enumerating algebras over a finite field

enumerating algebras over a finite field

‎we obtain the porc formulae for the number of non-associative algebras‎ ‎of dimension 2‎, ‎3 and 4 over the finite field gf$(q)$‎. ‎we also give some‎ ‎asymptotic bounds for the number of algebras of dimension $n$ over gf$(q)$.

Enumerating all the Irreducible Polynomials over Finite Field

In this paper we give a detailed analysis of deterministic and randomized algorithms that enumerate any number of irreducible polynomials of degree n over a finite field and their roots in the extension field in quasilinear∗ time cost per element. Our algorithm is based on an improved algorithm for enumerating all the Lyndon words of length n in linear delay time and the known reduction of Lynd...

Species Over a Finite Field

We generalize Joyal’s theory of species to the case of functors from the groupoid of finite sets to the category of varieties over Fq . These have cycle index series defined by counting fixed points of twisted Frobenius maps. We give an application to configuration spaces.

Enumerating permutation polynomials over finite fields by degree II

This note is a continuation of a paper by the same authors that appeared in 2002 in the same journal. First we extend the method of the previous paper proving an asymptotic formula for the number of permutations for which the associated permutation polynomial has d coefficients in specified fixed positions equal to 0. This also applies to the function Nq,d that counts the number of permutations...

Symmetry of Algebras over a Number Field

Enumerating Permutation Polynomials over Finite Fields by Degree

fσ has the property that fσ(a) = σ(a) for every a ∈ Fq and this explains its name. For an account of the basic properties of permutation polynomials we refer to the book of Lidl and Niederreiter [5]. From the definition, it follows that for every σ ∂(fσ) ≤ q − 2. A variety of problems and questions regarding Permutation polynomials have been posed by Lidl and Mullen in [3, 4]. Among these there...