Symmetric bilinear forms over finite fields of even characteristic

Abstract

Let S m be the set of symmetric bilinear forms on an m-dimensional vector space over GF(q), where q is a power of two. A subset Y of S m is called an (m, d)-set if the difference of every two distinct elements in Y has rank at least d. Such objects are closely related to certain families of codes over Galois rings of characteristic four.An upper bound on the size of (m, d)-sets is derived, and in certain cases, the rank distance distribution of an (m, d)-set is explicitly given. Constructions of (m, d)-sets are provided for all possible values of m and d.

On endomorphisms of alternating forms graph

“…Then after an appropriate congruence transformation which keeps (18) and N ′ 6 , without loss of generality we may assume N = N ′ 5 = ϕ(M 3 ). We claim that ϕ(N ′ 6 ) is type two.…”

On the correlation distribution of Delsarte–Goethals sequences

“…In the rest of this section, we consider a particular set of Z 4 -valued quadratic forms, which has been studied by the author in [6] following earlier work in [5].…”

“…Families Q(2), Q (3), and Q (5), as defined by Helleseth and Kumar [1, p. 1832], and are the largest known designs among all binary sequence families with asymptotically the same period and maximum nontrivial correlation. Table 1 shows that Families V (0) and V (1) compare favourably to the small and the large Kasami set, respectively.…”

Automorphism groups and new constructions of maximum additive rank metric codes with restrictions

“…Regarding upper bounds for such d-codes, parts of the following results can be found in [16,Theorem 3.3] and [15,Corollary 7,Remark 8], and the last open case that q and d both even was proved in [14].…”