The number of the non-full-rank Steiner triple systems

被引：5

作者：

Shi, Minjia ^{[1
]}

Xu, Li ^{[1
]}

Krotov, Denis S. ^{[2
]}

机构：

[1] Anhui Univ, Sch Math, Minist Educ, Key Lab Intelligent Comp & Signal Proc, Hefei, Anhui, Peoples R China

[2] Sobolev Inst Math, Pr Akad Koptyuga 4, Novosibirsk 630090, Russia

来源：

JOURNAL OF COMBINATORIAL DESIGNS | 2019年 / 27卷 / 10期

基金：

中国国家自然科学基金;

关键词：

2-rank; 3-rank; Steiner triple system; S(2 (M)-1; ORDER-19; M+2;

D O I：

10.1002/jcd.21663

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

The p-rank of a Steiner triple system (STS) B is the dimension of the linear span of the set of characteristic vectors of blocks of B, over GF(p). We derive a formula for the number of different STSs of order v and given 2-rank r2, r2<v, and a formula for the number of STSs of order v and given 3-rank r3, r3<v-1. Also, we prove that there are no STSs of 2-rank smaller than v and, at the same time, 3-rank smaller than v-1. Our results extend previous study on enumerating STSs according to the rank of their codes, mainly by Tonchev, V.A. Zinoviev, and D.V. Zinoviev for the binary case and by Jungnickel and Tonchev for the ternary case.

引用

页码：571 / 585

页数：15

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