Ghorbani, EbrahimEbrahimGhorbaniKamali, S.S.KamaliKhosrovshahi, Gholamreza B.Gholamreza B.Khosrovshahi2024-11-182024-11-182024-11-06Journal of Combinatorial Theory, Series A (2025)https://hdl.handle.net/11420/51940Motivated 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.en0097-31652024Elsevierhttp://rightsstatements.org/vocab/InC/1.0/Signed designsTotal tradesTradesNatural Sciences and Mathematics::510: MathematicsJournal Article10.1016/j.jcta.2024.105969Journal Article