Distance Theorems for Code Pairs

We shall report on recent results (by several authors) on problems concerning the distances of code words, where the words are from two binary codes A and B of length n. In the first two problems we suppose that there is a number 6 such that d(a, b) = 6 for every a E A and every b E B. In [l] Ahlswede et al. proved that this condition implies that I A I . I B I I 22rx’21. Although their proof is not long, it is not simple. We give a practically trivial proof of the same theorem and discuss the case of equality (cf. [3]). Next, we consider the same problem but with 6 prescribed. Recently, it was shown by Van Pul [7] that we then have