Research Article
BibTex RIS Cite
Year 2016, , 165 - 176, 09.08.2016
https://doi.org/10.13069/jacodesmath.40139

Abstract

References

  • A. Betten, R. Laue, A. Wassermann, Simple 7-designs with small parameters, J. Combin. Des. 7(2) (1999) 79–94.
  • M. Braun, T. Etzion, P. J. R. Östergard, A. Vardy, A. Wassermann, Existence of q-analogs of Steiner Systems, submitted, 2013.
  • M. Braun, A. Kerber, R. Laue, Systematic construction of q-analogs of t - $(v; k; lambda)$-designs, Des. Codes Cryptogr. 34(1) (2005) 55–70.
  • M. Braun, A. Kohnert, P. R. J. Östergard, A. Wassermann, Large sets of t-designs over finite fields, J. Combin. Theory Ser. A 124 (2014) 195–202.
  • P. J. Cameron, Generalisation of Fisher’s inequality to fields with more than one element, Lond. Math. Soc. Lecture Note Ser. 13 (1974) 9–13.
  • P. J. Cameron, Locally symmetric designs, Geometriae Dedicata 3(1) (1974) 65–76.
  • P. Delsarte, Association schemes and t-designs in regular semilattices, J. Combin. Theory Ser. A 20(2) (1976) 230–243.
  • A. Fazeli, S. Lovett, A. Vardy, Nontrivial t-designs over finite fields exist for all t, J. Combin. Theory Ser. A 127 (2014) 149–160.
  • T. Etzion, A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inf. Theory 57(2) (2011) 1165–1173.
  • T. Itoh, A new family of 2-designs over GF(q) admitting $SL_m(q^l)$, Geometriae Dedicata 69(3) (1998) 261–286.
  • R. Koetter, F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inf. Theory 54(8) (2008) 3579–3591.
  • E. S. Kramer, D. M. Mesner, t-designs on hypergraphs, Discrete Math. 15(3) (1976) 263–296.
  • E. S. Kramer, D. W. Leavitt, S. S. Magliveras, Construction procedures for t-designs and the existence of new simple 6-designs, Ann. Discrete Math. 26 (1985) 247–274.
  • G. Kuperberg, S. Lovett, R. Peled, Probabilistic existence of regular combinatorial structures, arXiv:1302.4295v2.
  • D. L. Kreher, D. R. Stinson, Combinatorial Algorithms : generation, enumeration and search, CRC Press, vol. 7, 1998.
  • R. Laue, S. S. Magliveras, A. Wassermann, New large sets of t-designs, J. Combin. Des. 9(1) (2001) 40–59.
  • M. Miyakawa, A. Munemasa, S. Yoshiara, On a class of small 2-designs over GF(q), J. Combin. Des. 3(1) (1995) 61–77.
  • D. K. Raychaudhuri, E. J. Schram, A large set of designs on vector spaces, J. Number Theory 47(3) (1994) 247–272.
  • H. Suzuki, 2-Designs over GF(q), Graphs Combin. 8(4) (1992) 381–389.
  • S. Thomas, Designs over finite fields, Geom. Dedicata 24(2) (1987) 237–242.

Some new large sets of geometric designs of type LS[3][2, 3, 2 8 ]

Year 2016, , 165 - 176, 09.08.2016
https://doi.org/10.13069/jacodesmath.40139

Abstract

Let $V$ be an  $n$-dimensional vector space over $\F_q$. By a {\textit {geometric}} $t$-$[q^n,k,\lambda]$ design we mean a collection $\mathcal{D}$ of $k$-dimensional subspaces of $V$, called blocks, such that every $t$-dimensional subspace $T$ of $V$ appears in exactly $\lambda$ blocks in $\mathcal{D}.$ A {\it large set}, LS[N]$[t,k,q^n]$, of
geometric designs, is a collection of N $t$-$[q^n,k,\lambda]$ designs which partitions the
collection $V \brack k$ of all $k$-dimensional subspaces of $V$.
Prior to recent article [4] only large sets of geometric 1-designs were known to exist. However in [4] M. Braun, A. Kohnert, P. \"{O}stergard, and A. Wasserman constructed the world's first large set of geometric 2-designs, namely an LS[3][2,3,$2^8$], invariant under a Singer subgroup in $GL_8(2)$. In this work we construct an additional 9 distinct, large sets LS[3][2,3,$2^8$], with the help of lattice basis-reduction.

References

  • A. Betten, R. Laue, A. Wassermann, Simple 7-designs with small parameters, J. Combin. Des. 7(2) (1999) 79–94.
  • M. Braun, T. Etzion, P. J. R. Östergard, A. Vardy, A. Wassermann, Existence of q-analogs of Steiner Systems, submitted, 2013.
  • M. Braun, A. Kerber, R. Laue, Systematic construction of q-analogs of t - $(v; k; lambda)$-designs, Des. Codes Cryptogr. 34(1) (2005) 55–70.
  • M. Braun, A. Kohnert, P. R. J. Östergard, A. Wassermann, Large sets of t-designs over finite fields, J. Combin. Theory Ser. A 124 (2014) 195–202.
  • P. J. Cameron, Generalisation of Fisher’s inequality to fields with more than one element, Lond. Math. Soc. Lecture Note Ser. 13 (1974) 9–13.
  • P. J. Cameron, Locally symmetric designs, Geometriae Dedicata 3(1) (1974) 65–76.
  • P. Delsarte, Association schemes and t-designs in regular semilattices, J. Combin. Theory Ser. A 20(2) (1976) 230–243.
  • A. Fazeli, S. Lovett, A. Vardy, Nontrivial t-designs over finite fields exist for all t, J. Combin. Theory Ser. A 127 (2014) 149–160.
  • T. Etzion, A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inf. Theory 57(2) (2011) 1165–1173.
  • T. Itoh, A new family of 2-designs over GF(q) admitting $SL_m(q^l)$, Geometriae Dedicata 69(3) (1998) 261–286.
  • R. Koetter, F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inf. Theory 54(8) (2008) 3579–3591.
  • E. S. Kramer, D. M. Mesner, t-designs on hypergraphs, Discrete Math. 15(3) (1976) 263–296.
  • E. S. Kramer, D. W. Leavitt, S. S. Magliveras, Construction procedures for t-designs and the existence of new simple 6-designs, Ann. Discrete Math. 26 (1985) 247–274.
  • G. Kuperberg, S. Lovett, R. Peled, Probabilistic existence of regular combinatorial structures, arXiv:1302.4295v2.
  • D. L. Kreher, D. R. Stinson, Combinatorial Algorithms : generation, enumeration and search, CRC Press, vol. 7, 1998.
  • R. Laue, S. S. Magliveras, A. Wassermann, New large sets of t-designs, J. Combin. Des. 9(1) (2001) 40–59.
  • M. Miyakawa, A. Munemasa, S. Yoshiara, On a class of small 2-designs over GF(q), J. Combin. Des. 3(1) (1995) 61–77.
  • D. K. Raychaudhuri, E. J. Schram, A large set of designs on vector spaces, J. Number Theory 47(3) (1994) 247–272.
  • H. Suzuki, 2-Designs over GF(q), Graphs Combin. 8(4) (1992) 381–389.
  • S. Thomas, Designs over finite fields, Geom. Dedicata 24(2) (1987) 237–242.
There are 20 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Michael R. Hurley This is me

Bal K. Khadka This is me

Spyros S. Magliveras This is me

Publication Date August 9, 2016
Published in Issue Year 2016

Cite

APA Hurley, M. R., Khadka, B. K., & Magliveras, S. S. (2016). Some new large sets of geometric designs of type LS[3][2, 3, 2 8 ]. Journal of Algebra Combinatorics Discrete Structures and Applications, 3(3), 165-176. https://doi.org/10.13069/jacodesmath.40139
AMA Hurley MR, Khadka BK, Magliveras SS. Some new large sets of geometric designs of type LS[3][2, 3, 2 8 ]. Journal of Algebra Combinatorics Discrete Structures and Applications. August 2016;3(3):165-176. doi:10.13069/jacodesmath.40139
Chicago Hurley, Michael R., Bal K. Khadka, and Spyros S. Magliveras. “Some New Large Sets of Geometric Designs of Type LS[3][2, 3, 2 8 ]”. Journal of Algebra Combinatorics Discrete Structures and Applications 3, no. 3 (August 2016): 165-76. https://doi.org/10.13069/jacodesmath.40139.
EndNote Hurley MR, Khadka BK, Magliveras SS (August 1, 2016) Some new large sets of geometric designs of type LS[3][2, 3, 2 8 ]. Journal of Algebra Combinatorics Discrete Structures and Applications 3 3 165–176.
IEEE M. R. Hurley, B. K. Khadka, and S. S. Magliveras, “Some new large sets of geometric designs of type LS[3][2, 3, 2 8 ]”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 3, no. 3, pp. 165–176, 2016, doi: 10.13069/jacodesmath.40139.
ISNAD Hurley, Michael R. et al. “Some New Large Sets of Geometric Designs of Type LS[3][2, 3, 2 8 ]”. Journal of Algebra Combinatorics Discrete Structures and Applications 3/3 (August 2016), 165-176. https://doi.org/10.13069/jacodesmath.40139.
JAMA Hurley MR, Khadka BK, Magliveras SS. Some new large sets of geometric designs of type LS[3][2, 3, 2 8 ]. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016;3:165–176.
MLA Hurley, Michael R. et al. “Some New Large Sets of Geometric Designs of Type LS[3][2, 3, 2 8 ]”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 3, no. 3, 2016, pp. 165-76, doi:10.13069/jacodesmath.40139.
Vancouver Hurley MR, Khadka BK, Magliveras SS. Some new large sets of geometric designs of type LS[3][2, 3, 2 8 ]. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016;3(3):165-76.

Cited By