An Optimal Binary CPCW(122,6,1) Code: A Four-Block Construction

Abstract

We give an explicit optimal binary cyclically permutable constant-weight code with parameters (v,k,λ) = (122,6,1). The construction consists of four 6-subsets of Z_122: B1 = {0,3,9,95,103,107},B2 = {0,7,32,45,65,79},B3 = {0,1,11,55,60,81},B4 = {0,2,31,48,71,87}. Exact independent verification shows that difference 61 occurs zero times and every other nonzero directed difference modulo 122 occurs exactly once across the four blocks. Since the standard difference bound is floor(121/30) = 4, the construction is optimal. The existence of an optimal (122,6,1) CPCW code was listed as open by Baicheva and Topalova, “On the Existence of Optimal (v,5,1) and (v,6,1) Binary Cyclically Permutable Constant-Weight Codes,” Axioms 15(1), 35 (2026), DOI: 10.3390/axioms15010035. This record contains the explicit certificate, a concise mathematical research note, and a standalone exact Python verifier requiring no solver database or external packages.