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Volume 15
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Article

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

by
Tsonka Baicheva
1,2,* and
Svetlana Topalova
1,2
1
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
2
Centre of Excellence in Informatics and Information and Communication Technologies, 1113 Sofia, Bulgaria
*
Author to whom correspondence should be addressed.
Axioms 2026, 15(1), 35; https://doi.org/10.3390/axioms15010035 (registering DOI)
Submission received: 3 December 2025 / Revised: 22 December 2025 / Accepted: 31 December 2025 / Published: 1 January 2026
(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications, 2nd Edition)
Versions Notes

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.
Keywords: coding theory; binary cyclically permutable constant-weight code; optical orthogonal code; cyclic design coding theory; binary cyclically permutable constant-weight code; optical orthogonal code; cyclic design

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MDPI and ACS Style

Baicheva, T.; Topalova, S. On the Existence of Optimal (v,5,1) and (v,6,1) Binary Cyclically Permutable Constant-Weight Codes. Axioms 2026, 15, 35. https://doi.org/10.3390/axioms15010035

AMA Style

Baicheva T, Topalova S. On the Existence of Optimal (v,5,1) and (v,6,1) Binary Cyclically Permutable Constant-Weight Codes. Axioms. 2026; 15(1):35. https://doi.org/10.3390/axioms15010035

Chicago/Turabian Style

Baicheva, Tsonka, and Svetlana Topalova. 2026. "On the Existence of Optimal (v,5,1) and (v,6,1) Binary Cyclically Permutable Constant-Weight Codes" Axioms 15, no. 1: 35. https://doi.org/10.3390/axioms15010035

APA Style

Baicheva, T., & Topalova, S. (2026). On the Existence of Optimal (v,5,1) and (v,6,1) Binary Cyclically Permutable Constant-Weight Codes. Axioms, 15(1), 35. https://doi.org/10.3390/axioms15010035

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