The best possible upper bound on the probability of undetected error for linear codes of full support

TL;DR

This paper establishes the optimal upper bound on the probability of undetected error for linear codes with full support and characterizes codes that meet this bound.

Contribution

It extends the known bounds to full support linear codes and characterizes the codes achieving this bound.

Findings

Derived the best possible upper bound for full support codes.

Characterized codes that meet the bound.

Enhanced understanding of error detection capabilities in linear codes.

Abstract

There is a known best possible upper bound on the probability of undetected error for linear codes. The $[n,k;q]$ codes with probability of undetected error meeting the bound have support of size $k$ only. In this note, linear codes of full support ($=n$) are studied. A best possible upper bound on the probability of undetected error for such codes is given, and the codes with probability of undetected error meeting this bound are characterized.