New Two-Stage Automorphism Group Decoders for Cyclic Codes

Chanki Kim,Jong-Seon No

doi:10.1109/access.2020.3022460

Abstract

Recently, error correcting codes in the erasure channel have drawn great attention for various applications such as distributed storage systems and wireless sensor networks, but many of their decoding algorithms are not practical because they have higher decoding complexity and longer delay. Thus, the automorphism group decoder (AGD) for cyclic codes in the erasure channel was introduced, which has good erasure decoding performance with low decoding complexity. In this paper, we propose new two-stage AGDs (TS-AGDs) for cyclic codes in the erasure channel by modifying the parity-check matrix and introducing the preprocessing stage to the AGD scheme. The proposed TS-AGD is analyzed for binary extended Golay and BCH codes. Also, TS-AGD can be used in the error channel using ordered statistics. Through numerical analysis, it is shown that the proposed decoding algorithm has good erasure decoding performance with lower decoding complexity than the conventional AGD. For some cyclic codes, it is shown that the proposed TS-AGD achieves the performance nearly identical to the maximum likelihood (ML) decoder in the erasure channel and the ordered statistics decoder (OSD) in the error channel.

Highlights

Research on error correcting codes in the erasure channel is one of the major subjects in information theory
2) automorphism group decoder (AGD) FOR ERASURE CHANNEL [8] AGD can be applied to cyclic codes, where AGD consists of the repeated iterative erasure decoder (IED) and cyclic shift operations for the received vectors
two-stage AGDs (TS-AGDs) IN THE ERROR CHANNEL In this subsection, we propose a new method of lowcomplexity TS-AGD in the error channel motivated by ordered statistics decoder (OSD)

Summary

INTRODUCTION

Research on error correcting codes in the erasure channel is one of the major subjects in information theory. One approach to overcome the inferior decoding performance of IED for the algebraic codes in the erasure channel was proposed, called the automorphism group decoder (AGD) for cyclic codes [8]. We propose a new decoding algorithm, referred to as a two-stage AGD (TS-AGD) which includes a construction algorithm of good parity-check matrix with polynomial-time complexity and has excellent decoding performance with low decoding complexity for cyclic codes. The proposed TS-AGD algorithm can be implemented by the modified parity-check matrix for the (n, k) cyclic code such that some of the (n − k)-tuple column vectors in the paritycheck matrix are standard vectors in the appropriate column indices and Hamming weight of the row vectors in the paritycheck matrix becomes as low as possible, which requires polynomial-time complexity.

PRELIMINARY

ERASURE DECODER

ERROR DECODER

SOME DEFINITIONS

CONCLUSION