Designs on vector spaces constructed using quadratic forms |
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Authors: | D K Ray-Chaudhuri Erin J Schram |
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Institution: | (1) Department of Mathematics, The Ohio State University, 43210 Columbus, Ohio, USA |
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Abstract: | Let N be the set of nonnegative integers, let , t, v be in N and let K be a subset of N, let V be a v-dimensional vector space over the finite field GF(q), and let W
Kbe the set of subspaces of V whose dimensions belong to K. A t-v, K, ; q]-design on V is a mapping : W
K N such that for every t-dimensional subspace, T, of V, we have
(B)= . We construct t-v, {t, t+1}, ; q-designs on the vector space GF(q
v) over GF(q) for t 2, v odd, and q
t(q–1)2 equal to the number of nondegenerate quadratic forms in t+1 variables over GF(q). Moreover, the vast majority of blocks of these designs have dimension t+1. We also construct nontrivial 2-v, k, ; q]-designs for v odd and 3 k v–3 and 3-v, 4, q
6+q
5+q
4; q]-designs for v even. The distribution of subspaces in the designs is determined by the distribution of the pairs (Q, a) where Q is a nondegenerate quadratic form in k variables with coefficients in GF(q) and a is a vector with elements in GF(q
v) such that Q(a)=0.This research was partly supported by NSA grant #MDA 904-88-H-2034. |
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