An Approximate Coding-Rate Versus Minimum Distance Formula for Binary Codes

TL;DR

This paper introduces a simple, invertible approximate formula linking the minimum distance and maximum code-rate of binary codes, aiding in the analysis and design of communication systems.

Contribution

It provides a novel, easy-to-use approximate formula for the relationship between code minimum distance and rate, with high accuracy across various scenarios.

Findings

Approximate formula closely matches known minimum distances (~97% accuracy).

Formula simplifies analysis and design of binary codes.

Example shows improved estimate of minimum distance for specific code parameters.

Abstract

We devise an analytically simple as well as invertible approximate expression, which describes the relation between the minimum distance of a binary code and the corresponding maximum attainable code-rate. For example, for a rate-(1/4), length-256 binary code the best known bounds limit the attainable minimum distance to 65<d(n=256,k=64)<90, while our solution yields d(n=256,k=64)=74.4. The proposed formula attains the approximation accuracy within the rounding error for ~97% of (n,k) scenarios, where the exact value of the minimum distance is known. The results provided may be utilized for the analysis and design of efficient communication systems.