Majority Logic Decoding of Affine Grassmann Codes Over Nonbinary Fields

TL;DR

This paper introduces a majority logic decoding method for affine Grassmann codes over nonbinary fields, leveraging automorphism groups and orthogonal parity checks to correct many errors efficiently.

Contribution

It develops a novel majority logic decoding approach for affine Grassmann codes, extending previous methods used for Grassmann codes.

Findings

Decoding capability matches that of existing Grassmann code decoders.

Constructs large orthogonal parity check sets using automorphisms.

Achieves efficient error correction for affine Grassmann codes.

Abstract

In this article, we consider the decoding problem of affine Grassmann codes over nonbinary fields. We use matrices of different ranks to construct a large set consisting of parity checks of affine Grassmann codes, which are orthogonal with respect to a fixed coordinate. By leveraging the automorphism groups of these codes, we generate a set of orthogonal parity checks for each coordinate. Using these parity checks, we perform majority logic decoding to correct a large number of errors in affine Grassmann codes. The order of error correction capability and the complexity of this decoder for affine Grassmann codes are the same as those of the majority logic decoder for Grassmann codes proposed in [BS21].