On k-wise set-intersections and k-wise Hamming-distances

We prove a version of the Ray-Chaudhuri–Wilson and Frankl–Wilson theorems for k-wise intersections and also generalize a classical code-theoretic result of Delsarte for k-wise Hamming distances. A set of code-words a1, a2,…,ak of length n have k-wise Hamming-distance ℓ, if there are exactly ℓ such coordinates, where not all of their coordinates coincide (alternatively, exactly n−ℓ of their coordinates are the same). We show a Delsarte-like upper bound: codes with few k-wise Hamming-distances must contain few code-words.