Shortened Hamming Codes Maximizing Double Error Detection

TL;DR

This paper constructs and proves the optimality of shortened Hamming codes that maximize double error detection for given parameters, enhancing error correction capabilities in digital communication.

Contribution

It introduces a new class of shortened Hamming codes that achieve maximal double error detection, with a proof of their optimality.

Findings

Constructed shortened Hamming codes with optimal double error detection

Proved the optimality of the constructed codes within specified parameters

Enhanced understanding of error detection limits in coding theory

Abstract

Given $r\geq 3$ and $2^{r-1}+1\leq n< 2^{r}-1$, an $[n,n-r,3]$ shortened Hamming code that can detect a maximal number of double errors is constructed. The optimality of the construction is proven.