A note on the Singleton bounds for codes over finite rings

TL;DR

This paper revisits the Singleton bounds for linear codes over finite commutative quasi-Frobenius rings, clarifying notation and establishing bounds for this class of rings.

Contribution

It introduces a notation for Singleton bounds over these rings and demonstrates the existence of a specific class of such rings with particular bounds.

Findings

Singleton bounds are extended to certain finite commutative quasi-Frobenius rings.

A class of these rings satisfying specific bounds is identified.

The bounds relate code parameters with ring properties.

Abstract

In this paper, we give a notation on the Singleton bounds for linear codes over a finite commutative quasi-Frobenius ring in the work of Shiromoto [5]. We show that there exists a class of finite commutative quasi-Frobenius rings. The Singleton bounds for linear codes over such rings satisfy \[ \frac{d(C)-1}{A}\leq n-\log_{|R|}|C|. \]