Association schemes of symmetric matrices over a finite field of characteristic two

Throughout this paper, we let q = 2W,﹜ w a positive integer, and for u = 1 or 2, we let GF(qu) denote the finite field of cardinality qu. Let - denote the involutory field automorphism of GF(q2) with GF(q) as fixed subfield, where ā = aQ for all a in GF﹛q2). Moreover, let | | denote the norm (multiplicative group homomorphism) mapping of GF(q2) onto GF(q), where |a| — a • ā = aQ+1.

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Bivectors over a Finite Field

Canadian Mathematical Bulletin ◽

10.4153/cmb-1981-073-8 ◽

1981 ◽

Vol 24 (4) ◽

pp. 489-490

Author(s):

J. A. MacDougall

Keyword(s):

Finite Field ◽

Vector Space ◽

Symmetric Matrices ◽

Dimensional Vector ◽

Dimensional Vector Space

AbstractLet U be an n -dimensional vector space over a finite field of q elements. The number of elements of Λ2U of each irreducible length is found using the isomorphism of Λ2U with Hn, the space of n x n skew-symmetric matrices, and results due to Carlitz and MacWilliams on the number of skew-symmetric matrices of any given rank.

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K3 of truncated polynomial rings over fields of characteristic two

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100066871 ◽

1987 ◽

Vol 101 (3) ◽

pp. 509-521 ◽

Cited By ~ 1

Author(s):

Janet Aisbett ◽

Victor Snaith

Keyword(s):

Finite Field ◽

Polynomial Ring ◽

Polynomial Rings ◽

Witt Vectors ◽

Characteristic Two

Write F for the finite field, , having 2m elements. Let W2(F) denote the Witt vectors of length two over F (for a definition, see [4] or [10], §10). Write F(q) for the truncated polynomial ring, F[t]/(tq).

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Pairs of Quadratic Forms over Finite Fields

The Electronic Journal of Combinatorics ◽

10.37236/4072 ◽

2016 ◽

Vol 23 (2) ◽

Cited By ~ 2

Author(s):

Alexander Pott ◽

Kai-Uwe Schmidt ◽

Yue Zhou

Keyword(s):

Finite Field ◽

Finite Fields ◽

Boolean Functions ◽

Quadratic Forms ◽

Symmetric Matrices ◽

Explicit Expressions

Let $\mathbb{F}_q$ be a finite field with $q$ elements and let $X$ be a set of matrices over $\mathbb{F}_q$. The main results of this paper are explicit expressions for the number of pairs $(A,B)$ of matrices in $X$ such that $A$ has rank $r$, $B$ has rank $s$, and $A+B$ has rank $k$ in the cases that (i) $X$ is the set of alternating matrices over $\mathbb{F}_q$ and (ii) $X$ is the set of symmetric matrices over $\mathbb{F}_q$ for odd $q$. Our motivation to study these sets comes from their relationships to quadratic forms. As one application, we obtain the number of quadratic Boolean functions that are simultaneously bent and negabent, which solves a problem due to Parker and Pott.

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