Resolving the Remaining Open CPCW Cases for k = 5 and k = 6: Nine Explicit Optimal Constructions

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Original abstract

We present explicit optimal binary cyclically permutable constant-weight codes for all nine 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 k = 6, the four cases are v = 122, 123, 124, 126. For k = 5, the five cases are v = 143, 144, 146, 162, 167. Explicit constructions attaining the standard difference upper bound are given for every one of these nine cases. k = 6 constructions 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} 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} 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} k = 5 constructions CPCW(143,5,1) {0,3,16,70,116} {0,7,26,86,95} {0,5,37,68,109} {0,4,24,66,118} {0,1,15,23,59} {0,2,35,47,53} {0,10,21,38,103} CPCW(144,5,1) {0,1,3,135,139} {0,7,18,86,130} {0,16,46,63,108} {0,20,51,77,105} {0,13,37,66,101} {0,19,42,69,103} {0,15,40,89,111} CPCW(146,5,1) {0,13,37,60,93} {0,5,48,59,100} {0,1,32,68,71} {0,15,34,84,106} {0,2,8,12,29} {0,7,35,49,65} {0,18,38,63,120} CPCW(162,5,1) {0,1,11,38,142} {0,3,9,116,132} {0,4,28,54,126} {0,13,32,73,91} {0,8,43,87,109} {0,17,42,105,128} {0,2,7,69,117} {0,12,68,82,97} CPCW(167,5,1) {0,8,24,73,93} {0,6,21,34,61} {0,1,4,37,63} {0,10,52,77,120} {0,7,19,103,114} {0,9,23,54,132} {0,2,32,50,88} {0,5,22,51,92} Optimality For each construction, all directed differences generated by the displayed base blocks are distinct. The number of blocks equals the standard difference bound floor((v - 1) / (k × (k - 1))), so every displayed code is optimal. Verification and reproducibility The constructions were discovered through exact and hybrid computational searches using mathematical reductions, symmetry methods, exact-cover techniques, compressed subset structures, and independently checked positive certificates. The positive mathematical claims do not depend on reproducing the original searches: this record includes a standalone verifier that recomputes the CPCW difference conditions directly from all nine displayed constructions. 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.22076107

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