Codes with symmetric distances
Gábor Hegedüs, Sho Suda, Ziqing Xiang
TL;DR
The paper investigates binary codes of length $2n$ with weight $n$ whose distance set is symmetric about $n/2$, proving the upper bound $|C|\le\binom{2n-1}{s}$ for codes of degree $s$ and symmetric inner distribution, interpreted via the Johnson scheme $J(2n,n)$. It then places these codes in the broader context of $Q$-bipartite $Q$-polynomial association schemes, deriving general upper bounds through annihilator polynomials and dual eigenvalues, and characterizing equality via the zeros of an associated polynomial $\Phi_s(z)$. Specializing to the Johnson scheme, the authors connect $\Psi^*_s$ to $\Phi_s$ and obtain a concrete bound framework, together with a strong structural criterion for tight codes. A substantial number-theoretic analysis of the zeros of $\Phi_s(z)$ yields conditional results: if all zeros are integral, either $s=1$ or $r=1$ or $s$ is extremely large relative to $r$, which in turn leads to corollaries validating a conjecture for $n\le 4\cdot 10^6$ and linking tight codes to Hadamard matrices ($s=1$) or maximal intersecting families ($s=n-1$). The work thus blends linear-algebraic, algebraic-combinatorial, and number-theoretic methods to address tight code structures in symmetric-distance scenarios and highlights clear avenues for identifying when equality can occur.
Abstract
For a code $C$ in a space with maximal distance $n$, we say that $C$ has symmetric distances if its distance set $S(C)$ is symmetric with respect to $n / 2$. In this paper, we prove that if $C$ is a binary code with length $2n$, constant weight $n$ 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.