发表机构
Moscow Institute of Physics and Technology (State University); Institute of Mathematics and Informatics, Bulgarian Academy of Sciences; Lomonosov Moscow State University; Adyghe State University（莫斯科物理技术学院; 保加利亚科学院数学与信息学研究所; 莫斯科罗蒙诺索夫国立大学; 阿迪格国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一个基于互信息和最优传输的框架，用于统一和推广二元及非二元常重码与常组合码的上界，并证明闭包算子幂等性及码率函数的Schur凹性，最终获得改进的理论和数值界。
AI 中文摘要
从现有界推导码大小的上界是编码理论中的经典方法，可追溯至Elias、Bassalygo和Levenshtein的开创性成果。我们研究了一个框架，该框架涵盖二元和非二元（常重）码的Bassalygo--Elias和Levenshtein不等式，并提供某些推广。在不同符号组合之间转移界的渐近代价以互信息表示，从而形成信息论最优传输公式。我们确定了任意组合之间在优化序意义下的最优置换传输代价，以其最小公共优超元表示。针对对称常重组合，我们得到显式传输剖面。作为副产品，我们证明了渐近常重码率作为相对重量的函数的单峰性。我们证明所得闭包算子是幂等的，并且在外层Bassalygo--Elias平均之前应用传输，会使由相同输入获得的无约束界保持不变。我们还建立了上界成为闭包算子不动点的充分必要条件。由于渐近码率函数是其自身的上界且是不动点，我们得出常组合码率函数的Schur凹性。最后，我们综述了二元和非二元常重码及常组合码的现有上界，在传输框架内将它们组合成优化包络，并获得改进的理论和数值界。
英文摘要
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.
Comments39 pages, 3 figures
