A MacWilliams type identity for m-spotty generalized Lee weight enumerators over $\mathbb{Z}_q$ q

TL;DR

This paper introduces a new MacWilliams type identity linking the weight enumerators of codes and their duals based on m-spotty generalized Lee weights over d_q, extending previous identities over Z2 and Z3.

Contribution

It generalizes the MacWilliams identity for m-spotty Lee weight enumerators over d_q, unifying and extending prior results over Z2 and Z3.

Findings

Established a MacWilliams type identity for m-spotty generalized Lee weights.

Unified previous identities over Z2 and Z3 as special cases.

Provided theoretical framework for analyzing error control codes in high-density memory systems.

Abstract

Burst errors are very common in practice. There have been many designs in order to control and correct such errors. Recently, a new class of byte error control codes called spotty byte error control codes has been specifically designed to fit the large capacity memory systems that use high-density random access memory (RAM) chips with input/output data of 8, 16, and 32 bits. The MacWilliams identity describes how the weight enumerator of a linear code and the weight enumerator of its dual code are related. Also, Lee metric which has attracted many researchers due to its applications. In this paper, we combine these two interesting topics and introduce the m-spotty generalized Lee weights and the m-spotty generalized Lee weight enumerators of a code over Z q and prove a MacWilliams type identity. This generalization includes both the case of the identity given in the paper [I. Siap, MacWilliams identity for m-spotty Lee weight enumerators, Appl. Math. Lett. 23 (1) (2010) 13-16] and the identity given in the paper [M. \"Ozen, V. \c{S}iap, The MacWilliams identity for m-spotty weight enumerators of linear codes over finite fields, Comput. Math. Appl. 61 (4) (2011) 1000-1004] over Z2 and Z3 as special cases.