Envelopes of upper bounds for nonbinary constant-weight and constant-composition codes

2026-09-20

AI Summary

该研究通过信息论最优传输方法改进了非二进制恒定重量和恒定组成码的上界问题。

Abstract

Deriving upper bounds on code size from existing bounds is a classical approach in coding theory, dating back to the seminal results of Elias, Bassalygo, and Levenshtein. We study a framework encompassing the Bassalygo--Elias and Levenshtein inequalities for binary and nonbinary (constant-weight) codes and provides certain generalizations. The asymptotic cost of transferring a bound between different symbol compositions is expressed in terms of mutual information, yielding an information-theoretic optimal transport formulation. We determine the optimal permutation-transport cost between arbitrary compositions in terms of their least common majorant in the majorization order. Specializing to symmetric constant-weight compositions yields explicit transport profiles. As a byproduct, we establish unimodality of the asymptotic constant-weight rate as a function of the relative weight. We prove that the resulting closure operators are idempotent and that applying transport before outer Bassalygo--Elias averaging leaves the unrestricted bound obtained from the same input unchanged. We also establish necessary and sufficient conditions for an upper bound to be a fixed point of the closure operator. Since the asymptotic rate function is an upper bound for itself and it is a fixed point, we conclude the Schur concavity of the constant-composition rate function. Finally, we survey existing upper bounds for binary and nonbinary constant-weight and constant-composition codes, combine them into optimized envelopes within the transport framework, and obtain improved theoretical and numerical bounds.

Problem

• nonbinary constant-weight codes
• constant-composition codes
• upper bounds
• information-theoretic optimal transport
• Bassalygo--Elias and Levenshtein inequalities

Innovation

• optimal transport
• majorization order
• closure operator
• Schur concavity
• constant-composition codes