Geometric construction of some families of two-class and three-class association schemes and codes from nondegenerate and degenerate Hermitian varieties

AbstractTaking a nondegenerate Hermitian variety as a projective set in a projective plane PG(2,s2), Mesner (1967) derived a two-class association scheme on the points of the affine space of dimension 3, for which the projective plane is the plane at infinity.We generalize his construction in two ways. We show how his construction works both for nondegenerate and degenerate Hermitian varieties in any dimension.We consider a projective space of dimension N, partitioned into an affine space of dimension N and a hyperplane H of dimension N − 1 at infinity.The points of the hyperplane are next partitioned into 2 or 3 subsets. A pair of points a,b of the affine space is defined to belong to class i if the line ab meets the subset i of H.In the first case, the two subsets of the hyperplane are a nondegenerate Hermitian variety and its complement. In this case, we show that the classification of pairs of affine points defines a family of two-class association schemes. This family of association schemes has the same set of parameters as those derived as restrictions of the Hamming association schemes to two-weight codes defined as linear spans of coordinate vectors of points on a nondegenerate Hermitian variety in a projective space of dimension N − 1. The relations of these codes to orthogonal arrays and difference sets are described in [5,6].In the second case, the three subsets are the singular point of the variety, the regular points of the variety and the complement of the variety defined by a Hermitian form of rank N − 1. This leads to a family of three-class association schemes on the points of the affine space. A geometric construction is first given for the case N = 3.Using a general algebraic method pointed out by the referee, we have also derived the three-class association scheme for general N

Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets

H. Cohn et. al. proposed an association scheme of 64 points in R^{14} which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal real mutually unbiased bases. These schemes also related to extremal line-sets in Euclidean spaces and Barnes-Wall lattices. D. de Caen and E. R. van Dam constructed two infinite series of formally dual 3-class association schemes. We explain this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes.Comment: 16 page

Strongly Regular Graphs Constructed from pp-ary Bent Functions

In this paper, we generalize the construction of strongly regular graphs in [Y. Tan et al., Strongly regular graphs associated with ternary bent functions, J. Combin.Theory Ser. A (2010), 117, 668-682] from ternary bent functions to $p$-ary bent functions, where $p$ is an odd prime. We obtain strongly regular graphs with three types of parameters. Using certain non-quadratic $p$-ary bent functions, our constructions can give rise to new strongly regular graphs for small parameters.Comment: to appear in Journal of Algebraic Combinatoric

Linear Codes from Some 2-Designs

A classical method of constructing a linear code over \gf(q) with a $t$-design is to use the incidence matrix of the $t$-design as a generator matrix over \gf(q) of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of $2$-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of $2$-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterisation of highly nonlinear Boolean functions is presented