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arXiv:2106.11146

On decoding of a specific type of self-dual codes

Radinka Yorgova

Published Jun 21, 2021• 1 citation
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Abstract

This work introduces a decoding strategy for binary self-dual codes possessing an automorphism of a specific type. The proposed algorithm is a hard decision iterative decoding scheme. The enclosed experiments show that the new decoding concept performs error correction beyond the upper bound for the code correction capability. Moreover, we prove that the requirements for the new algorithm hold for any binary self-dual code having an automorphism of the specific type, which makes decoding of this large group of codes possible.

DOI
10.48550/arxiv.2106.11146

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This work introduces a decoding strategy for binary self-dual codes possessing an automorphism of a specific type. The proposed algorithm is a hard decision iterative decoding scheme. The enclosed experiments show that the new decoding concept performs error correction beyond the upper bound for the code correction capability. Moreover, we prove that the requirements for the new algorithm hold for any binary self-dual code having an automorphism of the specific type, which makes decoding of this large group of codes possible.
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Binary cyclic codes achieve good error correction performance and allow the implementation of very simple encoder and decoder circuits. Among them, BCH codes represent a very important class of t-error correcting codes, with known structural properties and error correction capability. Decoding of binary cyclic codes is often accomplished through hard-decision decoders, although it is recognized that soft-decision decoding algorithms can produce significant coding gain with respect to hard-decision techniques. Several approaches have been proposed to implement iterative soft-decision decoding of binary cyclic codes. We study the technique based on "Extended Parity-Check Matrices", and show that such method is not suitable for high rates or long codes. We propose a new method, based on "Spread Parity-Check Matrices", that has the same complexity, but can achieve better correction performance in many practical cases.
The decoding error probability of a code CCC measures thequality of performance when CCC is used for error correction in data transmission.In this note we compare different types of codes with regard to the decoding errorprobability.
We present novel decoding schemes for hard and soft decision decoding of block codes using the minimal weight codewords of the dual code. The decoding schemes will be described for cyclic codes where polynomials can be used, however, the modification for non-cyclic codes is possible and straight forward. The hard decision decoding calculates syndrome polynomials which are the product of the received polynomial with dual codewords. Proper cyclic shifts of these syndrome polynomials are obtained and the non-zero positions are counted componentwise for these shifts. The values of this counting are a reliability measure and can be used for locating the error and also the non-error positions. This reliability measure is the basis for various variants of hard decision decoding algorithms. Decoding schemes with iterative error reduction are possible as well as information set decoding using the inherent reliability information of the measure even if there is no reliability information from the channel. Further, we will show how reliability information from the channel can be included in order to obtain soft decision decoding schemes. We derive the relation between bit flipping, believe propagation, and majority logic decoding to the novel schemes. As examples to illustrate the functioning we use BCH and Reed-Muller codes as examples for binary codes, and RS codes for non-binary codes. Besides the Plotkin construction we recall a known result that Reed-Muller codes punctured by one position are cyclic and thus, are equivalent to special cases of BCH codes. Simulation results for hard and soft decision decoding will be given for several examples and compared with results from literature. Finally, we analyze the soft decision decoding of the Plotkin construction and derive that one of the two codes uses a 333 dB better channel (also known as channel polarization).