Asymptotically optimal constant weight codes with even distance

TL;DR

This paper provides an asymptotically sharp estimate for the maximum size of constant weight codes with even minimum distance, extending previous results that focused on odd distances.

Contribution

It extends the asymptotic analysis of constant weight codes to the case where the minimum distance is even, answering an open question.

Findings

Derived an asymptotically sharp estimate for $A_q(n, d, w)$ with even $d$

Extended previous odd-distance results to even-distance case

Clarified the asymptotic behavior of constant weight codes for large $n$

Abstract

A $q$-ary code $C$ of length $n$ is a set of $n$-dimensional vectors (code words) with entries in $\{0, \ldots, q-1\}$. We say $C$ has constant weight $w$ if each code word has exactly $w$ nonzero entries. We say $C$ has minimum distance $d$ if any two distinct code words in $C$ differ in at least $d$ entries. We let $A_q(n, d, w)$ be the largest possible cardinality of any $q$-ary code of length $n$ with constant weight $w$ and minimum distance $d$. Very recently, Liu and Shangguan gave an asymptotically sharp estimate for $A_q(n, d, w)$ where $q, d, w$ are fixed, $d$ is odd and $n \rightarrow \infty$. In this note we answer a question of Liu and Shangguan by obtaining such an estimate in the case where $d$ is even.