An eigenvalue proof of Hegedüs's bound for codes with a single Hamming distance

2026-06-23

AI Summary

This study addresses the problem of determining the maximum size of a family of subsets of $\{1,\dots,n\}$ in which every pair of distinct elements is at a fixed Hamming distance not equal to $(n+1)/2$. By leveraging spectral analysis of the associated Gram matrix and exploiting structural properties of equiangular, equi-norm vectors, the authors provide a new proof of Hegedüs’s bound. Their approach offers, for the first time, a concise eigenvalue-based derivation showing that in the binary case the family size cannot exceed $n$. Moreover, the method naturally extends to alphabets of size $q$, confirming a refined version of Hegedüs’s conjecture and yielding the upper bound $n(q-1)$.

Abstract

We give a short, self-contained linear-algebra proof of a bound of Hegedüs [Australasian Journal of Combinatorics, 2026; arXiv:2409.07877]: if all pairwise Hamming distances in a family of subsets of $\{1,\ldots,n\}$ equal a fixed value $λ\ne(n+1)/2$, then the family has at most $n$ members. Our proof uses the same Gram matrix as in Hegedüs's argument, but reads its eigenvalues in place of its determinant, and keys off of a single fact about vectors of equal norm and equal pairwise inner product. That fact applies verbatim over an alphabet of size $q$, where it yields the bound $n(q-1)$ for $λ\ne\bigl((q-1)n+1\bigr)/q$ -- the corrected form of a conjecture of Hegedüs, recently established by Hu, Huang, and Yu [arXiv:2504.07036].

Problem

• Hamming distance
• eigenvalue
• code bound
• subset family
• Gram matrix

Innovation

• eigenvalue method
• Gram matrix
• Hamming distance
• code bounds
• linear algebra proof