# New Two-Stage Automorphism Group Decoders for Cyclic Codes in the Erasure Channel

## TL;DR

This paper introduces two-stage automorphism group decoders (TS-AGDs) for cyclic codes in erasure channels, improving decoding performance, reducing complexity and delay, and achieving ML decoding for some codes.

## Contribution

The paper proposes novel two-stage AGDs with modified parity check matrices and preprocessing, enhancing erasure decoding efficiency and performance for various cyclic codes.

## Key findings

- TS-AGDs outperform conventional AGDs in complexity and delay
- TS-AGDs achieve perfect decoding for some cyclic codes
- Proposed methods are effective for MDS codes with expanded parity check matrices

## 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 has been analyzed for the perfect codes, BCH codes, and maximum distance separable (MDS) codes. Through numerical analysis, it is shown that the proposed decoding algorithm has good erasure decoding performance with lower decoding complexity and delay than the conventional AGD. For some cyclic codes, it is shown that the proposed TS-AGD achieves the perfect decoding in the erasure channel, that is, the same decoding performance as the maximum likelihood (ML) decoder. For MDS codes, TS-AGDs with the expanded parity check matrix and the submatrix inversion are also proposed and analyzed.