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Computing the minimum distance of linear codes by the error impulse method

Claude Berrou, Sandrine Vaton, Michel Jézéquel, Catherine Douillard

Published Mar 22, 2004• 72 citations
Publisher

Abstract

A new method for computing the minimum distances of linear error-correcting codes is proposed and justified. Unlike classical techniques that rely on exhaustive or partial enumeration of codewords, this new method is based on the ability of the Soft-In decoder to overcome Error Impulse input patterns. It is shown that the maximum magnitude of the Error Impulse that can be corrected by the decoder is directly related to the minimum distance. This leads to a very fast algorithm to obtain minimum distances of any linear code whatever the block size and the code rate considered. In particular, the method can be advantageously worked out for turbo-like concatenated codes.

DOI
10.1109/glocom.2002.1188348

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A new method for computing the minimum distances of linear error-correcting codes is proposed and justified. Unlike classical techniques that rely on exhaustive or partial enumeration of codewords, this new method is based on the ability of the Soft-In decoder to overcome Error Impulse input patterns. It is shown that the maximum magnitude of the Error Impulse that can be corrected by the decoder is directly related to the minimum distance. This leads to a very fast algorithm to obtain minimum distances of any linear code whatever the block size and the code rate considered. In particular, the method can be advantageously worked out for turbo-like concatenated codes.
Abstract The evaluation of the minimum distance of low‐density parity‐check (LDPC) codes remains an open problem due to the rather large dimension of the parity check matrix H associated with any practical code. In this article, we propose an effective modification of the error impulse (EI) technique for computation of the minimum distance of the LDPC codes. The EI method is successfully applied to sub‐optimum decoding algorithms such as the iterative MAP decoding algorithm for turbo codes. We present novel modifications and extensions of this method to the sub‐optimum iterative sum‐product algorithm for LDPC codes. The performance of LDPC codes may be limited by pseudo‐codewords. There are, however, cases when the LDPC decoder behaves as a maximum‐likelihood (ML) decoder. This is specially so for randomly constructed LDPC codes operating at medium to high SNR values. In such cases, estimation of the minimum distance d m using the error‐impulse method can be useful to assess asymptotic performance. In short, apart from theoretical interest in achievable d m , the technique is useful in checking whether an LDPC code is poor by virtue of having a low d m . But, if a code has a high d m , simulations would still be needed to assess real performance. Copyright © 2006 AEIT.
This dissertation investigates the complexity of minimum-distance decoding (MDD) algorithms for general binary linear block codes, and also investigates the use of such codes in constructing error correcting/detecting codes that are effective against both random and unidirectional errors. A new MDD algorithm is proposed, and its complexity is analyzed and compared to that of other such algorithms. The question of whether all good codes satisfy certain properties required for efficient implementation of the algorithm is also considered. The performance of decoding algorithms, with the information bit error rate as the criterion instead of the block error rate, is also studied. New decoding rules which can yield lower information bit error rates than the MDD rule are proposed and evaluated. A new class of binary codes is developed which corrects random errors and detects all unidirectional errors. These codes are obtained by appending a suffix to each codeword of a linear block code. To evaluate the efficiency of the new codes, a thorough comparison with the best previously obtained codes is made.
The evaluation of the minimum distance of low-density parity-check (LDPC) codes remains an open problem due to the rather large dimension of the parity check matrix H associated with any practical code. In this article, we propose an effective modification of the error impulse (EI) technique for estimation of the minimum distance of the LDPCs. The EI method is successfully applied to suboptimum decoding algorithms such as the iterative MAP decoding algorithm for turbo codes. We present novel modifications and extensions of this method to the suboptimum iterative sum-product algorithm for LDPCs. Simulation results validate the functionality of the proposed technique. Simulations focus on a particular class of LDPC codes, but our approach is general and applies to any LDPC code.
For iterative soft-decision decoding algorithms for binary linear block codes, error performance and computational complexity are very important factors.In this thesis, suÆcient conditions of the optimality of decoded codeword and a suÆcient condition which is used for ruling-out some useless iterations of bounded distance decoding which reduces the computational complexity of iterative soft-decision decoding algorithms are shown.New iterative soft-decision decoding algorithms which uses the suÆcient conditions of the optimality of decoded codeword to achievegood error performance with small computational complexity are also shown.First, the suÆcient conditions are derived based on: (1) partial knowledge of the distance pro le of the code, and (2) previously generated candidate codewords.When the numberof previously generated candidate codewords is greater than 1, the conditions presented here are less stringent than those derived by Taipale and Pursley [4] and Kaneko et.al.[5].The condition based on three previously generated candidate codewords is the rst such s u Æ c i e n t condition that has ever been derived.It is shown by computer simulation that these suÆcient condions reduce the computational complexity of iterative decoding algorithms without degradation of error performance.Second, a suÆcient condition to rule-out some useless test error patterns is derived.In an iterative decoding algorithm, such as Chase Type-II decoding algorithm[2] and its improved versions, candidate codewords for a received vector are generated
A novel analytical approach to performance evaluation of soft-decoding algorithms for binary linear block codes based on probabilistic iterative error correction is presented. A convergence condition establishing the critical noise rate below which the expected bit-error probability tends to zero is theoretically derived. It explains the capability of iterative probabilistic decoding of binary linear block codes with sparse parity-check matrices to correct, with probability close to one, error patterns with the number of errors (far) beyond half the code minimum distance. Systematic experiments conducted on truncated simplex codes seem to agree well with the convergence condition. The method may also be interesting for the theoretical analysis of the so-called turbo codes.