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Year 2016, , 145 - 154, 09.08.2016

Dean Crnkovic Doris Dumicic Danilovic Sanja Rukavina

https://doi.org/10.13069/jacodesmath.29560

Abstract

References

  • R. W. Ahrens, G. Szekeres, On a combinatorial generalization of the 27 lines associated with a cubic surface, J. Austral. Math. Soc. 10 (1969) 485–492.
  • E. F. Assmus Jr., J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, 1992.
  • T. Beth, D. Jungnickel, H. Lenz, Design Theory, 2nd Edition, Cambridge University Press, Cambridge, 1999.
  • W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24(3-4) (1997) 235–265.
  • V. Cepulic, On symmetric block designs (45,12,3) with automorphisms of order 5, Ars Combin. 37 (1994) 33–48.
  • C. J. Colbourn, J. F. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, 2nd Edition, CRC Press, Boca Raton, 2007.
  • K. Coolsaet, J. Degraer, E. Spence, The strongly regular (45; 12; 3; 3) graphs, Electron. J. Combin. 13(1) (2006) Research Paper 32, 1–9.
  • D. Crnkovic, B. G. Rodrigues, S. Rukavina, L. Simcic, Ternary codes from the strongly regular (45,12,3,3) graphs and orbit matrices of 2-(45,12,3) designs, Discrete Math. 312(20) (2012) 3000– 3010.
  • D. Crnkovic, S. Rukavina, Construction of block designs admitting an abelian automorphism group, Metrika 62(2-3) (2005) 175–183.
  • D. Crnkovic, S. Rukavina, On some symmetric (45, 12, 3) and (40, 13, 4) designs, J. Comput. Math. Optim. 1(1) (2005) 55–63.
  • D. Crnkovic, S. Rukavina, L. Simcic, On triplanes of order twelve admitting an automorphism of order six and their binary and ternary codes, Util. Math., to appear.
  • U. Dempwolff, Primitive rank 3 groups on symmetric designs, Des. Codes Cryptogr. 22(2) (2001) 191–207.
  • C. E. Frasser, k-geodetic Graphs and Their Application to the Topological Design of Computer Networks, Argentinian Workshop in Theoretical Computer Science, 28 JAIIO-WAIT ’99 (1999) 187–
  • The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.2; 2013. (http://www.gap-system.org)
  • M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, Online available at http://www.codetables.de, Accessed on 2014-03-11.
  • Z. Janko, Coset enumeration in groups and constructions of symmetric designs, Combinatorics ’90 (Gaeta, 1990), Ann. Discrete Math. 52 (1992) 275–277.
  • P. Kaski, P. R. J. Östergård, Classification Algorithms for Codes and Designs, Springer, Berlin, 2006.
  • P. Kaski and P. R. J. Östergård, There are exactly five biplanes with k = 11, J. Combin. Des. 16(2) (2008) 117–127.
  • T. Kölmel, Einbettbarkeit symmetrischer (45,12,3) Blockplaene mit fixpunktfrei operierenden Automorphismen, Heidelberg, 1991.
  • C. W. H. Lam, G. Kolesova, L. H. Thiel, A computer search for finite projective planes of order 9, Discrete Math. 92(1-3) (1991) 187–195.
  • E. Lander, Symmetric Designs: An Algebraic Approach, Cambridge University Press, Cambridge, 1983.
  • R. Mathon, A. Rosa, 2-(v; k; ) Designs of Small Order, in: Handbook of Combinatorial Designs, 2nd ed., C.J. Colbourn, J.H. Dinitz (Editors), Chapman & Hall/CRC, Boca Raton (2007) 25–58.
  • R. L. MacFarland, A family of difference sets in non-cyclic groups, J. Combin. Theory Ser. A 15 (1973) 1–10.
  • V. Mandekic-Botteri, On symmetric block designs (45,12,3) with involutory automorphism fixing 15 points, Glas. Mat. Ser. III 36(56)(2) (2001) 193–222.
  • R. Mathon, E. Spence, On 2-(45,12,3) designs, J. Combin. Des. 4(3) (1996) 155–175.
  • MinT, Dept. of Mathematics, University of Salzburg, The online database for optimal parameters of (t,m,s)-nets, (t,s)-sequences, orthogonal arrays, linear codes, and OOAs, Online available at http://mint.sbg.ac.at/index.php, Accessed on 2014-03-11.
  • R. M. Ramos, J. Sicilia, M. T. Ramos, A generalization of geodetic graphs: K-geodetic graphs, Investigacion Operativa 6 (1998) 85–101.
  • L. Rudolph, A class of majority logic decodable codes, IEEE Trans. Inform. Theory 13(2) (1967) 305–307.
  • N. Srinivasan, J. Opatrný, V. S. Alagar, Construction of geodetic and bigeodetic blocks of connectivity k 3 and their relation to block designs, Ars Combin. 24 (1987) 101–114.
  • L.H. Soicher, DESIGN - a GAP package, Version 1.6, 23/11/2011.(http://www.gapsystem.org/Packages/design.html)

Enumeration of symmetric (45,12,3) designs with nontrivial automorphisms

Year 2016, , 145 - 154, 09.08.2016

Dean Crnkovic Doris Dumicic Danilovic Sanja Rukavina

https://doi.org/10.13069/jacodesmath.29560

Abstract

We show that there are exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms. Among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs. We describe the full automorphism groups of these designs and analyze their ternary codes. R. Mathon and E. Spence have constructed 1136 symmetric (45,12,3) designs with trivial automorphism group, which means that there are at least 5421 symmetric (45,12,3) designs. Further, we discuss trigeodetic graphs obtained from the symmetric $(45,12,3)$ designs. We prove that $k$-geodetic graphs constructed from mutually non-isomorphic designs are mutually non-isomorphic, hence there are at least 5421 mutually non-isomorphic trigeodetic graphs
obtained from symmetric $(45,12,3)$ designs.

Keywords

Symmetric design Linear code Automorphism group k-geodetic graph

References

  • R. W. Ahrens, G. Szekeres, On a combinatorial generalization of the 27 lines associated with a cubic surface, J. Austral. Math. Soc. 10 (1969) 485–492.
  • E. F. Assmus Jr., J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, 1992.
  • T. Beth, D. Jungnickel, H. Lenz, Design Theory, 2nd Edition, Cambridge University Press, Cambridge, 1999.
  • W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24(3-4) (1997) 235–265.
  • V. Cepulic, On symmetric block designs (45,12,3) with automorphisms of order 5, Ars Combin. 37 (1994) 33–48.
  • C. J. Colbourn, J. F. Dinitz (Eds.), The CRC Handbook of Combinatorial Designs, 2nd Edition, CRC Press, Boca Raton, 2007.
  • K. Coolsaet, J. Degraer, E. Spence, The strongly regular (45; 12; 3; 3) graphs, Electron. J. Combin. 13(1) (2006) Research Paper 32, 1–9.
  • D. Crnkovic, B. G. Rodrigues, S. Rukavina, L. Simcic, Ternary codes from the strongly regular (45,12,3,3) graphs and orbit matrices of 2-(45,12,3) designs, Discrete Math. 312(20) (2012) 3000– 3010.
  • D. Crnkovic, S. Rukavina, Construction of block designs admitting an abelian automorphism group, Metrika 62(2-3) (2005) 175–183.
  • D. Crnkovic, S. Rukavina, On some symmetric (45, 12, 3) and (40, 13, 4) designs, J. Comput. Math. Optim. 1(1) (2005) 55–63.
  • D. Crnkovic, S. Rukavina, L. Simcic, On triplanes of order twelve admitting an automorphism of order six and their binary and ternary codes, Util. Math., to appear.
  • U. Dempwolff, Primitive rank 3 groups on symmetric designs, Des. Codes Cryptogr. 22(2) (2001) 191–207.
  • C. E. Frasser, k-geodetic Graphs and Their Application to the Topological Design of Computer Networks, Argentinian Workshop in Theoretical Computer Science, 28 JAIIO-WAIT ’99 (1999) 187–
  • The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.2; 2013. (http://www.gap-system.org)
  • M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, Online available at http://www.codetables.de, Accessed on 2014-03-11.
  • Z. Janko, Coset enumeration in groups and constructions of symmetric designs, Combinatorics ’90 (Gaeta, 1990), Ann. Discrete Math. 52 (1992) 275–277.
  • P. Kaski, P. R. J. Östergård, Classification Algorithms for Codes and Designs, Springer, Berlin, 2006.
  • P. Kaski and P. R. J. Östergård, There are exactly five biplanes with k = 11, J. Combin. Des. 16(2) (2008) 117–127.
  • T. Kölmel, Einbettbarkeit symmetrischer (45,12,3) Blockplaene mit fixpunktfrei operierenden Automorphismen, Heidelberg, 1991.
  • C. W. H. Lam, G. Kolesova, L. H. Thiel, A computer search for finite projective planes of order 9, Discrete Math. 92(1-3) (1991) 187–195.
  • E. Lander, Symmetric Designs: An Algebraic Approach, Cambridge University Press, Cambridge, 1983.
  • R. Mathon, A. Rosa, 2-(v; k; ) Designs of Small Order, in: Handbook of Combinatorial Designs, 2nd ed., C.J. Colbourn, J.H. Dinitz (Editors), Chapman & Hall/CRC, Boca Raton (2007) 25–58.
  • R. L. MacFarland, A family of difference sets in non-cyclic groups, J. Combin. Theory Ser. A 15 (1973) 1–10.
  • V. Mandekic-Botteri, On symmetric block designs (45,12,3) with involutory automorphism fixing 15 points, Glas. Mat. Ser. III 36(56)(2) (2001) 193–222.
  • R. Mathon, E. Spence, On 2-(45,12,3) designs, J. Combin. Des. 4(3) (1996) 155–175.
  • MinT, Dept. of Mathematics, University of Salzburg, The online database for optimal parameters of (t,m,s)-nets, (t,s)-sequences, orthogonal arrays, linear codes, and OOAs, Online available at http://mint.sbg.ac.at/index.php, Accessed on 2014-03-11.
  • R. M. Ramos, J. Sicilia, M. T. Ramos, A generalization of geodetic graphs: K-geodetic graphs, Investigacion Operativa 6 (1998) 85–101.
  • L. Rudolph, A class of majority logic decodable codes, IEEE Trans. Inform. Theory 13(2) (1967) 305–307.
  • N. Srinivasan, J. Opatrný, V. S. Alagar, Construction of geodetic and bigeodetic blocks of connectivity k 3 and their relation to block designs, Ars Combin. 24 (1987) 101–114.
  • L.H. Soicher, DESIGN - a GAP package, Version 1.6, 23/11/2011.(http://www.gapsystem.org/Packages/design.html)
There are 30 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Dean Crnkovic This is me

Doris Dumicic Danilovic This is me

Sanja Rukavina This is me

Publication Date August 9, 2016
Published in Issue Year 2016

Cite

APA Crnkovic, D., Danilovic, D. D., & Rukavina, S. (2016). Enumeration of symmetric (45,12,3) designs with nontrivial automorphisms. Journal of Algebra Combinatorics Discrete Structures and Applications, 3(3), 145-154. https://doi.org/10.13069/jacodesmath.29560
AMA Crnkovic D, Danilovic DD, Rukavina S. Enumeration of symmetric (45,12,3) designs with nontrivial automorphisms. Journal of Algebra Combinatorics Discrete Structures and Applications. August 2016;3(3):145-154. doi:10.13069/jacodesmath.29560
Chicago Crnkovic, Dean, Doris Dumicic Danilovic, and Sanja Rukavina. “Enumeration of Symmetric (45,12,3) Designs With Nontrivial Automorphisms”. Journal of Algebra Combinatorics Discrete Structures and Applications 3, no. 3 (August 2016): 145-54. https://doi.org/10.13069/jacodesmath.29560.
EndNote Crnkovic D, Danilovic DD, Rukavina S (August 1, 2016) Enumeration of symmetric (45,12,3) designs with nontrivial automorphisms. Journal of Algebra Combinatorics Discrete Structures and Applications 3 3 145–154.
IEEE D. Crnkovic, D. D. Danilovic, and S. Rukavina, “Enumeration of symmetric (45,12,3) designs with nontrivial automorphisms”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 3, no. 3, pp. 145–154, 2016, doi: 10.13069/jacodesmath.29560.
ISNAD Crnkovic, Dean et al. “Enumeration of Symmetric (45,12,3) Designs With Nontrivial Automorphisms”. Journal of Algebra Combinatorics Discrete Structures and Applications 3/3 (August 2016), 145-154. https://doi.org/10.13069/jacodesmath.29560.
JAMA Crnkovic D, Danilovic DD, Rukavina S. Enumeration of symmetric (45,12,3) designs with nontrivial automorphisms. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016;3:145–154.
MLA Crnkovic, Dean et al. “Enumeration of Symmetric (45,12,3) Designs With Nontrivial Automorphisms”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 3, no. 3, 2016, pp. 145-54, doi:10.13069/jacodesmath.29560.
Vancouver Crnkovic D, Danilovic DD, Rukavina S. Enumeration of symmetric (45,12,3) designs with nontrivial automorphisms. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016;3(3):145-54.