Sum-Rank BCH Codes and Cyclic-Skew-Cyclic Codes

TL;DR

This paper introduces sum-rank BCH and cyclic-skew-cyclic codes, characterizes their algebraic structure, establishes bounds on their minimum distance, and demonstrates their superior performance over existing codes with an efficient decoding method.

Contribution

It defines and analyzes sum-rank BCH codes and cyclic-skew-cyclic codes, providing new bounds, structural insights, and practical decoding algorithms for these codes.

Findings

Sum-rank BCH codes outperform previous codes in the binary 2x2 matrix setting.

A lower bound on the minimum sum-rank distance is established.

An efficient decoder up to half the minimum distance is developed.

Abstract

In this work, cyclic-skew-cyclic codes and sum-rank BCH codes are introduced. Cyclic-skew-cyclic codes are characterized as left ideals of a suitable non-commutative finite ring, constructed using skew polynomials on top of polynomials (or vice versa). Single generators of such left ideals are found, and they are used to construct generator matrices of the corresponding codes. The notion of defining set is introduced, using pairs of roots of skew polynomials on top of poynomials. A lower bound (called sum-rank BCH bound) on the minimum sum-rank distance is given for cyclic-skew-cyclic codes whose defining set contains certain consecutive pairs. Sum-rank BCH codes, with prescribed minimum sum-rank distance, are then defined as the largest cyclic-skew-cyclic codes whose defining set contains such consecutive pairs. The defining set of a sum-rank BCH code is described, and a lower bound on its dimension is obtained. Thanks to it, tables are provided showing that sum-rank BCH codes beat previously known codes for the sum-rank metric for binary $ 2 \times 2 $ matrices (i.e., codes whose codewords are lists of $ 2 \times 2 $ binary matrices, for a wide range of list lengths that correspond to the code length). Finally, a decoder for sum-rank BCH codes up to half their prescribed distance is obtained.