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

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 $\{1,\dots ,n\}$ equal a fixed value $\lambda \mathrm{\ne }(n+1)\mathrm{/}2$, 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 $\lambda \mathrm{\ne }((q-1)n+1)\mathrm{/}q$ – the corrected form of a conjecture of Hegedüs, recently established by Hu, Huang, and Yu [arXiv:2504.07036].