A two-family extremal problem in Hamming space
The pair (A,B); \(A,B\subseteq \{0,1\}^ m\); is called an \((m,\delta)\)- system, if for the Hamming distance function d, \(d(a,b)=\delta\), \(\forall a\in A\), \(\forall b\in B\). Let \(S^ m_{\delta}\) denote the set of those systems. We consider the function \(M(m,\delta)=\max \{| A| | B|:(A,B)\in S^ m_{\delta}\}\) and prove the following Theorem: For \(m=1,2,...,\max_{0\leq \delta \leq m}M(m,\delta)=2^{2n}\), if \(m=2n\) or \(m=2n+1\). The maximum is assumed for \(\delta =n.\) Improvements and simpler proofs were given by \textit{P. Delsarte} and \textit{P. Pirot}: ''An extension of an inequality by Ahlswede, El Gamal and Pang for pairs of binary codes'' and by \textit{J. I. Hall} and \textit{J. H. van Lint}: ''Consstant distance code pairs''. Both papers are submitted to Discrete Math. Recently \textit{R. Ahlswede} and \textit{M. Moers}: ''Inequalities for code pairs'' [submitted to Eur. J. Comb.] discovered a much more general inequality.
scientific article; zbMATH DE number 3845499