Inequalities for code pairs (Q800361)

The main result is the following inequality: If \(A,B\subset \{1,...,\alpha\}^ m\); \(m\in {\mathbb{N}}\); satisfies for the Hamming distance d \(d(a,b)-d(a,b')+d(a',b')-d(a',b)\neq 1,2\) for all \(a,a'\in A\) and \(b,b'\in B\), then \(| A|\;| B|\leq \alpha^{*m}\), where \(\alpha^*=\alpha\) for \(\alpha =2,3,4,\) \(\alpha^*= \frac{\alpha}{2} \frac{\ulcorner\alpha \urcorner}{2}\) for \(\alpha \geq 4\), and the bound is best. It is a generalization of the first author, \textit{A. El Gamal} and \textit{K. F. Pang} [Discrete Math. 49, 1-5 (1984; Zbl 0532.94013)], \textit{P. Delsarte} and \textit{P. Piret} [An extension of an inequality by Ahlswede, El Gamal and Pang for pairs of binary codes, submitted to Discrete Math.]).