On the Existence of Optimal (v, 5, 1) and (v, 6, 1) Binary Cyclically Permutable Constant-Weight Codes

Abstract

The problem of the existence of optimal (v,k,1) binary cyclically permutable constant-weight (CPCW) codes has been completely solved for codeword weights k<5. We consider the smallest open cases, namely k=5 and k=6. We present such codes for small values of the code length v and derive necessary conditions for the existence of optimal (k(k−1)t+2,k,1) CPCW codes. These necessary conditions can be used to construct such codes, as well as to show that optimal codes with some parameters do not exist. In particular, we use them to prove that an optimal (92,6,1) CPCW code does not exist.