Hartmann--Tzeng bound and Skew Cyclic Codes of Designed Hamming Distance

TL;DR

This paper extends the Hartmann-Tzeng bound to skew cyclic codes, providing methods for constructing codes with specific Hamming distances and discussing decoding algorithms for skew BCH codes.

Contribution

It introduces a generalized Hartmann-Tzeng bound for skew cyclic codes and offers practical construction and decoding methods for these codes.

Findings

Extended Hartmann-Tzeng bound applies to skew cyclic codes

Practical construction methods for codes with designed Hamming distance

Decoding algorithms for skew BCH codes are discussed

Abstract

The use of skew polynomial rings allows to endow linear codes with cyclic structures which are not cyclic in the classical (commutative) sense. Whenever these skew cyclic structures are carefully chosen, some control over the Hamming distance is gained, and it is possible to design efficient decoding algorithms. In this paper, we give a version of the Hartmann-Tzeng bound that works for a wide class of skew cyclic codes. We also provide a practical method for constructing them with designed distance. For skew BCH codes, which are covered by our constructions, we discuss decoding algorithms. Detailed examples illustrate both the theory as the constructive methods it supports.