The vector space generated by permutations of a trade or a design

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-$(v,k,\lambda )$ design, its permutations can span the vector space generated by all t-$(v,k,\lambda )$ designs for sufficiently large values of v. In other words, any t-$(v,k,\lambda )$ 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.