Uniformly packed codes and more distance regular graphs from crooked functions

van Dam, E.R. ; Fon-Der-Flaass, D.

Journal of Algebraic Combinatorics, Tome 12 (2000) no. 2, pp. 115-121.

Résumé

Summary: Let $V$ and $W$ be $n$-dimensional vector spaces over $GF(2)$. A function $Q : V Q(0) = 0; Q( x) + Q( y) + Q( z) + Q( x + y + z) \textonesuperior 0for any three distinct x, y, z; Q( x) + Q( y) + Q( z) + Q( x + a) + Q( y + a) + Q( z + a) textonesuperior0textif a textonesuperior0text( x, y, ztextarbitrary)$. begingathered $Q(0) = 0; \hfill \\ Q(x) + Q(y) + Q(z) + Q(x + y + z) \ne 0$text for any three distinct x,y,z; $\hfill \\ Q(x) + Q(y) + Q(z) + Q(x + a) + Q(y + a) + Q(z + a) \ne 0$text if a $\ne 0$text (x,y,ztext arbitrary). $\hfill $ endgathered We show that crooked functions can be used to construct distance regular graphs with parameters of a Kasami distance regular graph, symmetric 5-class association schemes similar to those recently constructed by de Caen and van Dam from Kasami graphs, and uniformly packed codes with the same parameters as the double error-correcting BCH codes and Preparata codes.

Détail

Keywords: crooked function, distance-regular graph, association scheme, uniformly packed code