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 Probl. Peredachi Inf., 2004, Volume 40, Issue 4, Pages 48–67 (Mi ppi150)

Coding Theory

Classification of Steiner Quadruple Systems of Order 16 and of Rank at Most 13

V. A. Zinov'ev, D. V. Zinov'ev

Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: A Steiner quadruple system SQS($v$) of order $v$ is a 3-design $T(v;4;3;\lambda)$ with $\lambda=1$. In this paper we describe all nonisomorphic systems SQS(16) that can be obtained by the generalized concatenated construction (GC-construction). These Steiner systems have rank at most 13 over $\mathbb F_2$. In particular, there is one system SQS(16) of rank 11 (points and planes of the a fine geometry AG(4;2)), fifteen systems of rank 12, and 4131 systems of rank 13. All these Steiner systems are resolvable.

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English version:
Problems of Information Transmission, 2004, 40:4, 337–355

Bibliographic databases:

UDC: 621.394.74:512
Revised: 17.06.2004

Citation: V. A. Zinov'ev, D. V. Zinov'ev, “Classification of Steiner Quadruple Systems of Order 16 and of Rank at Most 13”, Probl. Peredachi Inf., 40:4 (2004), 48–67; Problems Inform. Transmission, 40:4 (2004), 337–355

Citation in format AMSBIB
\Bibitem{ZinZin04} \by V.~A.~Zinov'ev, D.~V.~Zinov'ev \paper Classification of Steiner Quadruple Systems of Order~16 and of Rank at Most~13 \jour Probl. Peredachi Inf. \yr 2004 \vol 40 \issue 4 \pages 48--67 \mathnet{http://mi.mathnet.ru/ppi150} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2105853} \zmath{https://zbmath.org/?q=an:1077.05015} \transl \jour Problems Inform. Transmission \yr 2004 \vol 40 \issue 4 \pages 337--355 \crossref{https://doi.org/10.1007/s11122-004-0003-1} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. A. Zinov'ev, D. V. Zinov'ev, “Vasil'ev Codes of Length $n=2^m$ and Doubling of Steiner Systems $S(n,4,3)$ of a Given Rank”, Problems Inform. Transmission, 42:1 (2006), 10–29
2. V. A. Zinov'ev, D. V. Zinov'ev, “Classification of Steiner Quadruple Systems of Order 16 and of Rank 14”, Problems Inform. Transmission, 42:3 (2006), 217–229
3. Kaski P., Östergård P.R.J., Pottonen O., “The Steiner quadruple systems of order 16”, J. Combin. Theory Ser. A, 113:8 (2006), 1764–1770
4. V. A. Zinov'ev, D. V. Zinov'ev, “On Resolvability of Steiner Systems $S(v=2^m,4,3)$ of Rank $r\le v-m+1$ over $\mathbb F_2$”, Problems Inform. Transmission, 43:1 (2007), 33–47
5. Solov'eva F.I., “On perfect binary codes”, Discrete Appl. Math., 156:9 (2008), 1488–1498
6. V. A. Zinoviev, D. V. Zinoviev, “On one transformation of Steiner quadruple systems $S(v,4,3)$”, Problems Inform. Transmission, 45:4 (2009), 317–332
7. S. V. Syrovatskaya, “O $P$-algebrakh na osnove 4-sistem”, Chebyshevskii sb., 12:2 (2011), 110–117
8. V. A. Zinoviev, D. V. Zinoviev, “Steiner triple systems $S(2^m-1,3,2)$ of rank $2^m-m+1$ over $\mathbb F_2$”, Problems Inform. Transmission, 48:2 (2012), 102–126
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