837 research outputs found
A Characterization of Primitive Polynomials over Finite Fields
AbstractWe characterize primitive polynomials, among irreducible polynomials, by the number of terms in a certain quotient. We apply this to BCH codes of maximal designed distance
Norms of Sums of Squares
For a finite separable extension K/F of fields of characteristic not 2, the norm of a sum of 2n squares in K is a sum of 2n squares in F. We find explicit identities
Octonions
We present the basic properties of the octonions and construct the five exceptional simple Lie algebras
Invariants of Trace Forms over Finite Fields of Characteristic 2
AbstractLet K be a finite extension of F2. We compute the invariants of the quadratic form Q(x)=trK/F2(x(x2a+x2b)) and so determine the number of zeros in K. This is applied to finding the cross-correlation of certain binary sequences
Highly Degenerate Quadratic Forms over Finite Fields of Characteristic 2
Let K/F be an extension of finite fields of characteristic two. We consider quadratic forms written as the trace of xR(x), where R(x) is a linearized polynomial. We show all quadratic forms can be so written, in an essentially unique way. We classify those R, with coefficients 0 or 1, where the form has a codimension 2 radical. This is applied to maximal Artin-Schreier curves and factorizations of linearized polynomials
Norm Euclidean Quaternionic Orders
We determine the norm Euclidean orders in a positive definite quaternion algebra over Q
Gorenstein Witt Rings II
The abstract Witt rings which are Gorenstein have been classified when the dimension is one and the classification problem for those of dimension zero has been reduced to the case of socle degree three. Here we classify the Gorenstein Witt rings of fields with dimension zero and socle degree three. They are of elementary type
Trace Forms over Finite Fields of Characteristic 2 with Prescribed Invariants
AbstractWe determine the possible pairs of invariants of a trace form and construct forms with these invariants. We use this to construct new maximal Artin–Schreier curves
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