Asymptotically optimal constant weight codes with even distance

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.