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

Generalization of a theorem of Carlitz

AbstractWe generalize Carlitzʼ result on the number of self-reciprocal monic irreducible polynomials over finite fields by showing that similar explicit formula holds for the number of irreducible polynomials obtained by a fixed quadratic transformation. Our main tools are a combinatorial argument and Hurwitz genus formula

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

Enumeration of Linear Transformation Shift Registers

We consider the problem of counting the number of linear transformation shift registers (TSRs) of a given order over a finite field. We derive explicit formulae for the number of irreducible TSRs of order two. An interesting connection between TSRs and self-reciprocal polynomials is outlined. We use this connection and our results on TSRs to deduce a theorem of Carlitz on the number of self-reciprocal irreducible monic polynomials of a given degree over a finite field.Comment: 16 page