On the minimality of Hamming compatible metrics

TL;DR

This paper introduces a new Hamming compatible metric, analyzes its properties, and proves its minimality among well-behaved metrics, contributing to the understanding of distance measures in coding theory.

Contribution

The paper defines a novel Hamming compatible metric, calculates the size of its spheres, and demonstrates its minimality within a specific class of metrics.

Findings

Computed the cardinality of spheres under the new metric

Proved the metric's minimality among well-behaved Hamming compatible metrics

Provided insights into metric design for coding theory

Abstract

A Hamming compatible metric is an integer-valued metric on the words of a finite alphabet which agrees with the usual Hamming distance for words of equal length. We define a new Hamming compatible metric, compute the cardinality of a sphere with respect to this metric, and show this metric is minimal in the class of all "well-behaved" Hamming compatible metrics.