Structure of steiner triple systems S(2 m − 1, 3, 2) of rank 2 m − m + 2 over $\mathbb{F}_2 $

On the number of Steiner triple systems S(2m−1,3,2) of rank Zinoviev1 2016Discrete Mathematics8140View full textAdd to dashboardCite

“…The similar formula for the rank v − m + 2 was obtained in [8], and for the rank v − m + 1 in [5]. The main result of this paper is the following theorem: …”

“…, P 7 }. It is known (see, for example Lemma 5 of [9]) that there exist Γ V = 6240 different partitions which form 6 orbits under the action of the permutation group S 8 of coordinate positions. Let V (1) , V (2) , .…”

“…Split it into seven (trivial) subcodes P i , where i ranges from 1 to 7 (i.e. into seven parallel classes of length 8 and weight 2) where each subcode has parameters (8,2,4,4), i.e.…”

“…Theorem 3 of [9] provides the estimate of the number M v of different Steiner triple systems S(v, 3, 2) of order v = 2 m − 1 = 8u + 7 ≥ 31 of rank not greater than v − m + 3, whose orthogonal code A m is of the form (2):…”

“…In that paper all different Steiner triple systems STS(v) of this rank were enumerated and the exact number of different triple systems was obtained [8]. The next value of rank 2 m − m + 2 was treated in [9], where all Steiner triple systems of this rank were enumerated and the number of different systems was estimated from above. For more on Steiner systems see the survey on the Steiner systems [1,2].…”