MacWilliams type identities for some new $m$-spotty weight enumerators over finite commutative Frobenius rings

TL;DR

This paper extends MacWilliams identities to various $m$-spotty weight enumerators for byte error-control codes over finite commutative Frobenius rings, enhancing error detection and correction analysis.

Contribution

It introduces new $m$-spotty weight enumerators and derives MacWilliams type identities for codes over Frobenius rings, broadening theoretical understanding.

Findings

Derived MacWilliams identities for $m$-spotty Hamming weight enumerators

Established identities for joint and split $m$-spotty Hamming weight enumerators

Extended $m$-spotty Lee weight enumerator to infinite rings

Abstract

Past few years have seen an extensive use of RAM chips with wide I/O data (e.g. 16, 32, 64 bits) in computer memory systems. These chips are highly vulnerable to a special type of byte error, called an $m$-spotty byte error, which can be effectively detected or corrected using byte error-control codes. The MacWilliams identity provides the relationship between the weight distribution of a code and that of its dual. This paper introduces $m$-spotty Hamming weight enumerator, joint $m$-spotty Hamming weight enumerator and split $m$-spotty Hamming weight enumerator for byte error-control codes over finite commutative Frobenius rings as well as $m$-spotty Lee weight enumerator over an infinite family of rings. In addition, MacWilliams type identities are also derived for these enumerators.