Classification of maximum-size q-ary equidistant codes

Classify all q-ary equidistant codes C contained in H_q^n that attain the upper bound (q-1)n+1, beyond the finite-geometric constructions given in Example 1.

Background

The paper proves that every q-ary equidistant code C⊆H_qn with distance d different from ((q-1)n+1)/q has size at most (q-1)n. Consequently, the general Delsarte bound |C|≤(q-1)n+1 can be attained only at the exceptional distance d=((q-1)n+1)/q. For prime-power q, Example 1 constructs codes meeting this bound using finite-field projective geometries. The authors explicitly ask whether additional, inequivalent or otherwise distinct constructions attaining the same maximum size exist.

References

Are there other equidistant codes $C \subseteq H_qn$ that attain the upper bound $(q-1)n + 1$, except those presented in Example \ref{example_equidistant_code_maximum_size}?

— Hegedus' Conjecture and Tighter Upper Bounds for Equidistant Codes in Hamming Spaces  (2504.07036 - Hu et al., 9 Apr 2025) in Section 4, Discussions; first Question after the heading “Discussions”