= gcd Q−1
q−1, Nand
(u1,...,ub)∈Fb
q\ {0}. If 06=β∈C(N1,Q)
i, then the b-symbol weight of c(β)is
wb(c(β)) = (qb−1)(Q−1)
qbN−N1
qbN
N1−1
X
i=0
#U(b, i, N1)η(N1,Q)
i.
Proof. The desired result follows from the definition of U(b, i, N1) and Theorem 8.
The two keys to determine the b-symbol weight distribution of irreducible cyclic
codes are #U(b, i, N1) and η(N1,Q)
i. By Lemma 10, #U(1, i, N1) are determined for
i∈ {0,1,...,N1−1}. However, there are very few known results on #U(b, i, N1) for
b≥2. Under some restrictions on b, i, N1and k0, Lemma 10 gives an upper bound for
#U(b, i, N1), a lower bound for #U(b, 0, N1). When b≤k0and N1=Q−1
(q−1)b, the value of
#U(b, i, N1) is determined.
Open Problem 13. Determine the value of #U(b, i, N1) for all i∈ {0,1,...,N1−1}.
Or give some strong bounds for #U(b, i, N1).
The following theorem gives a necessary and sufficient condition for an irreducible
cyclic code to be a constant b-symbol weight code under some assumption.
Theorem 14. Let 1≤b≤m−1.Assume that the matrix Adefined as in (3) is invertible.
Then the irreducible cyclic code C(Q, N)is a constant b-symbol code with length Q−1
Nand
dimension mif and only if #U(b, i, N1) = #U(b, j, N1)for all i6=j. Moreover, the
constant b-symbol weight is (qb−1)Q
qbN.
Proof. We prove the sufficient condition at first. Assume that #U(b, i, N1) = #U(b, j, N1)
for all i6=j. According to Theorem 8 and the definition of U(b, i, N1), for any two nonzero
elements β1, β2∈FQ,
•if β1and β2belong to the same cyclotomic class, then wb(c(β1)) = wb(c(β2));
•if β1∈C(N1,Q)
iand β2∈C(N1,Q)
jwhere i6=j, then
wb(c(β1)) = (qb−1)(Q−1)
qbN−N1
qbN
N1−1
X
k=0
#U(b, k, N1)η(N1,Q)
i+k(mod N1)
=(qb−1)(Q−1)
qbN−N1
qbN#U(b, 0, N1)
N1−1
X
k=0
η(N1,Q)
i+k(mod N1)
=(qb−1)(Q−1)
qbN−N1#U(b, 0, N1)
qbN(−1) (from Lemma 4)
=(qb−1)(Q−1)
qbN+N1#U(b, 0, N1)
qbN.
11
Similarly, we can prove that
wb(c(β2)) = (qb−1)(Q−1)
qbN−N1
qbN
N1−1
X
k=0
#U(b, k, N1)η(N1,Q)
j+k(mod N1)
=(qb−1)(Q−1)
qbN+N1#U(b, 0, N1)
qbN
=wb(c(β1)).
We now prove the necessity of the condition. Assume that C(Q, N ) is a constant b-symbol
weight code and the constant b-symbol weight is ζ1. Then for any i∈ {0,1,...,N1−1},
PN1−1
k=0 #U(b, k, N1)η(N1,Q)
i+k(mod N1)is a constant which equals ζ2.Assume that βi∈C(N1,Q)
i
for i∈ {0,1,...,N1−1}and A(i) denotes the matrix
A(i) =
η(N1,Q)
0··· η(N1,Q)
i−2ζ2η(N1,Q)
i··· η(N1,Q)
N1−1
η(N1,Q)
N1−1··· η(N1,Q)
i−3ζ2η(N1,Q)
i−1··· η(N1,Q)
N1−2
.
.
.....
.
..
.
..
.
.....
.
.
η(N1,Q)
1··· η(N1,Q)
i−1ζ2η(N1,Q)
i+1 ··· η(N1,Q)
0
N1×N1
.
According to Lemma 4, we have
N1−1
X
i=0
wb(c(βi)) = (qb−1)(Q−1)N1
qbN−N1
qbN
N1−1
X
i=0
N1−1
X
k=0
#U(b, k, N1)η(N1,Q)
i+k(mod N1)
=(qb−1)(Q−1)N1
qbN−N1
qbN(qb−1) ·(−1)
=(qb−1)QN1
qbN.
Then ζ1=(qb−1)Q
qbNand ζ2=−qb+ 1. Therefore, the constant b-symbol weight is (qb−1)Q
qbN.
Solving the following system of equations, we obtain
wb(c(β0)) = ζ1,
wb(c(β1)) = ζ1,
.
.
.
wb(c(βN1−1)) = ζ1,
=⇒A·
#U(b, 0, N1)
#U(b, 1, N1)
.
.
.
#U(b, N1−1, N1)
=
ζ2
ζ2
.
.
.
ζ2
,(9)
where the matrix Ais defined as in Lemma 4. Combing the assumption that Ais invertible
and the Cramer’s rule, the solutions of (9) are
#U(b, i, N1) = det(A(i))
det(A)for all i∈ {0,1,...,N1−1}.
The desired result follows since det(A(0)) = ···= det(A(N1−1)).
12
The b-symbol weight distribution of irreducible cyclic codes and related consequences
Theorem 14 gives a complete characterization of constant b-symbol weight irreducible
cyclic codes in the general case that Nis any divisor of Q−1. It easy to check that
#U(b, i, N1) = #U(b, j, N1) for all i6=jif N1= 1 or b=m. Shi et al. [31] give the
b-symbol weight enumerator of the irreducible cyclic codes C(Q, N ) when N1= 1. By
Theorem 1, for any cyclic code Cwith dimension m,Cis a constant b-symbol weight code
with b-symbol weight nif b≥m.
Recall that f(x) = Pk−1
i=0 η(k,Q)
ixi.From a number of numerical results computed by
Magma, we find the assumption f(wi)6= 0 always holds for all i∈ {0,1,...,N1−1}. It is
reasonable to conjecture that Ais invertible by the part (4) of Lemma 4. This conjecture
is of interest for cyclotomy.
Conjecture 15. The circulant matrix Adefined as in (3) is invertible.
3.1 The complete b-symbol weight enumerator when N1= 2
Zhu et al. [41] considered the b-symbol weight enumerator of the irreducible cyclic codes
C(Q, N) where N1= 2. For completeness, we list these results as follows.
Theorem 16. [41] Let P(b)be the subset of cardinality qb−1
q−1in F∗
Qdefined as
P(b) = Sb−1
j=1 θ(j−1) +x1θj+···+xb−jθ(b−1)|(x1,...,xj)∈Fj
q∪θ(b−1)
and let
µ(b) = # x∈ P(b)|xis a square in F∗
Q.
If N1= 2 and 1≤b≤m−1, then the b-symbol weight distribution of C(Q, N )is
1 + Q−1
2(Tu1+Tu2),
where
u1=
qb−1
N(q−1)qb−1Q−Q+(q−1)Q1
2
q+2µ(b)(q−1)Q1
2
Nqbif p≡1 mod 4,
qb−1
N(q−1)qb−1Q−Q+(−1) sm
2(q−1)Q1
2
q+2µ(b)(−1) sm
2(q−1)Q1
2
Nqbif p≡3 mod 4,
and
u2=
qb−1
N(q−1)qb−1Q−Q−(q−1)Q1
2
q−2µ(b)(q−1)Q1
2
Nqbif p≡1 mod 4,
qb−1
N(q−1)qb−1Q−Q−(−1) sm
2(q−1)Q1
2
q−2µ(b)(−1) sm
2(q−1)Q1
2
Nqbif p≡3 mod 4.
The expressions of Theorem 16 depends on the invariant µ(b). By the definitions of
#U(b, 0,2) and µ(b), we obtain #U(b, 0,2) = (q−1)µ(b).For the sake of consistency, we
give the expression by using #U(b, 0,2) rather than µ(b) in the following.
13
Theorem 17. If N1= 2 and 1≤b≤m−1, then the b-symbol weight distribution of
C(Q, N)is
1 + Q−1
2(Tu1+Tu2),
where
u1=
(qb−1)(Q−Q1
2)
Nqb+2Q1
2
Nqb#U(b, 0,2),if p≡1(mod 4),
(qb−1)(Q−(−1) sm
2Q1
2)
Nqb+2(−1) sm
2Q1
2
Nqb#U(b, 0,2),if p≡3(mod 4),
and
u2=
(qb−1)(Q+Q1
2)
Nqb−2Q1/2
Nqb#U(b, 0,2),if p≡1(mod 4),
(qb−1)(Q+(−1) sm
2Q1
2)
Nqb−2(−1) sm
2Q1
2
Nqb#U(b, 0,2),if p≡3(mod 4).
Moreover,
wb(c(β)) =
0,if β= 0;
u1,if β∈C(2,Q)
0;
u2,if β∈C(2,Q)
1.
3.2 The complete b-symbol weight enumerator when N1= 3
Theorem 18. Let N1= 3 and 1≤b≤m−1. When p≡2(mod 3), the b-symbol weight
enumerator of C(Q, N)is
Ab(T) = 1 + Q−1
3(Tu1+Tu2+Tu3),
where
ui=
(qb−1)(Q−Q1
2)
qbN+3Q1
2
qbN#U(b, j, 3),if sm
2is even,
(qb−1)(Q+Q1
2)
qbN−3Q1
2
qbN#U(b, j, 3),if sm
2is odd,
and i+j≡0 ( mod 3).Moreover,
wb(c(β)) =
0,if β= 0;
u1,if β∈C(3,Q)
0;
u2,if β∈C(3,Q)
1;
u3,if β∈C(3,Q)
2.
Proof. Assume the 0 6=β∈C(3,Q)
iand j≡ −i(mod 3).When p≡2 ( mod 3), from the
known results on the Gaussian periods, η(3,Q)
itake only two values. Combining the values
14
The b-symbol weight distribution of irreducible cyclic codes and related consequences
of η(3,Q)
i(see (5) of Lemma 5) and Theorem 8, we have
wb(c(β)) = (qb−1)(Q−1)
qbN−#U(b, j, 3)3η(3,Q)
0
qbN
−(qb−1−#U(b, j, 3))3η(3,Q)
j1
qbN(where j16= 0)
=
(qb−1)(Q−Q1
2)
qbN+3Q1
2
qbN#U(b, j, 3),if sm
2is even;
(qb−1)(Q+Q1
2)
qbN−3Q1
2
qbN#U(b, j, 3),if sm
2is odd.
This completes the proof.
3.3 The complete b-symbol weight enumerator when N1= 4
Theorem 19. Let N1= 4 and 1≤b≤m−1. When p≡3(mod 4), the b-symbol weight
enumerator of C(Q, N)is
Ab(T) = 1 + Q−1
4(Tu1+Tu2+Tu3+Tu4),
where
ui=
(qb−1)(Q−Q1
2)
qbN+4Q1
2
qbN#U(b, j, 4) if sm
2is even,
(qb−1)(Q+Q1
2)
qbN−4Q1
2
qbN#U(b, j, 4) if sm
2is odd,
and i+j≡0 ( mod 4).Moreover,
wb(c(β)) =
0,if β= 0;
u1,if β∈C(4,Q)
0;
u2,if β∈C(4,Q)
1;
u3,if β∈C(4,Q)
2;
u4,if β∈C(4,Q)
3.
Proof. Assume that 0 6=β∈C(4,Q)
iand i+j≡0(mod 4).From (6) of Lemma 5, the
values of η(4,Q)
ifor i= 0,1,2,3 are known. When p≡3(mod 4), we obtain
wb(c(β)) = (qb−1)(Q−1)
qbN−4η(4,Q)
0
qbN#U(b, j, 4)
−4η(4,Q)
j1
qbN(qb−1−#U(b, j, 4)) (where j16= 0)
=
(qb−1)(Q−Q1
2)
qbN+4Q1
2
qbN#U(b, j, 4),if sm
2is even;
(qb−1)(Q+Q1
2)
qbN−4Q1
2
qbN#U(b, j, 4),if sm
2is odd.
This completes the proof.
15
The values of Gaussian periods of some small order, such as, η(5,Q)
i,η(6,Q)
i,η(8,Q)
iand
η(12,Q)
iare determined in [16, 18]. Mimicking the proofs above, we can give the b-symbol
weight enumerators of C(Q, N) when N1= 5,6,8 and 12. These results depend on the
values of #U(b, j, N1) and have very tedious expression of the Gaussian periods. For the
sake of brevity, we omit these cases.
3.4 The complete b-symbol weight enumerator in the semi-primitive cases
The following theorem gives the b-symbol weight enumerators for a class of irreducible
cyclic codes.
Theorem 20. Let s·mbe even and N1>2. Assume that there exists a positive integer
jsuch that pj≡ −1(mod N1), and the jis the least such. Let Q=p2jγ for some integer
γ.
If γ, p and pj+1
N1are all odd, we have the following result. Assume that N1≤Q1
2.
•If β= 0, then wb(c(β)) = 0;
•If β∈C(N1,Q)
i, then
wb(c(β)) = (qb−1)(Q+Q1
2)
qbN−N1Q1
2
qbN#U(b, j1, N1),
where i+j1≡N1
2(mod N1).
Moreover, the b-symbol weight enumerator of C(Q, N )is
Ab(T) = 1 + Q−1
N1
N1
X
j=1
Tuj,
where
ui=(qb−1)(Q+Q1
2)
qbN−N1Q1
2
qbN#U(b, j1, N1).
In all other cases, we have the following. Assume that Assume that N1≤Q1
2if γis
odd.
•If β= 0, then wb(c(β)) = 0;
•If β∈C(N1,Q)
i, then
wb(c(β)) = (qb−1)(Q−(−1)γQ1
2)
qbN+(−1)γN1Q1
2
qbN#U(b, j2, N1),
where i+j2≡0(mod N1).
16
The b-symbol weight distribution of irreducible cyclic codes and related consequences
Moreover, the b-symbol weight enumerator of C(Q, N )is
Ab(T) = 1 + Q−1
N1
N1
X
j=1
Tuj,
where
ui=(qb−1)(Q−(−1)γQ1
2)
qbN+(−1)γN1Q1
2
qbN#U(b, j2, N1).
Proof. To ensure the dimension of C(Q, N ) is 2jγ, there is not any β∈F∗
Qsuch that
wb(c(β)) = 0. Then wb(c(β)) ≥bfor any β∈F∗
Q.
If γ, p and pj+1
kare all odd, by (7) and (8) of Lemma 5, then
wb(c(β)) = (qb−1)(Q−1)
qbN−
N1η(N1,Q)
N1
2
qbN#U(b, j1, N1)
−N1η(N1,Q)
j′
1
qbN(qb−1−#U(b, j1, N1)) (where j′
16=N1
2)
=(qb−1)(Q+Q1
2)
qbN−N1Q1
2
qbN#U(b, j1, N1),
where i+j1≡N1
2(mod N1).We need the assumption that N1≤Q1
2=Q−b
Q1
2+ 1 since
#U(b, j1, N1)≤qb−1 and wb(c(β)) ≥b.
In all other cases, we obtain
wb(c(β)) = (qb−1)(Q−1)
qbN−N1η(N1,Q)
0
qbN#U(b, j2, N1)
−N1η(N1,Q)
j′
2
qbN(qb−1−#U(b, j2, N1)) (where j′
26= 0)
=(qb−1)(Q−(−1)γQ1
2)
qbN+(−1)γN1Q1
2
qbN#U(b, j2, N1),
where i+j2≡0(mod N1).Similarly, we need the assumption that N1≤Q1
2=Q−b
Q1
2+ 1
to ensure wb(c(β)) ≥bif γis odd.
It is easy to check that Theorem 17, Theorem 18, and Theorem 19 are special cases
of Theorem 20. When N1=N, this is the classical semi-primitive case. The Hamming
weight enumerator of the semi-primitive irreducible cyclic codes are studied by Delsarte
and Goethals [8], McEliece [26], and Baumert and McEliece [1]. Ding and Yang [13]
considered the more flexible case where N1= gcd Q−1
q−1, N>2.Theorem 20 generalizes
their results to b-symbol metric.
17
4 The b-symbol weight hierarchy of some irreducible cyclic codes
Yang et al. [39] studied Hamming weight hierarchy db(C) of irreducible cyclic codes in
2015. In this section, we study the b-symbol weight hierarchy db(C) of some classes of
irreducible cyclic codes and compare the two hierarchies. Since db(C) = db(C) for any
cyclic code if b= 1 or b= dim(C), we omit the two trivial cases in this section.
4.1 The b-symbol weight hierarchy when N1= 1
The following result on the b-th generalized weight hierarchy of C(Q, N ) is from [39,
Corollary 7].
Corollary 21. [39] If N1= 1 and 2≤b≤m−1, then
db(C(Q, N)) = qb−1Q
qbN.
Combining Theorem 14 and Corollary 21, we obtain the following result directly.
Corollary 22. If N1= 1 and 2≤b≤m−1, then
db(C(Q, N)) = db(C(Q, N )) = qb−1Q
qbN.
4.2 The b-symbol weight hierarchy when N1= 2
The following result is also from [39].
Theorem 23. If N1= 2, then 2≤b≤m−1, then
db(C(Q, N)) =
(qb−1)Q−Q1
2
Nqb,for 2≤b≤m
2;
Qqb−2Q+qb
Nqb,for m
2< b ≤m−1.
When N1= 2, the two types of the minimum distance of the same irreducible cyclic
code are equal under some restrictions.
Theorem 24. If N1= 2 and 1≤b≤m−1, then
db(C(Q, N)) =
(qb−1)Q+Q1
2
Nqb−2Q1
2
Nqb#U(b, 0,2),if #U(b, 0,2) ≥qb−1
2;
(qb−1)Q−Q1
2
Nqb+2Q1
2
Nqb#U(b, 0,2),if #U(b, 0,2) <qb−1
2.
Moreover, db(C(Q, N)) = db(C(Q, N )) if and only if one of the following two statement
holds:
(1) 1 ≤b≤m
2and #U(b, 0,2) = qb−1.
18
The b-symbol weight distribution of irreducible cyclic codes and related consequences
(2) m
2< b ≤mand #U(b, i, 2) = (qb−Q1
2)(Q1
2+1)
2Q1
2for some i∈ {0,1}.
Proof. Assume that β0∈C(2,Q)
0and β1∈C(2,Q)
1. If #U(b, 0,2) ≥qb−1
2, then we have
wb(c(β0)) = (qb−1)(Q−Q1
2)
Nqb+2Q1
2
qbN#U(b, 0,2)
≥wb(c(β1)) = (qb−1)(Q+Q1
2)
Nqb−2Q1
2
qbN#U(b, 0,2),if p≡1(mod 4) or sm
2is even;
wb(c(β0)) = (qb−1)(Q+Q1
2)
Nqb−2Q1
2
qbN#U(b, 0,2)
≥wb(c(β1)) = (qb−1)(Q−Q1
2)
Nqb+2Q1
2
qbN#U(b, 0,2),if p≡3(mod 4) and sm
2is odd.
If #U(b, 0,2) <qb−1
2, then we have
wb(c(β0)) = (qb−1)(Q−Q1
2)
NQb+2Q1
2
qbN#U(b, 0,2)
< wb(c(β1)) = (qb−1)(Q+Q1
2)
NQb−2Q1
2
qbN#U(b, 0,2),if p≡1(mod 4) or sm
2is even;
wb(c(β0)) = (qb−1)(Q+Q1
2)
NQb−2Q1
2
qbN#U(b, 0,2)
> wb(c(β1)) = (qb−1)(Q−Q1
2)
NQb+2Q1
2
qbN#U(b, 0,2),if p≡3(mod 4) and sm
2is odd.
Therefore, the minimum b-symbol distance of C(Q, N) is
db(C(Q, N)) =
(qb−1)(Q+Q1
2)
Nqb−2Q1
2
Nqb#U(b, 0,2),if #U(b, 0,2) ≥qb−1
2;
(qb−1)(Q−Q1
2)
Nqb+2Q1
2
Nqb#U(b, 0,2),if #U(b, 0,2) <qb−1
2.
The two types of the minimum distance are equal if and only if the following equations
hold:
•If 2 ≤b≤m
2, then
(qb−1)(Q−Q1
2) = ((qb−1)(Q+Q1
2)−2Q1
2#U(b, 0,2),if #U(b, 0,2) ≥qb−1
2;
(qb−1)(Q−Q1
2) + 2Q1
2#U(b, 0,2),if #U(b, 0,2) <qb−1
2;
⇐⇒ #U(b, 0,2) = 0 or #U(b, 0,2) = qb−1.
According to the part (5) of Lemma 10, #U(b, 0,2) can not be zero, then #U(b, 0,2) =
qb−1.
•If m
2< b ≤m−1, then
Qqb−2Q+qb=((qb−1)(Q+Q1
2)−2Q1
2#U(b, 0,2),if #U(b, 0,2) ≥qb−1
2;
(qb−1)(Q−Q1
2) + 2Q1
2#U(b, 0,2),if #U(b, 0,2) <qb−1
2;
⇐⇒ #U(b, 0,2) = (qb+Q1
2)(Q1
2−1)
2Q1
2
or #U(b, 0,2) = (qb−Q1
2)(Q1
2+ 1)
2Q1
2
.
This completes the proof.
Example 25. When q= 3, m= 10 and b= 2, we have #U(2,0,2) = 8 = qb−1.Then the
minimal symbol-pair distance of C(310,2) equals d2(C(310,2)) = (32−1)(35−1)·35
2·32= 26136.
19
4.3 The b-symbol weight hierarchy in the semi-primitive cases
Theorem 26. Let s·mbe even and N1>2. Assume that there exists a positive integer
jsuch that pj≡ −1(mod N1), and the jis the least such. Let Q=p2jγ for some integer
γ. Then
db(C(Q, N )) =
(qb−1)(Q+Q1
2)
qbN−N1Q1
2
qbN·max{#U(b, i, N1)|i∈ZN1},if γis odd and N1≤Q1
2,
(qb−1)(Q−Q1
2)
qbN+N1Q1
2
qbN·min{#U(b, i, N1)|i∈ZN1},if γis even.
Proof. From Theorem 20, if γ, p and pj+1
N1are all odd, we have the following result. Assume
that N1≤Q1
2. Then
db(C(Q, N)) = (qb−1)(Q+Q1
2)
qbN−N1Q1
2
qbNmax{#U(b, i, N1)|i∈ZN1}.
In all other cases, we have the following result. Assume that N1≤Q1
2if γis odd. Then
db(C(Q, N )) =
(qb−1)(Q+Q1
2)
qbN−N1Q1
2
qbN·max{#U(b, i, N1)|i∈ZN1},if γis odd,
(qb−1)(Q−Q1
2)
qbN+N1Q1
2
qbN·min{#U(b, i, N1)|i∈ZN1},if γis even.
This completes the proof.
Remark 27.From the Magma experimental data, #U(b, 0, N1) is always the maximum
value of the set {#U(b, i, N1)|i∈ZN1}. So it is important to determine the value of
#U(b, 0, N1).
5 Shortened codes from irreducible cyclic codes
In this section, we introduce a technique for constructing new codes from old codes, which
we call b-symbol shortened construction. To this end, we need the following definition of
b-symbol support of a vector.
Definition 28. The b-symbol support of a vector xis defined by
Ib(x) = supp(πb(x)) =
b−1
[
i=0
supp(τi(x)),
where supp(x) denotes the support of the vector x.Let Ib(x) = {1,2,...,n}\Ib(x).
Remark 29.In the b-th generalized Hamming metric, there is a definition about the
support of the subcode Dof Cwhich defined to be
χ(D) = {i: 0 ≤i≤n−1|ci6= 0 for some (c0, c1,...,cn−1)∈D}.
The two definitions on the support can be viewed as two different generalizations for the
support of a vector.
20
The b-symbol weight distribution of irreducible cyclic codes and related consequences
The following proposition gives an interesting shortening technique.
Proposition 30. Let Cbe a cyclic code with parameters [n, K, dH(C)] over Fq. Let
1≤b≤Kand let cbe a codeword with the minimal b-symbol weight of C. Then the
shortened code CIb(c)has parameters [db(C), b, ≥dH(C)].
Proof. Let Gb(c) be the matrix
Gb(c) =
c
τ(c)
.
.
.
τb−1(c)
b×n
=r1r2··· rnb×n,
where riare column vectors belonging to Fb
qfor i∈ {1,...,n}.From [32, Lemma 16], the
rank of Gb(c) equals b. Let
G′
b(c) = rj1rj2... rjmb×m,
where rj1,...,rjmare all the nonzero columns of Gb(c). By the definition of the b-symbol
weight metric, m=db(C).This yields the desired result, since the shortened code CIb(c)
is generated by G′
b(c).
The following bound is the famous Griesmer bound. It was proved by Griesmer [15]
for binary codes, and later generalized by Solomon and Stiffler [33] for q > 2.
Theorem 31. Let Cbe a linear codes with parameters [n, K, dH(C)] over Fqwith K≥1.
Then
n≥
K−1
X
i=0 dH(C)
qi.
A code which achieves the Griesmer bound is called a Griesmer code. The following
theorem gives a class of Griesmer codes from C(Q, N ) that shorten some proper coordi-
nates.
Theorem 32. Let N1= 1 and 1≤b≤m. Let cbe a nonzero codeword of C(Q, N).
Then the parameters of the shortened codes C(Q, N)Ib(c)are
(qb−1)Q
qbN, b, (q−1)Q
qN .
Moreover, C(Q, N )Ib(c)are Griesmer codes if N|q−1.
Proof. When N1= 1, any nonzero codeword of C(Q, N) has the minimal b-symbol weight.
Combining Proposition 30 and Theorem 14, we obtain the desired parameters. Recall that
Q=qm. Then we have
b−1
X
i=0 &(q−1)Q
qN
qi'=q−1
N
b
X
i=1
qm−i=n.
Therefore, C(Q, N)Ib(c)are Griesmer codes if N|q−1.
21
Remark 33.When N=q−1, the code C(Q, N ) is the Simplex code. Liu et al. [23]
considered the code from the Simplex code S(m, q) which shortened any one or two
coordinates. Theorem 32 consider the case where the size of shortening set is greater than
2.
Example 34. As we all know, the parameters of the Simplex code S(m, q) over Fqare
[qm−1
q−1, m, qm−1].Let cbe a nonzero codeword of S(m, q). The parameters of the shortened
codes S(m, q)Ib(c)are given in Table 2. These codes are all Griesmer codes. From the
numerical example, we can see that when the size of the shortened set exceeds 2, we still
get the codes with very good parameters.
Table 2: Numeral examples of Theorem 31
Shortened code Parameters
S(4,2)I3(c)[14,3,8]2
S(4,2)I2(c)[12,2,8]2
S(5,2)I4(c)[30,4,16]2
S(5,2)I3(c)[28,3,16]2
S(5,2)I2(c)[24,2,16]2
S(4,3)I3(c)[39,3,27]3
S(4,3)I2(c)[36,2,27]3
S(5,3)I4(c)[120,4,81]3
S(5,3)I3(c)[117,3,81]3
S(5,3)I2(c)[108,2,81]3
S(4,4)I3(c)[84,3,64]4
S(4,4)I2(c)[80,2,64]4
S(5,4)I4(c)[340,4,256]4
S(5,4)I3(c)[336,3,256]4
S(5,4)I2(c)[320,2,256]4
6 Summary and concluding remarks
The main contributions of this paper are the following:
•A general formula for computing the b-symbol weight of a nonzero codeword of an
irreducible cyclic code is given. It is a generalization of the formula for computing
the Hamming weight of a nonzero codeword of an irreducible cyclic code from [13].
•The b-symbol weight hierarchies of some irreducible cyclic codes are given. The
two types of weight hierarchies of the same irreducible cyclic code mentioned in
this paper are compared. In particular, the two weight hierarchies are equal under
certain conditions.
22
The b-symbol weight distribution of irreducible cyclic codes and related consequences
•we present an application of the b-symbol weight hierarchy of cyclic codes in the
shortening technique and construct some new shortened codes with nice parameters.
Compared to the work in [23], when the object code is an irreducible cyclic code,
the size of the shortened set can be greater than 2.
•The results that the b-symbol weight hierarchies of irreducible cyclic codes provide
nice upper bounds on the generalized weight hierarchies of irreducible cyclic codes.
Acknowledgement
This research is supported by Natural Science Foundation of China (12071001), Excellent
Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20).
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