Transformations on irreducible binary polynomials

Abstract

Using the natural action of GL2(double-struck F2) ≃ script G3 over double-struck F2[X], one can define different classes of polynomials strongly analogous to self-reciprocal irreducible polynomials. We give transformations to construct polynomials of each kind of invariance and we deal with the question of explicit infinite sequences of invariant irreducible polynomials in double-struck F 2[X]. We generalize results obtained by Varshamov, Wiedemann, Meyn and Cohen and we give sequences of invariant irreducible polynomials. Moreover we explain what happens when the given constructions fail. We also give a result on the order of the polynomials of one of the classes: the alternate irreducible polynomials. © 2010 Springer-Verlag.