Hegedüs' conjecture on non-exceptional q-ary equidistant codes

The 2024 formulation asserts the bound for all non-exceptional distances using $\frac{(q-1)(n+1)}{q}$. Hu, Huang, and Yu showed that this threshold is incorrect for general $q$: for $n=1$ and $q\ge 3$, all $q$ words form a counterexample.

April 2025 correction; June 2026 independent proof

• Hu, Huang, and Yu prove the corrected theorem: if $d\ne\frac{(q-1)n+1}{q}$, then $|C|\le n(q-1)$; at the corrected exceptional distance, $n(q-1)+1$ is tight for prime-power $q$.
• A June 2026 preprint gives an eigenvalue proof of the same corrected result and explicitly identifies it as the corrected $q$-ary form of Hegedüs’ conjecture.
• No retrieved source reports a standing objection, withdrawal, or retraction.

Current status (as of August 2026): The original statement is disproved for general $q$, while the corrected assertion with exceptional distance $\frac{(q-1)n+1}{q}$ is proved and independently reproved.