An upper bound on the size of a code with $s$ distances

TL;DR

This work bounds the maximum size $A_2(n,\mathcal{D})$ of binary codes of length $n$ with $s$ distinct pairwise distances without restricting the distances $d_i$. It introduces a polynomial-method approach using $P_{\boldsymbol{u}}(\boldsymbol{x})$ to obtain the universal bound $|C|\le {n+s\choose s}$, and then refines it by examining even-degree monomials to derive a tighter bound involving partition numbers: $|C|\le {n+s\choose s}-\sum_{t:2t\le s}\big({t+n-1\choose t}-p(t)\big)$. The results are asymptotically optimal and linked to combinatorial designs, with conjectures that extremal codes for general $s$ have sizes $\binom{n}{s}$ or $\binom{n}{s}+1$, guiding future understanding of the structure of $s$-distance codes.

Abstract

Let $C$ be a binary code of length $n$ with distances $0<d_1<\cdots<d_s\le n$. In this note we prove a general upper bound on the size of $C$ without any restriction on the distances $d_i$. The bound is asymptotically optimal.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 5
• proof
• Theorem 6
• Theorem 7