Universal Bounds for Size and Energy of Codes of Given Minimum and Maximum Distances

TL;DR

This paper develops universal bounds on the size and energy of codes with specified minimum and maximum distances using positive definite measures, providing conditions for optimality and characterizing extremal codes.

Contribution

It introduces Levenshtein-type bounds for code size and energy considering both minimum and maximum distances, extending previous bounds to more general code parameters.

Findings

Derived upper bounds on code size for given distance constraints.

Established universal lower bounds on potential energy for certain codes.

Characterized conditions for when bounds are tight and optimal.

Abstract

We employ signed measures that are positive definite up to certain degrees to establish Levenshtein-type upper bounds on the cardinality of codes with given minimum and maximum distances, and universal lower bounds on the potential energy (for absolutely monotone interactions) for codes with given maximum distance and cardinality. The distance distributions of codes that attain the bounds are found in terms of the parameters of Levenshtein-type quadrature formulas. Necessary and sufficient conditions for the optimality of our bounds are derived. Further, we obtain upper bounds on the energy of codes of fixed minimum and maximum distances and cardinality.