Skew codes of prescribed distance or rank

In this paper we propose two methods to produce block codes of prescribed rank or distance. Following [4, 5] we work with skew polynomial rings of automorphism type and the codes we investigate are ideals in quotients of this ring. There is a strong connection with linear difference operators and with linearized polynomials (or q-polynomials) reviewed in the first section. 1 Galois theory of difference equations over finite fields A finite difference field (Fq, θ) is a field together with an automorphism θ. A difference (or recurrence) equation over (Fq, θ) is an equation of the form L(y) = an θ (y) + . . . + a1θ(y) + a0 y = 0 Let q = p and θ(x) = x i with i in {0, ..., r − 2}. A finite difference field (Fqs ,Θ) is a difference field extension of (Fq, θ) if Fq ⊆ Fqs and Θ defined by Θ(x) = x i is an extension of θ to an automorphism of Fqs . Note that even if there are several ways to extend a field automorphism, we keep the expression θ for the extension Θ. A solution of the difference equation L(y) = 0 is an element β in a finite difference field extension of (Fq, θ) such that L(β) = 0. We call (Fq) the field of constants. The solution space of the difference equation L(y) = 0 is a vector space over (Fq) of dimension ≤ n. There is a difference Galois theory of difference rings [2, 14] where the existence of a difference splitting ring (of PicardVessiot ring) is proven under the assumption that the field of constants is algebraically closed. In our special situation of a finite coefficient field we do not want to work with an algebraically closed field of constants and we will show that if a0 6= 0, then a finite PV-field always exists. In connection with coding theory the equivalent notion of p-polynomial or linearized polynomial is more common ([10, 13]). In this section we recall the basic facts and connections between those notions. ∗IRMAR (UMR 6625), Université de Rennes 1, Campus de Beaulieu, F-35042 Rennes Cedex †IRMAR (UMR 6625) et DGA/CELAR, La Roche Marguerite, 35174 Bruz Cedex, France ‡IRMAR (UMR 6625), Université de Rennes 1, Campus de Beaulieu, F-35042 Rennes Cedex 1 ha l-0 02 67 03 4, v er si on 1 26 M ar 2 00 8 Author manuscript, published in "Designs Codes and Cryptography / Designs Codes and Cryptography An International Journal 50, 3 (2009) 267-284" DOI : 10.1007/s10623-008-9230-6