Hegedus' Conjecture and Tighter Upper Bounds for Equidistant Codes in Hamming Spaces

TL;DR

The work addresses tight upper bounds for equidistant codes in $H_q^n$, building on Delsarte-type bounds and refining Hegedüs' binary results to the general $q$-ary setting. It introduces a simplex-based embedding to translate code-distance structure into spherical geometry, establishing a corrected bound: for equidistant $C$ with distance $d$, if $d \neq \frac{(q-1)n+1}{q}$, then $|C| \le n(q-1)$. A $q$-ary generalization of Deza's theorem is developed via $\Delta_q(n,k,l)$-systems and a binary lifting technique, yielding distance-dependent and asymptotically tight bounds $|C| \le \max\{d^2+d+2, q, \lfloor 2n/d \rfloor\}$ and $|C| \le \left\lfloor \frac{2n}{d} \right\rfloor$ for large $n$, with constructions achieving these bounds. The results connect binary and $q$-ary constructions and open several avenues for exact threshold and extremal-code questions.

Abstract

An equidistant code is a code in the Hamming space such that two distinct codewords have the same Hamming distance. This paper investigates the bounds for equidistant codes in Hamming spaces.

Theorems & Definitions (29)

• Theorem 1: Delsarte delsarte1975association, 1975
• Corollary 2
• Theorem 3: Hegedüs hegedus_2024_a_new_upper, 2024
• Conjecture 1
• Theorem 4: Barg and Musin barga_musin_2011_bounds, 2011
• Theorem 5: Deza Deza1973, 1973
• Definition 1
• Proposition 1
• proof
• Proposition 2