Resolving the Four Remaining Open k=6 CPCW Cases: Explicit Optimal Constructions for v = 122, 123, 124, and 126

Abstract

We present explicit optimal binary cyclically permutable constant-weight codes for CPCW(122,6,1), CPCW(123,6,1), CPCW(124,6,1), and CPCW(126,6,1). These are the four k = 6 parameter cases identified as unresolved by Baicheva and Topalova in On the Existence of Optimal (v,5,1) and (v,6,1) Binary Cyclically Permutable Constant-Weight Codes, Axioms 15(1), 35 (2026). For each value of v, four explicit base blocks of size 6 are given. Each construction has 120 distinct directed differences and attains the standard difference upper bound floor((v - 1) / 30) = 4. Hence all four constructions are optimal. CPCW(122,6,1) {0,3,9,95,103,107} {0,7,32,45,65,79} {0,1,11,55,60,81} {0,2,31,48,71,87} CPCW(123,6,1) {0,1,3,8,34,74} {0,4,23,41,55,102} {0,6,17,30,84,94} {0,12,27,65,87,107} The unique missing circular difference class is 9. CPCW(124,6,1) {0,1,4,96,98,108} {0,11,34,58,80,99} {0,6,15,48,79,87} {0,5,18,68,75,89} The self-inverse difference 62 is absent and circular class 38 is the unique unused usable class. CPCW(126,6,1) {0,1,6,10,108,110} {0,3,34,42,77,88} {0,7,27,57,71,86} {0,12,33,65,78,101} The self-inverse difference 63 is absent and circular classes 51 and 56 are unused. Verification and reproducibility The constructions were discovered through exact computational searches using mathematical reductions, symmetry methods, and exact subset-search techniques. Each certificate was checked independently of the discovery search. This record includes a standalone verifier, so the existence claims can be checked directly without reproducing the searches. The configurations were found with substantial help from AI systems, specifically OpenAI's ChatGPT and Anthropic's Claude, under human direction, with exact computational search and independent verification. Reference:T. Baicheva and S. 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. Version DOI: 10.5281/zenodo.22070814