Codes as Modules over Skew Polynomial Rings

In previous works we considered codes defined as ideals of quotients of non commutative polynomial rings, so called Ore rings of automorphism type. In this paper we consider codes defined as modules over non commutative polynomial rings, removing therefore some of the constraints on the length of the codes defined as ideals. The notion of BCH codes can be extended to this new approach and the codes whose duals are also defined as modules can be characterized. We show that under some restriction, self dual module codes must be constacyclic ideal codes and found two non equivalent Euclidean self-dual [56, 28, 15]4 codes which improve the best previously known distance 14 for self-dual codes of this length over F4. 1 Coding with skew polynomial rings Starting from the finite field Fq and an automorphism θ of Fq, we define a ring structure on the set: R = Fq[X, θ] = {anX n + . . .+ a1X + a0 | ai ∈ Fq and n ∈ N} . The addition in R is defined to be the usual addition of polynomials and the multiplication is defined by the basic rule X a = θ(a)X (a ∈ Fq) and extended to all elements of R by associativity and distributivity (cf. [1, 7, 8]). The ring R is a left and right Euclidean ring whose left and right ideals are principal [8]. In the following we denote Fq ⊂ Fq the fixed field of θ. 1.1 Ideal θ-codes In [4] we defined codes as ideals of quotient rings of R. If I = (f) is a two sided ideal of R, then, in analogy to classical cyclic codes, we associate to an element a(X) = an−1X n−1 + . . .+ a1X + a0 in R/(f) the ‘word‘ a = (a0, a1, . . . , an−1) ∈ Fq . 1 Definition 1 (cf. [4]) Let f ∈ R be of degree n. If I = (f) is a two sided ideal of R, then an ideal θ-code C is a left ideal (g)/(f) ⊂ R/(f), where g ∈ R is a right divisor of f in R. If the order of θ divides n then, 1. If f = X+ c with c ∈ Fq, then we call the ideal θ-code corresponding to the left ideal (g)/(X + c) ⊂ R/(X + c) an ideal θ-constacyclic code. 2. If f = X − 1 , then we call the ideal θ-code corresponding to the left ideal (g)/(X − 1) ⊂ R/(X − 1) an ideal θ-cyclic code. An ideal θ-cyclic code C has the following property ([3], Theorem 1) (a0, a1, . . . , an−1) ∈ C ⇒ (θ(an−1), θ(a0), θ(a1), . . . , θ(an−2)) ∈ C. If θ is not the identity, then the non commutative ring R is not a unique factorisation ring and there are much more right factors of f ∈ R than in the commutative case, leading to a huge number of linear codes that are not cyclic codes (cf. [3, 4]). Example. Let α be a generator of the multiplicative group of F4 and θ the Frobenius automorphism given by θ(a) = a. The polynomial X + α X + α is a right divisor of X − 1 ∈ F4[X, θ] so it generates a [4, 2]4 ideal θ-cyclic code. Note that there are seven different monic right factors of degree two of X − 1 in F4[X, θ] ([3], Example 2). In order to generate a two sided ideal of R, a monic polynomial f must be of the form X t f̃ where f̃ is a monic polynomial belonging to the center Fq[X ] of R, where m is the order of θ. If f is in the center of R, then we call the ideal θ-code, corresponding to the left ideal (g)/(f) ⊂ R/(f), an ideal θ-central code (cf [4]). The length of an ideal θ-code is determined by the degree of f , while the code itself is given by the generator matrix