MacWilliams type identities for some new $m$-spotty weight enumerators

TL;DR

This paper develops MacWilliams type identities for new $m$-spotty weight enumerators of byte error-control codes over finite rings and fields, aiding in error detection and correction in high-density RAM systems.

Contribution

It introduces joint and split $m$-spotty weight enumerators and derives MacWilliams identities for these, expanding coding theory tools for byte error correction.

Findings

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

Extended weight enumerator concepts to codes over rings and fields.

Discussed applications in error detection and correction.

Abstract

Past few years have seen an extensive use of high-density 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. In this paper, we present joint $m$-spotty weight enumerator and split $m$-spotty weight enumerator for byte error-control codes over the ring of integers modulo $\ell$ ($\ell \geq 2$ is an integer) and over arbitrary finite fields. We also derive MacWilliams type identities for each of the aforementioned enumerators and discuss some of their applications.