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Cyclic relative difference families with block size four and their applications.
- Source :
-
Journal of Combinatorial Theory - Series A . Aug2024, Vol. 206, pN.PAG-N.PAG. 1p. - Publication Year :
- 2024
-
Abstract
- Given a subgroup H of a group (G , +) , a (G , H , k , 1) difference family (DF) is a set F of k -subsets of G such that { f − f ′ : f , f ′ ∈ F , f ≠ f ′ , F ∈ F } = G ∖ H. Let g Z g h be the subgroup of order h in Z g h generated by g. A (Z g h , g Z g h , k , 1) -DF is called cyclic and written as a (g h , h , k , 1) -CDF. This paper shows that for h ∈ { 2 , 3 , 6 } , there exists a (g h , h , 4 , 1) -CDF if and only if g h ≡ h (mod 12) , g ⩾ 4 and (g , h) ∉ { (9 , 3) , (5 , 6) }. As a corollary, it is shown that a 1-rotational Steiner system S (2 , 4 , v) exists if and only if v ≡ 4 (mod 12) and v ≠ 28. This solves the long-standing open problem on the existence of a 1-rotational S (2 , 4 , v). As another corollary, we establish the existence of an optimal (v , 4 , 1) -optical orthogonal code with ⌊ (v − 1) / 12 ⌋ codewords for any positive integer v ≡ 1 , 2 , 3 , 4 , 6 (mod 12) and v ≠ 25. We also give applications of our results to cyclic group divisible designs with block size four and optimal cyclic 3-ary constant-weight codes with weight four and minimum distance six. [ABSTRACT FROM AUTHOR]
- Subjects :
- *FAMILY size
*STEINER systems
*CYCLIC groups
*ORTHOGONAL codes
*DIVISIBILITY groups
Subjects
Details
- Language :
- English
- ISSN :
- 00973165
- Volume :
- 206
- Database :
- Academic Search Index
- Journal :
- Journal of Combinatorial Theory - Series A
- Publication Type :
- Academic Journal
- Accession number :
- 177353048
- Full Text :
- https://doi.org/10.1016/j.jcta.2024.105890