It is now well known that the performance of a linear code Copf under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for Copf. Several recent papers refer to this parameter as the stopping distance s of Copf. This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for Copf depends on the corresponding choice of a parity-check matrix. It is easy to see that s les d, where d is the minimum Hamming distance of Copf, and we show that it is always possible to choose a parity-check matrix for Copf (with sufficiently many dependent rows) such that s=d. We thus introduce a new parameter, the stopping redundancy of Copf, defined as the minimum number of rows in a parity- check matrix H for Copf such that the corresponding stopping distance s(H) attains its largest possible value, namely, s(H)=d. We then derive general bounds on the stopping redundancy of linear codes. We also examine several simple ways of constructing codes from other codes, and study the effect of these constructions on the stopping redundancy. Specifically, for the family of binary Reed-Muller codes (of all orders), we prove that their stopping redundancy is at most a constant times their conventional redundancy. We show that the stopping redundancies of the binary and ternary extended Golay codes are at most 34 and 22, respectively. Finally, we provide upper and lower bounds on the stopping redundancy of MDS codesIt is now well known that the performance of a linear code Copf under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for Copf. Several recent papers refer to this parameter as the stopping distance s of Copf. This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for Copf depends on the corresponding choice of a parity-check matrix. It is easy to see that s les d, where d is the minimum Hamming distance of Copf, and we show that it is always possible to choose a parity-check matrix for Copf (with sufficiently many dependent rows) such that s=d. We thus introduce a new parameter, the stopping redundancy of Copf, defined as the minimum number of rows in a parity- check matrix H for Copf such that the corresponding stopping distance s(H) attains its largest possible value, namely, s(H)=d. We then derive general bounds on the stopping redundancy of linear codes. We also examine several simple ways of constructing codes from other codes, and study the effect of these constructions on the stopping redundancy. Specifically, for the family of binary Reed-Muller codes (of all orders), we prove that their stopping redundancy is at most a constant times their conventional redundancy. We show that the stopping redundancies of the binary and ternary extended Golay codes are at most 34 and 22, respectively. Finally, we provide upper and lower bounds on the stopping redundancy of MDS codes

IEEE Transactions on Information TheoryMar 1, 2006178 citations

On the Stopping Distance and the Stopping Redundancy of Codes

Schwartz, Moshe, Alexander Vardy

It is now well known that the performance of a linear code $C$ under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for $C$ . Several recent papers refer to this parameter as the \emph{stopping distance} $s$ of $C$ . This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for $C$ depends on the corresponding choice of a parity-check matrix. It is easy to see that $s \le d$ , where $d$ is the minimum Hamming distance of $C$ , and we show that it is always possible to choose a parity-check matrix for $C$ (with sufficiently many dependent rows) such that $s = d$ . We thus introduce a new parameter, termed the \emph{stopping redundancy} of $C$ , defined as the minimum number of rows in a parity-check matrix $H$ for $C$ such that the corresponding stopping distance $s(H)$ attains its largest possible value, namely $s(H) = d$ . We then derive general bounds on the stopping redundancy of linear codes. We also examine several simple ways of constructing codes from other codes, and study the effect of these constructions on the stopping redundancy. Specifically, for the family of binary Reed-Muller codes (of all orders), we prove that their stopping redundancy is at most a constant times their conventional redundancy. We show that the stopping redundancies of the binary and ternary extended Golay codes are at most 35 and 22, respectively. Finally, we provide upper and lower bounds on the stopping redundancy of MDS codes.It is now well known that the performance of a linear code $C$ under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for $C$. Several recent papers refer to this parameter as the \emph{stopping distance} $s$ of $C$. This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for $C$ depends on the corresponding choice of a parity-check matrix. It is easy to see that $s \le d$, where $d$ is the minimum Hamming distance of $C$, and we show that it is always possible to choose a parity-check matrix for $C$ (with sufficiently many dependent rows) such that $s = d$. We thus introduce a new parameter, termed the \emph{stopping redundancy} of $C$, defined as the minimum number of rows in a parity-check matrix $H$ for $C$ such that the corresponding stopping distance $s(H)$ attains its largest possible value, namely $s(H) = d$. We then derive general bounds on the stopping redundancy of linear codes. We also examine several simple ways of constructing codes from other codes, and study the effect of these constructions on the stopping redundancy. Specifically, for the family of binary Reed-Muller codes (of all orders), we prove that their stopping redundancy is at most a constant times their conventional redundancy. We show that the stopping redundancies of the binary and ternary extended Golay codes are at most 35 and 22, respectively. Finally, we provide upper and lower bounds on the stopping redundancy of MDS codes.

arXiv:cs/0503058Mar 23, 20050 citations

On the Stopping Redundancy of Reed–Muller Codes

Tuvi Etzion

The stopping redundancy of the code is an important parameter which arises from analyzing the performance of a linear code under iterative decoding on a binary erasure channel. In this paper, we will consider the stopping redundancy of Reed-Muller codes and related codes. Let R(lscr,m) be the Reed-Muller code of length 2mand order lscr. Schwartz and Vardy gave a recursive construction of parity-check matrices for the Reed-Muller codes, and asked whether the number of rows in those parity-check matrices is the stopping redundancy of the codes. We prove that the stopping redundancy of R(m-2,m), which is also the extended Hamming code of length 2m, is 2m-1 and thus show that the recursive bound is tight in this case. We prove that the stopping redundancy of the simplex code equals its redundancy. Several constructions of codes for which the stopping redundancy equals the redundancy are discussed. We prove an upper bound on the stopping redundancy of R(1,m). This bound is better than the known recursive bound and thus gives a negative answer to the question of Schwartz and VardyThe stopping redundancy of the code is an important parameter which arises from analyzing the performance of a linear code under iterative decoding on a binary erasure channel. In this paper, we will consider the stopping redundancy of Reed-Muller codes and related codes. Let R(lscr,m) be the Reed-Muller code of length 2mand order lscr. Schwartz and Vardy gave a recursive construction of parity-check matrices for the Reed-Muller codes, and asked whether the number of rows in those parity-check matrices is the stopping redundancy of the codes. We prove that the stopping redundancy of R(m-2,m), which is also the extended Hamming code of length 2m, is 2m-1 and thus show that the recursive bound is tight in this case. We prove that the stopping redundancy of the simplex code equals its redundancy. Several constructions of codes for which the stopping redundancy equals the redundancy are discussed. We prove an upper bound on the stopping redundancy of R(1,m). This bound is better than the known recursive bound and thus gives a negative answer to the question of Schwartz and Vardy

IEEE Transactions on Information TheoryOct 30, 200642 citations

Code representation and performance of graph -based decoding

Junsheng Han

Key to the success of modern error correcting codes is the effectiveness of message-passing iterative decoding (MPID). Unlike maximum-likelihood (ML) decoding, the performance of MPID depends not only on the code, but on how the code is represented. In particular, the performance of MPID is potentially improved by using a redundant representation. We focus on Tanner graphs and study combinatorial structures therein that help explain the performance disparity among different representations of the same code. Emphasis is placed on the complexity- performance tradeoff, as more and more check nodes are allowed in the graph. Our discussion applies to MPID as well as linear programming decoding (LPD), which we collectively refer to as graph-based decoding. On an erasure channel, it is well-known that the performance of MPID or LPD is determined by stopping sets. Following Schwartz and Vardy, we define the stopping redundancy as the smallest number of check nodes in a Tanner graph such that smallest size of a non-empty stopping set is equal to the minimum Hamming distance of the code. Roughly speaking, stopping redundancy measures the complexity requirement (in number of check nodes) for MPID of a redundant graph representation to achieve performance comparable to ML decoding (up to a constant factor for small channel erasure probability). General upper bounds on stopping redundancy are obtained. One of our main contribution is a new upper bound based on probabilistic analysis, which is shown to be by far the strongest. From this bound, it can be shown, for example, that for a fixed minimum distance, the stopping redundancy grows just linearly with the redundancy (codimension). Specific results on the stopping redundancy of Golay and Reed-Muller codes are also obtained. We show that the stopping redundancy of maximum distance separable (MDS) codes is bounded in between a Turan number and a single-exclusion (SE) number --- a purely combinatorial quantity that we introduce. By studying upper bounds on the SE number, new results on the stopping redundancy of MDS codes are obtained. Schwartz and Vardy conjecture that the stopping redundancy of an MDS code should only depend on its length and minimum distance. Our results provide partial confirmation, both exact and asymptotic, to this conjecture. Stopping redundancy can be large for some codes. We observe that significantly fewer checks are needed if a small number of small stopping sets are allowed. These small stopping sets can then be dealt with by ̀g̀uessing'' during the iterative decoding process. Correspondingly, the guess-g stopping redundancy is defined and it is shown that the savings in number of required check nodes are potentially significant. Another theoretically interesting question is when MPID of a Tanner graph achieves the same word error rate an ML decoder. This prompts us to define and study ML redundancy. Applicability and possible extensions of the current work to a non-erasure channel are discussed. A framework based on pseudo-codewords is considered and shown to be relevant. However, it is also observed that the polytope characterization of pseudo-codewords is not complete enough to be an accurate indicator of MPID performance. Finally, in a separate piece of work, the probability of undetected error (PUE) for over-extended Reed-Solomon codes is studied through the weight distribution bounds of the code. The resulting PUE expressions are shown to be tight in a well-defined senseKey to the success of modern error correcting codes is the effectiveness of message-passing iterative decoding (MPID). Unlike maximum-likelihood (ML) decoding, the performance of MPID depends not only on the code, but on how the code is represented. In particular, the performance of MPID is potentially improved by using a redundant representation. We focus on Tanner graphs and study combinatorial structures therein that help explain the performance disparity among different representations of the same code. Emphasis is placed on the complexity- performance tradeoff, as more and more check nodes are allowed in the graph. Our discussion applies to MPID as well as linear programming decoding (LPD), which we collectively refer to as graph-based decoding. On an erasure channel, it is well-known that the performance of MPID or LPD is determined by stopping sets. Following Schwartz and Vardy, we define the stopping redundancy as the smallest number of check nodes in a Tanner graph such that smallest size of a non-empty stopping set is equal to the minimum Hamming distance of the code. Roughly speaking, stopping redundancy measures the complexity requirement (in number of check nodes) for MPID of a redundant graph representation to achieve performance comparable to ML decoding (up to a constant factor for small channel erasure probability). General upper bounds on stopping redundancy are obtained. One of our main contribution is a new upper bound based on probabilistic analysis, which is shown to be by far the strongest. From this bound, it can be shown, for example, that for a fixed minimum distance, the stopping redundancy grows just linearly with the redundancy (codimension). Specific results on the stopping redundancy of Golay and Reed-Muller codes are also obtained. We show that the stopping redundancy of maximum distance separable (MDS) codes is bounded in between a Turan number and a single-exclusion (SE) number --- a purely combinatorial quantity that we introduce. By studying upper bounds on the SE number, new results on the stopping redundancy of MDS codes are obtained. Schwartz and Vardy conjecture that the stopping redundancy of an MDS code should only depend on its length and minimum distance. Our results provide partial confirmation, both exact and asymptotic, to this conjecture. Stopping redundancy can be large for some codes. We observe that significantly fewer checks are needed if a small number of small stopping sets are allowed. These small stopping sets can then be dealt with by ̀g̀uessing'' during the iterative decoding process. Correspondingly, the guess-g stopping redundancy is defined and it is shown that the savings in number of required check nodes are potentially significant. Another theoretically interesting question is when MPID of a Tanner graph achieves the same word error rate an ML decoder. This prompts us to define and study ML redundancy. Applicability and possible extensions of the current work to a non-erasure channel are discussed. A framework based on pseudo-codewords is considered and shown to be relevant. However, it is also observed that the polytope characterization of pseudo-codewords is not complete enough to be an accurate indicator of MPID performance. Finally, in a separate piece of work, the probability of undetected error (PUE) for over-extended Reed-Solomon codes is studied through the weight distribution bounds of the code. The resulting PUE expressions are shown to be tight in a well-defined sense

Jan 1, 20080 citations

On the probabilistic computation of the stopping redundancy of LDPC codes

Masanori Hirotomo, Yoshiho Konishi, Masakatu Morii

On the binary erasure channel (BEC), the performance of low-density parity-check (LDPC) codes decoded by iterative algorithms is estimated by the small stopping sets in the codes. The size of smallest stopping set is called the stopping distance. Since it depends on the structure of parity-check matrix for the code, we can construct the parity-check matrix by adding dependent rows so that the stopping distance is equal to the minimum distance. The stopping redundancy is defined as the smallest number of rows in such a parity-check matrix. In this paper, we propose a probabilistic algorithm for computing the stopping redundancy of LDPC codes. Additionally, we show numerical results of computing the stopping redundancy of (155,64), (504,252) and (1008,504) LDPC codes.On the binary erasure channel (BEC), the performance of low-density parity-check (LDPC) codes decoded by iterative algorithms is estimated by the small stopping sets in the codes. The size of smallest stopping set is called the stopping distance. Since it depends on the structure of parity-check matrix for the code, we can construct the parity-check matrix by adding dependent rows so that the stopping distance is equal to the minimum distance. The stopping redundancy is defined as the smallest number of rows in such a parity-check matrix. In this paper, we propose a probabilistic algorithm for computing the stopping redundancy of LDPC codes. Additionally, we show numerical results of computing the stopping redundancy of (155,64), (504,252) and (1008,504) LDPC codes.

IEEE International Symposium on Information Theory (ISIT)Jun 1, 20091 citation

The Trapping Redundancy of Linear Block Codes

Stefan Laendner, Thorsten Hehn, Olgica Milenković +1 more

We generalize the notion of the stopping redundancy in order to study the smallest size of a trapping set in Tanner graphs of linear block codes. In this context, we introduce the notion of the trapping redundancy of a code, which quantifies the relationship between the number of redundant rows in any parity-check matrix of a given code and the size of its smallest trapping set. Trapping sets with certain parameter sizes are known to cause error-floors in the performance curves of iterative belief propagation (BP) decoders, and it is therefore important to identify decoding matrices that avoid such sets. Bounds on the trapping redundancy are obtained using probabilistic and constructive methods, and the analysis covers both general and elementary trapping sets. Numerical values for these bounds are computed for the [2640, 1320] Margulis code and the class of projective geometry codes, and compared with some new code-specific trapping set size estimates.We generalize the notion of the stopping redundancy in order to study the smallest size of a trapping set in Tanner graphs of linear block codes. In this context, we introduce the notion of the trapping redundancy of a code, which quantifies the relationship between the number of redundant rows in any parity-check matrix of a given code and the size of its smallest trapping set. Trapping sets with certain parameter sizes are known to cause error-floors in the performance curves of iterative belief propagation (BP) decoders, and it is therefore important to identify decoding matrices that avoid such sets. Bounds on the trapping redundancy are obtained using probabilistic and constructive methods, and the analysis covers both general and elementary trapping sets. Numerical values for these bounds are computed for the [2640, 1320] Margulis code and the class of projective geometry codes, and compared with some new code-specific trapping set size estimates.

arXiv:cs/0701006IEEE Transactions on Information TheoryJan 1, 200932 citations

It is now well known that the performance of a linear code

C

under iterative decoding on a binary erasure channel (and other channels) is determined by the size of the smallest stopping set in the Tanner graph for

C

. Several recent papers refer to this parameter as the \emph{stopping distance}

s

C

. This is somewhat of a misnomer since the size of the smallest stopping set in the Tanner graph for

C

depends on the corresponding choice of a parity-check matrix. It is easy to see that

s \le d

, where

d

is the minimum Hamming distance of

C

, and we show that it is always possible to choose a parity-check matrix for

C

(with sufficiently many dependent rows) such that

s = d

. We thus introduce a new parameter, termed the \emph{stopping redundancy} of

C

, defined as the minimum number of rows in a parity-check matrix

H

for

C

such that the corresponding stopping distance

s(H)

attains its largest possible value, namely

s(H) = d

. We then derive general bounds on the stopping redundancy of linear codes. We also examine several simple ways of constructing codes from other codes, and study the effect of these constructions on the stopping redundancy. Specifically, for the family of binary Reed-Muller codes (of all orders), we prove that their stopping redundancy is at most a constant times their conventional redundancy. We show that the stopping redundancies of the binary and ternary extended Golay codes are at most 35 and 22, respectively. Finally, we provide upper and lower bounds on the stopping redundancy of MDS codes.