An eigenvalue proof of Heged\"{u}s's bound for codes with a single Hamming distance (opens in new tab)

We give a short, self-contained linear-algebra proof of a bound of Heged\"{u}s [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 $\lambda\ne(n+1)/2$, then the family has at most $n$ members. Our proof uses the same Gram matrix as in Heged\"{u}s's argument, but reads its eigenvalues in place of its determinant, and keys off of a singl...