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On the gaps of the spectrum of volumes of trades. (English) Zbl 1391.05055

Summary: A pair \(\{T_0,T_1\}\) of disjoint collections of \(k\)-subsets (blocks) of a set \(V\) of cardinality \(v\) is called a \(t-(v,k)\) trade or simply a \(t\)-trade if every \(t\)-subset of \(V\) is included in the same number of blocks of \(T_0\) and \(T_1\). The cardinality of \(T_0\) is called the volume of the trade. Using the weight distribution of the Reed-Muller code, we prove the conjecture that for every \(i\) from 2 to \(t\), there are no \(t\)-trades of volume greater than \(2^{t+1}-2^i\) and less than \(2^{t+1}-2^{i-1}\) and derive restrictions on the \(t\)-trade volumes that are less than \(2^{t+1}+2^{t-1}\).

MSC:

05B05 Combinatorial aspects of block designs
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