An extension of an inequality by Ahlswede, El Gamal and Pang for pairs of binary codes (Q1065758)

\textit{R. Ahlswede, A. El Gamal} and \textit{K. F. Pang} [Discrete Math. 49, 1-5 (1984; Zbl 0532.94013)] proved that if A, B are binary codes of length n such that the Hamming distance between codewords of A and B has a constant value, then the product of the cardinalities of A and B is at most \(2^ n\) when n is even and \(2^{n-1}\) if n is odd. This paper proves the same result by assuming that the distance taken modulo 4 has a constant value. The proof is based on a lemma which states that if the codes A and B satisfy the above mentioned property, then the translated codes \(A+a_ 0\) and \(B+b_ 0\) are orthogonal for any vectors \(a_ 0\) in A and \(b_ 0\) in B. (Two codes A and B are said to be orthogonal if the inner product (x,y) vanishes for all vectors x in A and y in B.)