If $S$ is a transitive metric space, then $|C|\cdot|A| \le |S|$ for any distance-$d$ code $C$ and a set $A$, ``anticode'', of diameter less than $d$. For every Steiner S$(t,k,n)$ system $S$, we show the existence of a $q$-ary constant-weight code $C$ of length~$n$, weight~$k$ (or $n-k$), and distance $d=2k-t+1$ (respectively, $d=n-t+1$) and an anticode $A$ of diameter $d-1$ such that the pair $(C,A)$ attains the code--anticode bound and the supports of the codewords of $C$ are the blocks of $S$ (respectively, the complements of the blocks of $S$). We study the problem of estimating the minimum value of $q$ for which such a code exists, and find that minimum for small values of $t$. Keywords: diameter perfect codes, anticodes, constant-weight codes, code--anticode bound, Steiner systems.

翻译：设$S$是一个传递度量空间，则对于任意距离为$d$的码$C$和直径小于$d$的集合$A$（即"反码"），有$|C|\cdot|A| \le |S|$。对于每个斯坦纳S$(t,k,n)$系统$S$，我们证明了存在一个$q$元等重码$C$，其长度为~$n$，重量为$k$（或$n-k$），距离为$d=2k-t+1$（或$d=n-t+1$），以及一个直径为$d-1$的反码$A$，使得对$(C,A)$达到码-反码界，且$C$的码字支撑集是$S$的区组（或$S$区组的补集）。我们研究了估计存在此类码时$q$最小值的问题，并针对较小的$t$值求出了该最小值。关键词：直径完美码，反码，等重码，码-反码界，斯坦纳系统。