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Advances in Mathematics of Communications

On the performance of binary extremal self-dual codes

Stefka Bouyuklieva, Anton Malevich, Wolfgang M. Willems

Published Jan 1, 2011• 4 citations
Publisher

Abstract

The decoding error probability of a code $C$ measures thequality of performance when $C$ is used for error correction in data transmission.In this note we compare different types of codes with regard to the decoding errorprobability.

DOI
10.3934/amc.2011.5.267

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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.
Assume that a binary linear code is transmitted over a channel with the symbol error probability p and that bounded distance decoding is used. In this case it is known [3] that for sufficiently small p the performance of the code is only a question of the weight distribution (weight enumerators of more efficient codes are lexicographically smaller). Based on this result we study the efficiency of extremal self-dual codes. Extremal doubly-even codes have unique weight enumerators for each length, and at first glance it may seem that these codes should perform worse than other codes with the same minimum distance. However we show that this is not always the case when comparing to singly-even codes. We also show that a doubly-even code may perform better than a non self-dual code. We find the most efficient codes among extremal singly-even codes of length 24m + 8 and 24m + 16. We also consider some families of extremal singly-even codes with uniquely determined weight enumerators (s-extremal codes, codes with minimal shadow). It appears that they always perform better than extremal doublyeven codes of the same length. The results are joint work with Wolfgang Willems (Magdeburg).
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.
The development of computationally-efficient algorithms is crucial to support the extremely-low latency and ultra-high reliability requirements of the next-generation radio communication systems. Error-correction codes (ECC) are used in communication systems to maintain the reliability of data transmissions. In this paper, several commonly used artificial intelligence (AI) methods are used to design algorithms for bitwise decoding of (ECC). The AI-based decoding algorithms are analysed and compared using their error-correction performance, training time and computational intensity. The efficacy is assessed using the benchmark codes: extended binary Golay and the Hamming code for varying Signal-to-Noise Ratios (SNR) over the Additive White Gaussian Noise (AWGN) channel and the decoding performance is evaluated using the block error rate (BLER). It is envisaged that the results from this comparative study would help to identify AI models most suitable for developing computationally efficient and practically-implementable algorithms for decoding longer ECC.