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

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 r 2, r 2 < v, and a formula for the number of STSs of order v and...

Full description

Saved in:
Bibliographic Details
Published inJournal of combinatorial designs Vol. 27; no. 10; pp. 571 - 585
Main Authors Shi, Minjia, Xu, Li, Krotov, Denis S.
Format Journal Article
LanguageEnglish
Published Hoboken Wiley Subscription Services, Inc 01.10.2019
Subjects
2‐rank
3‐rank
Steiner triple system
Online AccessGet full text

Cover

More Information
Summary: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 r 2, r 2 < v, and a formula for the number of STSs of order v and given 3‐rank r 3, r 3 < 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.
ISSN:1063-8539
1520-6610
DOI:10.1002/jcd.21663