‎s‎. ‎kim et al‎. ‎have been analyzed the girth of some algebraically‎ ‎structured quasi-cyclic (qc) low-density parity-check (ldpc)‎ ‎codes‎, ‎i.e‎. ‎tanner $(3,5)$ of length $5p$‎, ‎where $p$ is a prime of‎ ‎the form $15m+1$‎. ‎in this paper‎, ‎by extension this method to‎ ‎tanner $(3,7)$ codes of length $7p$‎, ‎where $p$ is a prime of the‎ ‎form $21m‎+ ‎1$‎, ‎the girth values of tanner $(3,7...

The maximum-likelihood decoding of linear block codes by the Wagner rule decoding is discussed. In this approach, the Wagner rule decoding which has been primarily applied to single parity check codes is employed on acyclic Tanner graphs. Accordingly, a coset decoding equipped with the Wagner rule decoding is applied to the decoding of a code C having a Tanner graph with cycles. A subcode C1 of...

If a linear block code C of length n has a Tanner graph without cycles, then maximum-likelihood soft-decision decoding of C can be achieved in time O(n 2). However, we show that cycle-free Tanner graphs cannot support good codes. Specically, let C be an (n;k;d) linear code of rate R = k=n that can be represented by a Tanner graph without cycles. We prove that if R 0:5 then d 2, while if R < 0:5...

If a linear block code C of length n has a Tanner graph without cycles, then maximum-likelihood soft-decision decoding of C can be achieved in time O(n). However, we show that cycle-free Tanner graphs cannot support good codes. Specifically, let C be an (n, k, d) linear code of rate R = k/n that can be represented by a Tanner graph without cycles. We prove that if R ≥ 0.5 then d ≤ 2, while if R...

Two decision problems related to the computation of stopping sets in Tanner graphs are shown to be NP-complete. NP-hardness of the problem of computing the stopping distance of a Tanner graph follows as a consequence.

Low-density parity check codes (LDPC codes) were introduced by Gallager [2] and have received intensive study in the last few years, as advances in technology have made their implementation far more practical than when they were originally proposed. Basically, an LDPC code is a binary linear block code defined by a sparse m × n parity-check matrix H . Equivalently such a code can be thought of ...

This paper presents a new approach to decode turbo codes using a nonbinary belief propagation decoder. The proposed approach can be decomposed into two main steps. First, a nonbinary Tanner graph representation of the turbo code is derived by clustering the binary parity-check matrix of the turbo code. Then, a group belief propagation decoder runs several iterations on the obtained nonbinary Ta...

In this paper we study a class of graph codes with BCH component codes as affine variety codes. We are able to find some optimal binary and ternary codes as Tanner codes with BCH component codes. We choose a special subgraph of the point-line incidence plane of P(2, q) as the Tanner graph, and we are able to describe the codes using Gröbner basis.

Low Density Parity Check (LDPC) Codes are the class of linear block codes which provide near capacity performance on large collection of data transmission channels while simultaneously feasible for implementable decoders. LDPC codes were first proposed by Gallger in 1967 [1] and were rarely used until their rediscovery by Mackay, Luby and others [9][10][11]. Much research is devoted to characte...