Codes with symmetric distances
G\'abor Heged\"us, Sho Suda, Ziqing Xiang

TL;DR
This paper investigates binary codes with symmetric distance sets, establishing upper bounds on their size and providing a framework for studying such codes within Johnson association schemes.
Contribution
It introduces a new bound for binary codes with symmetric distances and extends the analysis to Q-bipartite Q-polynomial association schemes.
Findings
Upper bound |C| ≤ binomial(2n - 1, |S(C)|) for certain codes
Framework for studying symmetric distance codes in association schemes
Characterization of equality cases using number theory
Abstract
For a code in a space with maximal distance , we say that has symmetric distances if its distance set is symmetric with respect to . In this paper, we prove that if is a binary code with length , constant weight and symmetric distances, then \[ |C| \leq \binom{2 n - 1}{|S(C)|}. \] This result can be interpreted using the language of Johnson association schemes. More generally, we give a framework to study codes with symmetric distances in Q-bipartite Q-polynomial association schemes, and provide upper bounds for such codes. Moreover, we use number theoretic techniques to determine when the equality holds.
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Taxonomy
TopicsCoding theory and cryptography · graph theory and CDMA systems · Mathematical Approximation and Integration
