Content uploaded by Mohammad Javad Amoshahy
Author content
All content in this area was uploaded by Mohammad Javad Amoshahy on Oct 24, 2014
Content may be subject to copyright.
3488 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 8, AUGUST 2009
On the Stopping Distance of Array Code
Parity-Check Matrices
Morteza Esmaeili, Member, IEEE, and Mohammad Javad Amoshahy
Abstract—For
q
an odd prime and
1
m
q
, we study two bi-
nary
qm
2
q
2
parity check matrices for binary array codes. For
both parity check matrices, we determine the stopping distance
and the minimum distance of the associated code for
2
m
3
,
and for
(
m; q
)=(4
;
5)
. In the case
(
m;q
)=(4
;
7)
, the stopping
distance and the related minimum distance are also determined for
one of the given parity check matrices. Moreover, we give a lower
bound on the stopping distances for
m>
3
and
q>
3
.
Index Terms—Stopping distance, array code, low-density parity-
check (LDPC) code.
I. INTRODUCTION
LET be a parity-check matrix, with possibly dependent
rows, for a given binary code , where and
denote the code’s length, dimension, and minimum Hamming
distance, respectively. Thus, the number of rows of is at least
, the redundancy of , denoted . A stopping set of
size for is an -elements set of column indices such
that the associated -columns submatrix of does not contain a
weight-one row. From Tanner graph [1] perspective, a stopping
set in is a subset of the variable nodes in TG , the Tanner
graph representing , such that all the neighbors of are con-
nected to at least twice. The size of the smallest nonempty
stopping sets of , denoted , is called the stopping dis-
tance of [2]–[9]. The stopping distance is an important pa-
rameter for iterative decoding on binary erasure channels (BEC)
and should be maximized for better performance. The role of the
stopping distance of on the performance of under iterative
decoding algorithms is very similar to that of , the min-
imum distance of , under maximum-likelihood decoding over
a BEC. Although the minimum distance is a fixed param-
eter, the stopping distance depends on the chosen parity-check
matrix for . It is easy to verify that .
Low-density parity-check (LDPC) codes, were discovered
first by Gallager [10], [11] in the early 1960s. It has been shown
that these codes achieve a remarkable performance very close
to the Shannon limit with iterative decoding [12], [13]. Array
codes form a well-structured class of quasi-cyclic LDPC codes
[14], [15]. Due to their quasi-cyclic structure, these codes can
Manuscript received June 22, 2008; revised February 16, 2009. Current ver-
sion published July 15, 2009. This work was supported in part by a grant from
Iran Telecommunication Research Center (ITRC).
The authors are with the Department of Mathematical Sciences, Isfahan Uni-
versity of Technology, 84156-83111, Isfahan, Iran (e-mail: emorteza@cc.iut.
ac.ir; ajam@math.iut.ac.ir).
Communicated by L. M. G. M. Tolhuizen, Associate Editor for Coding
Theory.
Digital Object Identifier 10.1109/TIT.2009.2023736
be encoded with simple shift registers [16], [17]. The lattice
codes constructed in [18], [19] are equivalent to array codes.
For high rate and moderate length, say up to about 5000,
these codes perform as well as the best comparable randomly
constructed regular LDPC codes given in [20]. Array codes are
well studied from the minimum distance and girth perspective
[18], [19], [21]–[23]. The Tanner graph of an array code has
girth [21].
In general, it is very hard to give a full characterization of
stopping sets of an arbitrary code, and harder to analyze the re-
lationships that exist between minimum distance and stopping
distance of a given code. It is known that the stopping distance
of a Berlekamp-Justesen based LDPC code is not smaller than
the best known lower bound of its minimum distance [24]. It
has been shown in [9] that a previously known lower bound
on the minimum distance of finite-geometry LDPC codes is
also a lower bound on the stopping distance of these codes,
implying that finite-geometry LDPC codes have considerably
large stopping distance. As mentioned, the stopping distance of
any parity-check matrix for a given code satisfies
. Hence finding a code with parity-check matrix , ei-
ther full-rank or with few linearly dependent rows, satisfying
would be of great interest.
In this paper, we consider two classes of codes referred to
as proper and improper array codes. For a given odd prime
and integer , these codes are denoted by and
, respectively, and represented by specific parity-check
matrices and , respectively. The stopping
distance of and is determined for
and . We also determine stopping distance of
. Interestingly, for each of these cases the stopping
distance of the considered parity-check matrix is equal to
the minimum distance of the associated code. We show that
if .
Section II is devoted to code construction. In Subsection III.A
the stopping distance of matrices
and is determined. The stopping distance of
and is examined in Section III-B.
The paper concludes with few open problems stated in
Section IV.
II. ARRAY CODES
Let denote the identity matrix and be the circulant
permutation matrix obtained from by cyclically shifting its
rows positions to the left, that is where
s are all zero matrices of appropriate sizes. We define
when the number , introducing the size of the matrix, is known;
and .
0018-9448/$25.00 © 2009 IEEE
Authorized licensed use limited to: UNIVERSITY OF ALBERTA. Downloaded on July 15, 2009 at 18:28 from IEEE Xplore. Restrictions apply.
ESMAEILI AND AMOSHAHY: ARRAY CODE PARITY-CHECK MATRICES 3489
Let be a prime, be a circulant
permutation matrix, and . Then the LDPC code
with the following parity-check matrix is known as an array
LDPC code [15]
.
.
..
.
..
.
..
.
..
.
.
Considering as a matrix, with the circulant per-
mutation matrices as its entries, each of its rows ( columns)
is referred to as a block-row (respectively, block-column).
A generalization of matrix has been given in [21]. The
parity-check matrix given in [21] is in the form matrix ,
given below, where
.
.
..
.
..
.
..
.
.....
.
.
where is some sequence of distinct integers
satisfying . If the sequence
forms an arithmetic progression (A.P.), i.e., if
, for some , then the corresponding
code is called a proper array (PA) code, else an improper array
(IA) code. Note that if , then the PA code is simply the
ordinary array code defined by parity-check matrix .
In this paper by a PA code we mean the PA codes with
, and by IA code we refer to an IA code defined by
distinct integers and for .
Also, the term array code refers to both PA codes and IA
codes. The first cyclotomic coset of -modulo is the set
where is the smallest positive
integer for which . Thus the parity-check
matrix of an IA code of length considered in this paper has
at most block-rows.
The parity-check matrix of a PA code (respectively, IA code)
of length and block-rows, (respectively, ),
is denoted by (respectively, ). The associ-
ated LDPC codes are denoted by and , re-
spectively. In this paper, the term “array code with
parity-check matrix ” refers to both proper and im-
proper array codes specified above.
It is obvious that for we have
. This is also the case when where
. For simplicity, in the rest of the paper we work
with circulant permutation matrix .
Example 1: For we have . The
matrices and are as follows. These matrices
are not row-equivalent. However, and are
equivalent since .
Definition 1: and ): Consider a given
matrix as a matrix whose entries are
circulant permutation matrices. Let the circulant per-
mutation matrix be an arbitrary entry of this ma-
trix . Denote , by the string
and the identity matrix by
. The matrix obtained from by this proce-
dure is denoted by . The matrix is defined
in the same manner.
As an example, associated with we have
Theorem 1 ([21]): The Tanner graphs and
, have girth .
III. STOPPING DISTANCE OF ARRAY CODES
The main results of the paper are given in this section. The
first subsection contains the results on and , and
Section III-B consists of the rest of the work: determination of
stopping distance of for and . Prior to
studying and , we give an upper bound and a
lower bound on the stopping distance of .
Proposition 1: Let be an odd prime and .For
any integer , the parity-check matrix
has at least one stopping set of size . Thus .
We also have if .
Proof: The matrix is formed by circulant
permutation matrices and hence each row of the matrix con-
sisting of the first columns of has at least two
nonzero components. The second statement is also obvious.
Lemma 1 ([24]): Let be a binary code with parity-
check matrix . Suppose has girth at least 6 and that
the minimum column-weight in is . Then [24,
Lemma 2]. Furthermore, if has constant even column-weight
and consists of circulant permutation matrices then
[24, Th. 3].
Corollary 1: Let be an odd prime and . Then
if is odd
if is even
Proof: This follows from Theorem 1 and Lemma 1.
A. Stopping Distance of and
The value of is given by the following theorem.
Authorized licensed use limited to: UNIVERSITY OF ALBERTA. Downloaded on July 15, 2009 at 18:28 from IEEE Xplore. Restrictions apply.
3490 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 8, AUGUST 2009
Theorem 2: Let be an odd prime. Then we have
Proof: By Corollary 1 it suffices to display a stopping set
of size four. We know that . Denote the th
column of by and consider the set
. It follows from definition of that the
submatrix of consisting of columns ,is
As has no rows of weight one, the set is a stopping set for
.
To prove , applying the algebraic structure of
we first show that there is no stopping set of size four
for . Then in Lemma 3, we make use of the four-cycle
free property of , to show that has
no stopping set of size five. The statement is then completed by
providing a stopping set of size six.
Lemma 2: Let be an odd prime. Then ; thus,
for .
Proof: Note that we have .Itis
enough to show that . To do this, according to
Corollary 1, we just need to show that there is no stopping set
of size four for . Suppose, to the contrary, that there is a
stopping set of size four for . Therefore, up to a column
permutation, considering the fact that has girth 6,
the matrix contains a four-columns submatrix in
the form given below where
and .
The th block-column of , denoted ,is
given by (1) wherein the entries are considered modulo and
for a nonnegative integer we define if
and only if .
(1)
Assuming that the th, , column of is in the th
block-column of , we get the following form for
the matrix
wherein the entries are considered modulo . This results in the
following congruence relations modulo wherein as mentioned
above we have if and only if
.
Let , denote the th equation of this system.
Then gives , and
gives resulting in a
contradiction.
The fact that the girth of , is six implies
that has no stopping set of size five.
Lemma 3: Let be an odd prime and . There is
no stopping set of size five for .
Proof: Let be a five-column submatrix of .
In order for to define a stopping set of size five, each row
of needs to be a five-tuple consisting of precisely two dis-
tinct elements one of which repeated twice. Therefore, up to a
column permutation, we may assume that the first row of is
. Due to the structure of cycles of length four
and the fact that , has girth six, the first
two components of the second row of need to be distinct. De-
note these by and , left to right. As the replication number
of is 2 or 3, and since the last three components of the first
row are identical, we may assume that the third component of
the second row of is . Then the fourth component must be
. The obtained structure, however, accepts neither nor
as the fifth component of the second row.
According to Lemmas 2 and 3, the stopping distance of
is at least ; indeed we have .
Theorem 3: Let be an odd prime. Then .
Proof: By Lemmas 2 and 3 we just need to display a
stopping set of size six for . It is obvious that any two
block-columns of form a six-columns matrix with no
weight-one row. Thus we assume that . Consider the set
of column indices
that is the considered six-columns submatrix of
consists of the first two columns of the first block-column, the
first and last columns of the second block-column, the second
column of the th block-column, and the last column of the
th block-column of . These block-columns form
the matrix given here.
It is easily verified that the six-columns submatrix of
corresponding to the columns of matrix , is the matrix
given above, and hence is a stopping set.
Authorized licensed use limited to: UNIVERSITY OF ALBERTA. Downloaded on July 15, 2009 at 18:28 from IEEE Xplore. Restrictions apply.
ESMAEILI AND AMOSHAHY: ARRAY CODE PARITY-CHECK MATRICES 3491
B. Stopping Distance of
It is shown in this part that for any prime and
integer ,wehave and
.
Applying and the algebraic structure of ,
we show in Lemma 4 that has no stopping set of size
six. This together with Lemmas 2 and 3 implies .
Lemma 4: If a prime, then there is no stopping
set of size six for .
Proof: We first consider . The arguments given for
, except in one case, applies to .
Suppose a six-columns submatrix of represents
a stopping set of size six for . It is obvious that all
elements of the first row of cannot be identical and that
no element of this row can be with replication number one;
thus, the first row needs to contain at least two distinct elements
and each element must appear at least two times. Therefore,
up to a permutation, the first row of is limited to three
cases and as shown in
the equations at the bottom of the page for
some distinct integers . Note that this prop-
erty applies to each row of . Since has no
four-cycle, if then the second row of
needs to have at least four distinct elements, a contradiction.
Thus we need to consider only the last two choices for .
Suppose with . It is easy to
see that for this case the second row of is in the form
for some distinct integers
. The th column of the th block-column of is the
transpose of with
if and only if . Thus,
assuming that the th, , column of
belongs to the th (respectively, )) block-column of ,
the matrix will be in the form given by (+) wherein the entries
are considered modulo .
It follows from that
with . Thus from
the second row we have . The third row of
gives , and hence ,
meaning that is a multiple of , a contradiction.
Suppose with distinct integers
. The above argument shows that any row of
needs to contain precisely three distinct elements. There-
fore, since has no four-cycle, we may assume
that the second row of is for
some distinct integers . It is worth men-
tioning that though there are three other feasible arrangements
for , namely and
, but the four obtained two-rows matrices
are the same up to a column permutation.
With and
, due to the same reason, the third row of
can only be in the form
or
. With and , the fourth row
of has only two choices
and .
With the triple , the fourth row cannot
be different from and
. With the triple
there won’t be any choice for the fourth row. The triple
can be extended to a four-rows matrix with
girth six only with and
. Among these choices for
, matrices and are
row equivalent (interchanging and ). The same is true
with matrices and ,
and also between and .
Therefore, we need to consider only three matrices
and
.
Let and set and
. Suppose has the general form given by .
Assume that . It follows from
the second and third rows of that
(2)
Let , denote the th equation of the system given
in (2). Then gives
gives ; and
gives . The sum of these
equations gives , that is is a
multiple of which is a contradiction. Since this argument
does not make use of the fourth row of and that and
(+)
(*)
Authorized licensed use limited to: UNIVERSITY OF ALBERTA. Downloaded on July 15, 2009 at 18:28 from IEEE Xplore. Restrictions apply.
3492 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 8, AUGUST 2009
are the same in their first three rows, the result also applies to
.
Assume that . The second
rows in and are the same and hence the equations on
the top part of (2) hold for . The third and fourth rows
of equation result in the following two systems of
equations:
(3)
Let , and , denote the th and
th equations of the systems given by (2) and (3), respectively.
It follows from and that
, and from and that
, and hence .
On the other hand, and give
, and and give
; hence . These
two obtained equations give , a contradiction (note that
equations are all modulo ). Thus has no stopping set
of size six.
Consider now the matrix . The difference between
matrices and is limited only to their last rows.
Therefore, all the given arguments except the last part, that is
when wherein the fourth row involves, applies to
. The six-columns matrix given by is changed
to a submatrix of by simply replacing the numbers “4”
in its last row by number “3.” Consider this modified version of
and suppose . The equations on the lower part of
(3) are then changed to the following:
(4)
Let , denote the th equation of the systems
given by (4). Equations and give
, and and give
; hence,
. This together with
gives ,a
contradiction.
It follows from Lemmas 2, 3, and 4 that the stopping distance
of is at least seven if and . In the
following lemma the four-cycle free property of
is applied to show that for these values of and we have
.
Lemma 5: Let be a prime and let . Then there
is no stopping set of size seven for .
Proof: Let be a seven-columns submatrix of
representing a stopping set of size seven for . It is ob-
vious that each row of has at least two and at most three
distinct elements each of which with replication number at least
two. Accordingly, we are limited to the partitions , ,
and for number . By partition of a row in ,
for instance, we mean has only two distinct elements one of
which appeared two times and the other one five times. Since
has no four-cycle, it is easily verified that the ex-
istence of a row with partition or leads to a con-
tradiction and hence each row of can only be with partition
.
Suppose with
. It is easy to see that, up to a column permutation, can
only be in the form of or
. With the third row can only
be in the form of
or
. However, none of
can be extended to a four-rows matrix free of four-cycles using
a row with partition (2, 2, 3). The same process applies to the
pair .
Lemmas 4 and 5 have provided a good basis for the main state-
ment given here.
Theorem 4: Let be a prime and .
Then and
.
Proof: The first part of the statement follows from
Lemmas 4 and 5, and Proposition 1. It has been shown in [22]
that and have minimum distance 8. This
together with and the first part of the statement
implies
. As for , ac-
cording to and , it suffices
to determine a codeword of weight 8 in . The set
is a stopping set for .
The associated eight-columns matrix is given below. The
binary version of the columns of this matrix add to zero and
hence the set is the support of a codeword of weight 8.
Though the statements of the following corollary are known
[22], we rederive them using the results obtained above.
Corollary 2: Let be an odd prime. Then we have
Proof: For , according to Theorem 2 and
,wehave . On the other hand, the four-
columns matrix given in the proof of Theorem 2 gives four
linearly dependent columns, and hence . The-
orem 3 and the matrix given in its proof show .
Relation comes from Theorem 4 and
.
Authorized licensed use limited to: UNIVERSITY OF ALBERTA. Downloaded on July 15, 2009 at 18:28 from IEEE Xplore. Restrictions apply.
ESMAEILI AND AMOSHAHY: ARRAY CODE PARITY-CHECK MATRICES 3493
IV. SUMMARY
The stopping distance of two types of array codes referred
to as proper and improper codes with parameters and ,
denoted by and and represented by two
specific binary parity-check matrices and
, respectively, was examined for several values of
and . It was shown that
for and
where and de-
note the stopping distance of and the minimum dis-
tance of , respectively. The question of whether equations
hold for other values of and is left as an open problem.
Another related question is whether for any fixed ,if is a
sufficiently large prime, then .
ACKNOWLEDGMENT
The authors would like to thank the anonymous referees
whose comments greatly improved the manuscript.
REFERENCES
[1] R. M. Tanner, “A recursive approach to low complexity codes,” IEEE
Trans. Inf. Theory, vol. 27, pp. 533–547, Sep. 1981.
[2] C. Di, D. Proietti, I. E. Telatar, T. J. Richardson, and R. L. Urbanke,
“Finite-length analysis of low-density parity-check codes on the binary
erasure channel,” IEEE Trans. Inf. Theory, vol. 48, pp. 1570–1579, Jun.
2002.
[3] M. Esmaeili and V. Ravanmehr, “Stopping sets of binary parity-check
matrices with constant weight columns and stopping redundancy of
the associated codes,” Utilitas Mathematica, vol. 76, pp. 265–276, July
2008.
[4] M. Hivadi and M. Esmaeili, “On the stopping distance and stopping
redundancy of product codes,” IEICE Trans. Fund. Electron., Commun.
Comput. Sci., vol. E91-A, no. 8, pp. 2167–2173, Aug. 2008.
[5] N. Kashyap and A. Vardy, “Stopping sets in codes from designs,” in
Proc. IEEE Int. Symp. Inf. Theory, Yokohama, Japan, Jun. 29–Jul. 4
2003, pp. 122–.
[6] A. Orlitsky, K. Viswanathan, and J. Zhang, “Stopping set distribution
of LDPC code ensembles,” IEEE Trans. Inf. Theory, vol. 51, no. 3, pp.
929–953, Mar. 2005.
[7] M. Schwartz and A. Vardy, “On the stopping distance and the stop-
ping redundancy of codes,” IEEE Trans. Inf. Theory, vol. 52, no. 3, pp.
922–932, Mar. 2006.
[8] J. H. Weber and K. A. S. Abdel-Ghaffar, “Results on parity-check ma-
trices with optimal stopping and/or dead-end set enumerators,” IEEE
Trans. Inf. Theory, vol. 54, pp. 1368–1374, 2008.
[9] S.-T. Xia and F.-W. Fu, “On the stopping distance of finite geometry
LDPC codes,” IEEE Commun. Lett., vol. 10, no. 5, pp. 381–383, May
2006.
[10] R. G. Gallager, “Low density parity check codes,” IRE Trans. Inf.
Theory, vol. IT-8, pp. 21–28, Jan. 1962.
[11] R. G. Gallager, Low Density Parity Check Codes. Cambridge, MA:
MIT Press, 1963.
[12] D. J. C. MacKay and R. M. Neal, “Near Shannon limit performance
of low density parity check codes,” Electron. Lett., vol. 32, no. 18, pp.
1645–1646, 1996.
[13] T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, “Design of
capacity-approaching irregular low-density parity check codes,” IEEE
Trans. Inf. Theory, vol. 47, pp. 619–637, Feb. 2001.
[14] M. Blaum and R. Roth, “New array codes for multiple burst error cor-
rection,” IEEE Trans. Inf. Theory, vol. 39, no. 1, pp. 66–77, Jan. 1993.
[15] J. L. Fan, “Array codes as low-density parity-check codes,” in Proc.2nd
Int. Symp. Turbo Codes, Brest, France, Sep. 4–7, 2000, pp. 543–546.
[16] D. E. Hocevar, “Efficient encoding for a family of quasi-cyclic LDPC
codes,” in Proc. IEEE Global Telecommun. Conf., Dec. 1–5, 2003, vol.
7, pp. 3996–4000.
[17] Z. Li, L. Chen, L. Zeng, S. Lin, and W. H. Fong, “Efficient encoding
of QC-LDPC codes,” IEEE Trans. Commun., vol. 54, no. 1, pp. 71–81,
Jan. 2006.
[18] B. Vasic and O. Milenkovic, “Combinatorial constructions of low-den-
sity parity-check codes for iterative decoding,” IEEE Trans. Inf.
Theory, vol. 50, no. 6, pp. 1156–1176, June 2004.
[19] B. Vasic, K. Pedagani, and M. Ivkovic, “High-rate girth-eight low-den-
sity parity-check codes on rectangular integer lattices,” IEEE Trans.
Commun., vol. 52, no. 8, pp. 1248–1252, Aug. 2004.
[20] D. J. C. MacKay [Online]. Available: http://wol.ra.phy.cam.ac.uk/
mackay/codes/data.html, Encyclopedia of Sparse Graph Codes, [On-
line]. Available:
[21] O. Milenkovic, D. Leyba, and N. Kashyap, “Shortened array codes of
large girth,” IEEE Trans. Inf. Theory, vol. 52, no. 8, pp. 3707–3722,
Aug. 2006.
[22] K. Yang and T. Helleseth, “On the minimum distance of array codes as
LDPC codes,” IEEE Trans. Inf. Theory, vol. 49, no. 12, pp. 3268–3271,
Dec. 2003.
[23] K. Sugiyama and Y. Kaji, “On the minimum weight of simple full-
length array LDPC codes,” IEICE Trans. Fundamentals of Electron.,
Commun. Comput. Sci., vol. E91-A, no. 6, Jun. 2008.
[24] X. Ge and S. T. Xia, “LDPC codes based on Berlekamp-Justesen codes
with large stopping distances,” in Proc. 2006 IEEE Inf. Theory Work-
shop (ITW’06), pp. 214–218.
Morteza Esmaeili (M’03) received the M.S. degree in mathematics in 1988
from Teacher Training University of Tehran, Iran, and the Ph.D. degree in math-
ematics (coding theory) in 1996 from Carleton University, Ottawa, Canada.
The next two years, he was a Postdoctoral Fellow with the Department
of Electrical and Computer Engineering, University of Waterloo, Waterloo,
Canada. Since September 1998, he has been with the Department of Mathe-
matical Sciences, Isfahan University of Technology, Isfahan, Iran, where he
is now an Associate Professor. He joined the Department of Electrical and
Computer Engineering, University of Victoria, Victoria, BC, Canada, as an
Adjunct Professor in July 2009. His current research interest include coding
and information theory, cryptography, combinatorics, and its application to
communication theory.
Mohammad Javad Amoshahy received the B.S. and M.S. degrees in mathe-
matics in 2005 and 2008, respectively, both from Isfahan University of Tech-
nology, Isfahan, Iran.
His research interest includes applied mathematics, in particular, coding
theory.
Authorized licensed use limited to: UNIVERSITY OF ALBERTA. Downloaded on July 15, 2009 at 18:28 from IEEE Xplore. Restrictions apply.
