Some properties of generalized self-reciprocal polynomials over finite fields

Numerous results on self-reciprocal polynomials over finite fields have been studied. In this paper we generalize some of these to a-self reciprocal polynomials defined in [4]. We consider some properties of the divisibility of a-reciprocal polynomials and characterize the parity of the number of irreducible factors for a-self reciprocal polynomials over finite fields of odd characteristic

The parity of the number of irreducible factors for some pentanomials

AbstractIt is well known that the Stickelberger–Swan theorem is very important for determining the reducibility of polynomials over a binary field. Using this theorem the parity of the number of irreducible factors for some kinds of polynomials over a binary field, for instance, trinomials, tetranomials, self-reciprocal polynomials and so on was determined. We discuss this problem for Type II pentanomials, namely xm+xn+2+xn+1+xn+1∈F2[x] for even m. Such pentanomials can be used for the efficient implementation of multiplication in finite fields of characteristic two. Based on the computation of the discriminant of these pentanomials with integer coefficients, we will characterize the parity of the number of irreducible factors over F2 and establish necessary conditions for the existence of this kind of irreducible pentanomials.Our results have been obtained in an experimental way by computing a significant number of values with Mathematica and extracting the relevant properties

On the parity of the number of irreducible factors of self-reciprocal polynomials over finite fields

AbstractUsing the Stickelberger–Swan theorem, the parity of the number of irreducible factors of a self-reciprocal even-degree polynomial over a finite field will be hereby characterized. It will be shown that in the case of binary fields such a characterization can be presented in terms of the exponents of the monomials of the self-reciprocal polynomial

A Swan-like note for a family of binary pentanomials

In this note, we employ the techniques of Swan (Pacific J. Math. 12(3): 1099-1106, 1962) with the purpose of studying the parity of the number of the irreducible factors of the penatomial $X^n+X^{3s}+X^{2s}+X^{s}+1\in\mathbb{F}_2[X]$, where $s$ is even and $n>3s$. Our results imply that if $n \not\equiv \pm 1 \pmod{8}$, then the polynomial in question is reducible

On self-dual double circulant codes

Self-dual double circulant codes of odd dimension are shown to be dihedral in even characteristic and consta-dihedral in odd characteristic. Exact counting formulae are derived for them and used to show they contain families of codes with relative distance satisfying a modified Gilbert-Varshamov bound.Comment: 8 page

Construction of irreducible polynomials through rational transformations

Let $\mathbb F_q$ be the finite field with $q$ elements, where $q$ is a power of a prime. We discuss recursive methods for constructing irreducible polynomials over $\mathbb F_q$ of high degree using rational transformations. In particular, given a divisor $D>2$ of $q+1$ and an irreducible polynomial $f\in \mathbb F_{q}[x]$ of degree $n$ such that $n$ is even or $D\not \equiv 2\pmod 4$, we show how to obtain from $f$ a sequence $\{f_i\}_{i\ge 0}$ of irreducible polynomials over $\mathbb F_q$ with $\mathrm{deg}(f_i)=n\cdot D^{i}$.Comment: 21 pages; comments are welcome