On the Existence of Optimal (k(k-1)s2,k,1) Binary Cyclically Permutable Constant Weight Codes+

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Necessary conditions for the existence of optimal (k(k−1)s+2,k,1) binary cyclically permutable constant weight codes with length divisible by 2n (where n≥2) can be established by analyzing the incidence matrix of the corresponding partial cyclic design. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Tsonka Baicheva; Tsvetyana Yoveva; Svetlana Topalova

doi:10.3390/axioms15090679

September 11, 2026AxiomsOpen Access

On the Existence of Optimal (k(k-1)s2,k,1) Binary Cyclically Permutable Constant Weight Codes+

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Key Points

Establish necessary conditions for the existence of optimal (k(k-1)s+2, k, 1) binary cyclically permutable constant weight codes with lengths divisible by 2^n for n ≥ 2.
Formulated corresponding partial cyclic designs represented by block incidence matrices composed of circulants of order v/2^m.
Applied algebraic double-counting arguments to pairs of points within the incidence matrix structures.
Derived new necessary existence constraints governing code parameters when the block length is a multiple of 2^n (n ≥ 2).
Demonstrated that point-pair distributions among circulant submatrices restrict permissible configurations in optimal cyclic combinatorial designs.

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Implication

Mathematical analysis reveals necessary existence conditions for cyclically permutable codes with lengths divisible by powers of two, narrowing parameters for optimal combinatorial designs.

Abstract

We derive necessary conditions for the existence of optimal (k(k−1)s+2,k,1) binary cyclically permutable constant weight codes with length divisible by 2n, where n≥2. This is achieved by counting in different ways the pairs of points in the corresponding partial cyclic design whose incidence matrix is equivalent to a matrix containing circulants of order v2m for 0≤m

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Baicheva et al. (2026) studied this question.

synapsesocial.com/papers/6aa7b2b20926e14a848b1246 https://doi.org/10.3390/axioms15090679

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On the Existence of Optimal (k(k-1)s2,k,1) Binary Cyclically Permutable Constant Weight Codes+

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On the Existence of Optimal (k(k-1)s2,k,1) Binary Cyclically Permutable Constant Weight Codes+

Key Points

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Abstract

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