The zero short Covering Problem for finite rings

TL;DR

This paper determines the size of minimal zero short covers for all finite rings, removing previous restrictions, and applies the results to a family of rings relevant in coding theory.

Contribution

It extends the zero short covering problem solution to all finite rings by removing earlier restrictions using the structure theorem for Artinian rings.

Findings

Solved the zero short covering problem for all finite rings.

Applied results to the family of rings R_k in coding theory.

Removed the restriction D(A)^2=0 from previous work.

Abstract

In this work, we find the cardinality of minimal zero short covers of An for any finite local ring A, removing the restriction of D(A)^2 = 0 from the previous works in the literature. Using the structure theorem for Artinian rings, we conclude that we have solved the zero short covering problem for all finite rings. We demonstrate our results on R_k, an infinite family of finite commutative rings extensively studied in coding theory, which satisfy D(A)^2 \neq 0 for all k \geq 2.