Linear Binary Codes Correcting One or More Errors

TL;DR

This paper investigates linear binary codes that can correct multiple errors, providing bounds and constructions for single-error correction and insights into the structure of codes for multiple errors.

Contribution

It presents a constructive method achieving the Hamming bound for single-error correction and derives a lower bound for parameters of codes correcting multiple errors.

Findings

Hamming bound is achieved by a constructive method for single-error correction

An exact expression for minimal codeword length in single-error correction is derived

A simple lower bound for parameters of codes correcting multiple errors is established

Abstract

This paper examines linear binary codes capable of correcting one or more errors. For the single-error-correcting case, it is shown that the Hamming bound is achieved by a constructive method, and an exact expression for the minimal codeword length is derived. For the general case, a simple lower bound for the parameters of linear codes is derived from an analysis of the coset structure.