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Paper   IPM / M / 16033
School of Mathematics
  Title:   On the volumes and affine types of trades
1.  Ebrahimi Ghorbani
2.  S Kamali
3.  Gholamreza Khosrovshahi (Joint with D. S. Krotov)
  Status:   Publishied
  Journal: Electron. J. Combin.
  Vol.:  27
  Year:  2020
  Pages:   #P.1.29
  Supported by:  IPM
A [t]-trade is a pair T=(T+, T) of disjoint collections of subsets (blocks) of a v-set V such that for every 0 ≤ it, any i-subset of V is included in the same number of blocks of T+ and of T. It follows that |T+| = |T| and this common value is called the volume of T. If we restrict all the blocks to have the same size, we obtain the classical t-trades as a special case of [t]-trades. It is known that the minimum volume of a nonempty [t]-trade is 2t. Simple [t]-trades (i.e., those with no repeated blocks) correspond to a Boolean function of degree at most vt−1. From the characterization of Kasami-Tokura of such functions with small number of ones, it is known that any simple [t]-trade of volume at most 2·2t belongs to one of two affine types, called Type (A) and Type (B) where Type (A) [t]-trades are known to exist. By considering the affine rank, we prove that [t]-trades of Type (B) do not exist. Further, we derive the spectrum of volumes of simple trades up to 2.5·2t, extending the known result for volumes less than 2·2t. We also give a characterization of "small" [t]-trades for t=1,2. Finally, an algorithm to produce [t]-trades for specified t, v is given. The result of the implementation of the algorithm for t ≤ 4, v ≤ 7 is reported.

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