Classification of optimal (v, 4, 1) binary cyclically permutable constant-weight codes and cyclic 2-(v, 4, 1) designs with v ≤ 76

Abstract

We classify up to isomorphism optimal (v, 4, 1) binary cyclically permutable constantweight (CPCW) codes with v ? 76 and cyclic 2-(73, 4, 1) and 2-(76, 4, 1) designs. There is a one-to-one correspondence between optimal (v, 4, 1) CPCW codes, optimal cyclic binary constant-weight codes with weight 4 and minimum distance 6, (v, 4; ?(v ? 1)/12?) difference packings, and optimal (v, 4, 1) optical orthogonal codes. Therefore, the classification of CPCW codes holds for them too. Perfect (v, 4, 1) CPCWcodes are equivalent to (v, 4, 1) cyclic difference families, and thus (73, 4, 1) cyclic difference families are classified too.