Entropy Distance

TL;DR

This paper introduces the entropy distance, a new metric based on the logarithm of the surface area of Hamming balls, and explores its properties, bounds, and applications in coding theory.

Contribution

It defines the entropy distance as a novel metric related to entropy, derives bounds for linear codes, and analyzes its application in linear encoders.

Findings

Entropy distance is a metric for non-binary fields and a pseudometric for binary fields.

Derived bounds for the maximum size of linear codes with given entropy distance.

Established tight bounds for the entropy distance of linear encoders.

Abstract

Motivated by the approach of random linear codes, a new distance in the vector space over a finite field is defined as the logarithm of the "surface area" of a Hamming ball with radius being the corresponding Hamming distance. It is named entropy distance because of its close relation with entropy function. It is shown that entropy distance is a metric for a non-binary field and a pseudometric for the binary field. The entropy distance of a linear code is defined to be the smallest entropy distance between distinct codewords of the code. Analogues of the Gilbert bound, the Hamming bound, and the Singleton bound are derived for the largest size of a linear code given the length and entropy distance of the code. Furthermore, as an important property related to lossless joint source-channel coding, the entropy distance of a linear encoder is defined. Very tight upper and lower bounds are obtained for the largest entropy distance of a linear encoder with given dimensions of input and output vector spaces.