Alternating bilinear forms over GF(q)

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The set (m, q) of alternating bilinear forms on an m-dimensional vector space over GF(q) has the combinatorial structure of an association scheme with n = [m/2] classes, in the sense of Bose and Shimamoto [5] with respect to the n + 1 relations 6 k , k = 0, 1,. . ., n, 8 k = {(A, B) a B2(m, q) I rank(A-B) = 2i}. The properties of this association scheme are first studied in Section 2. We obtain in particular an explicit expression for the eigenvalues Pk (i) of the incidence matrices of the relations B k , and show that Pk can be considered as a generalized Krawtchouk polynomial (cf. [16, p. 35]). We shall mainly be concerned with subsets Y S B(m, q), called (m, d)-sets, having the property that, for any distinct A, B e Y, rank(A-B) > 2d holds, where d is a fixed integer, 1 < d < n. The properties of these subsets are studied in Section 3. We obtain in particular an upper bound on the cardinality I Y I of an (m, d)-set, I Y I , cn-d+1, with n = [m/2], c = gm(na-l)/2n, and show that, for subsets meeting the bound, and therefore called maximal (m, d)-sets, the numbers 1 Y2 n 0 k 1, k = 1, 2,. . ., n, are uniquely defined. Finally, in Section 4, we give some general constructions of maximal (m, d)-sets, over any field for the case when m is odd, and only over fields of characteristic 2 when m is even. It is still an open question whether maximal (m, d)-sets, with m even, do exist over fields of odd characteristic. This study was motivated by some applications to coding theory of the properties of alternating bilinear forms over GF(2), which will be the subject of a forthcoming paper. Hence, our emphasis is more on the combinatorial properties of the above-mentioned association scheme than 26




Delsarte, P., & Goethals, J. M. (1975). Alternating bilinear forms over GF(q). Journal of Combinatorial Theory, Series A, 19(1), 26–50.

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