On the 3-way (v,k,2)(v,k,2) Steiner trades

A 3-way (v,k,t)(v,k,t) trade of volume mm consists of three disjoint collections T1T1, T2T2 and T3T3, each of mm blocks of size kk, such that for every tt-subset of vv-set VV, the number of blocks containing this tt-subset is the same in each Ti(1≤i≤3). If any tt-subset of found(T) occurs at most once in T1(Tj,j≥2), then TT is called 3-way (v,k,t)(v,k,t) Steiner trade. In this paper the spectrum (that is, the set of allowable volumes) of 3-way (v,k,t)(v,k,t) Steiner trades is discussed. Here it is shown that the volume of a 3-way (v,k,2)(v,k,2) Steiner trade is at least 3(k−1)3(k−1) for k≠4k≠4. Also we show how to construct a 3-way (v,k,2)(v,k,2) Steiner trade of volume mm when m≥12(k−1)m≥12(k−1) for k≥15k≥15, or mm is multiple of three and 3(k−1)≤m≤12(k−1)3(k−1)≤m≤12(k−1).