Intersection of Isomorphic Linear Codes

Given an (n, k) linear code C over GF(q), the intersection of C with a codeπ(C), whereπ∈Sn, is an (n, k1) code, where max{0, 2k−n}⩽k1⩽k. The intersection problem is to determine which integers in this range are attainable for a given code C. We show that, depending on the structure of the generator matrix of the code, some of the values in this range are attainable. As a consequence we give a complete solution to the intersection problem for most of the interesting linear codes, e.g. cyclic codes, Reed–Muller codes, and most MDS codes.