On the gaps of the spectrum of volumes of trades

Abstract

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 $2^{t+1}+2^{t-1}$.