Options
The vector space generated by permutations of a trade or a design
Publikationstyp
Journal Article
Date Issued
2024-11-06
Sprache
English
Author(s)
Volume
210
Article Number
105969
Citation
Journal of Combinatorial Theory, Series A (2025)
Publisher DOI
Scopus ID
Publisher
Elsevier
Motivated by a classical result of Graver and Jurkat (1973) and Graham, Li, and Li (1980) in combinatorial design theory, which states that the permutations of t-(v,k) minimal trades generate the vector space of all t-(v,k) trades, we investigate the vector space spanned by permutations of an arbitrary trade. We prove that this vector space possesses a decomposition as a direct sum of subspaces formed in the same way by a specific family of so-called total trades. As an application, we demonstrate that for any t-(v,k,λ) design, its permutations can span the vector space generated by all t-(v,k,λ) designs for sufficiently large values of v. In other words, any t-(v,k,λ) design, or even any t-trade, can be expressed as a linear combination of permutations of a fixed t-design. This substantially extends a result by Ghodrati (2019), who proved the same result for Steiner designs.
Subjects
Signed designs
Total trades
Trades
DDC Class
510: Mathematics
Loading...
Name
Space-of-Designs15.pdf
Size
315.8 KB
Format
Adobe PDF