Maximal partial spreads and the modular n-queen problem III

AbstractMaximal partial spreads in PG(3,q)q=pk,p odd prime and q⩾7, are constructed for any integer n in the interval (q2+1)/2+6⩽n⩽(5q2+4q−1)/8 in the case q+1≡0,±2,±4,±6,±10,12(mod24). In all these cases, maximal partial spreads of the size (q2+1)/2+n have also been constructed for some small values of the integer n. These values depend on q and are mainly n=3 and n=4. Combining these results with previous results of the author and with that of others we can conclude that there exist maximal partial spreads in PG(3,q),q=pk where p is an odd prime and q⩾7, of size n for any integer n in the interval (q2+1)/2+6⩽n⩽q2−q+2

Linear codes meeting the Griesmer bound, minihypers and geometric applications

Coding theory and Galois geometries are two research areas which greatly influence each other. In this talk, we focus on the link between linear codes meeting the Griesmer bound and minihypers in finite projective spaces. Minihypers are particular (multiple) blocking sets. We present characterization results on minihypers, leading to equivalent characterization results on linear codes meeting the Griesmer bound. Next to being interesting from a coding-theoretical point of view, minihypers also are interesting for geometrical applications. We present results on maximal partial μ-spreads in PG(N, q), (μ + 1)|(N + 1), on minimal μ-covers in PG(N, q), (μ + 1)|(N + 1), on (N − 1)-covers of Q + (2N + 1, q), on partial ovoids and on partial spreads of finite classical polar spaces, and on partial ovoids of generalized hexagons, following from results on minihypers

Linear sections of GL(4, 2)

For V = V (n; q); a linear section of GL(V ) = GL(n; q) is a vector subspace S of the n 2 -dimensional vector space End(V ) which is contained in GL(V ) [ f0g: We pose the problem, for given (n; q); of classifying the di erent kinds of maximal linear sections of GL(n; q): If S is any linear section of GL(n; q) then dim S n: The case of GL(4; 2) is examined fully. Up to a suitable notion of equiv- alence there are just two classes of 3-dimensional maximal normalized linear sections M3;M0 3 , and three classes M4;M0 4 ;M00 4 of 4-dimensional sections. The subgroups of GL(4; 2) generated by representatives of these ve classes are respectively G3 = A7; G 0 3 = GL(4; 2); G4 = Z15; G 0 4 = Z3 A5; G 00 4 = GL(4; 2): On various occasions use is made of an isomorphism T : A8 ! GL(4; 2): In particular a representative of the class M3 is the image under T of a subset f1; ::: ; 7g of A7 with the property that 1 i j is of order 6 for all i =6 j: The classes M3;M0 3 give rise to two classes of maximal partial spreads of order 9 in PG(7; 2); and the classes M0 4 ;M00 4 yield the two isomorphism classes of proper semi eld planes of order 16

Maximal partial line spreads of non-singular quadrics

For n >= 9 , we construct maximal partial line spreads for non-singular quadrics of for every size between approximately and , for some small constants and . These results are similar to spectrum results on maximal partial line spreads in finite projective spaces by Heden, and by Gacs and SzAnyi. These results also extend spectrum results on maximal partial line spreads in the finite generalized quadrangles and by Pepe, Roing and Storme

Partial ovoids and partial spreads in symplectic and orthogonal polar spaces

We present improved lower bounds on the sizes of small maximal partial ovoids and small maximal partial spreads in the classical symplectic and orthogonal polar spaces, and improved upper bounds on the sizes of large maximal partial ovoids and large maximal partial spreads in the classical symplectic and orthogonal polar spaces. An overview of the status regarding these results is given in tables. The similar results for the hermitian classical polar spaces are presented in [J. De Beule, A. Klein, K. Metsch, L. Storme, Partial ovoids and partial spreads in hermitian polar spaces, Des. Codes Cryptogr. (in press)]

Partial ovoids and partial spreads in finite classical polar spaces

We survey the main results on ovoids and spreads, large maximal partial ovoids and large maximal partial spreads, and on small maximal partial ovoids and small maximal partial spreads in classical finite polar spaces. We also discuss the main results on the spectrum problem on maximal partial ovoids and maximal partial spreads in classical finite polar spaces

Constant rank-distance sets of hermitian matrices and partial spreads in hermitian polar spaces

In this paper we investigate partial spreads of $H(2n-1,q^2)$ through the related notion of partial spread sets of hermitian matrices, and the more general notion of constant rank-distance sets. We prove a tight upper bound on the maximum size of a linear constant rank-distance set of hermitian matrices over finite fields, and as a consequence prove the maximality of extensions of symplectic semifield spreads as partial spreads of $H(2n-1,q^2)$. We prove upper bounds for constant rank-distance sets for even rank, construct large examples of these, and construct maximal partial spreads of $H(3,q^2)$ for a range of sizes