Duality Preserving Gray Maps for Codes over Rings

TL;DR

This paper introduces duality preserving Gray maps for codes over rings using new basis concepts, enabling effective translation of codes from rings to subrings while maintaining duality properties.

Contribution

It defines pseudo-self-dual and symmetric bases for rings, generalizing field concepts, and constructs duality preserving maps from codes over rings to subrings.

Findings

Examples demonstrate advantages of the mappings.

Mappings preserve duality properties.

Abundance of such bases in various rings.

Abstract

Given a finite ring $A$ which is a free left module over a subring $R$ of $A$, two types of $R$-bases, pseudo-self-dual bases (similar to trace orthogonal bases) and symmetric bases, are defined which in turn are used to define duality preserving maps from codes over $A$ to codes over $R$. Both types of bases are generalizations of similar concepts for fields. Many illustrative examples are given to shed light on the advantages to such mappings as well as their abundance.