Michael R. Hurley, Bal K. Khadka, Spyros S. Magliveras

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

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.

Keywords:

Geometric t-designs, Large sets of geometric t-designs, t-designs over GF(q) Parallelisms, Lattice basis reduction, LLL algorithm,

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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.