The stopping distance and stopping redundancy of a linear code are important concepts in the analysis of the performance and complexity of the code under iterative decoding on a binary erasure channel. In this paper, we studied the stopping distance and stopping redundancy of Finite Geometry LDPC (FG-LDPC) codes, and derived an upper bound of the stopping redundancy of FG-LDPC codes. It is shown from the bound that the stopping redundancy of the codes is less than the code length. Therefore, FG-LDPC codes give a good trade-off between the performance and complexity and hence are a very good choice for practical applications.The stopping distance and stopping redundancy of a linear code are important concepts in the analysis of the performance and complexity of the code under iterative decoding on a binary erasure channel. In this paper, we studied the stopping distance and stopping redundancy of Finite Geometry LDPC (FG-LDPC) codes, and derived an upper bound of the stopping redundancy of FG-LDPC codes. It is shown from the bound that the stopping redundancy of the codes is less than the code length. Therefore, FG-LDPC codes give a good trade-off between the performance and complexity and hence are a very good choice for practical applications.

IEICE Transactions on Fundamentals of Electronics Communications and Computer SciencesAug 1, 20084 citations

The Correcting Capacities of Linear Codes

M. V. Kozlov

Soviet physics. DokladyNov 1, 19693 citations

Hulls of linear codes revisited with applications

Satanan Thipworawimon, Somphong Jitman

arXiv:1907.12026Journal of Applied Mathematics and ComputingSep 18, 201910 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

A decoding algorithm for linear codes

Martin Bossert, F. Hergert

Lecture notes in computer scienceJan 1, 19862 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.