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We generalize the notion of cyclic codes by using generator poly-nomials in (non commutative) skew polynomial rings. Since skew polynomial rings are left and right euclidean, the obtained codes share most properties of cyclic codes. Since there are much more skew-cyclic codes, this new class of codes allows to systematically search for codes with good… (More)

We generalize the construction of linear codes via skew polynomial rings by using Galois rings instead of finite fields as coefficients. The resulting non commutative rings are no longer left and right Euclidean. Codes that are principal ideals in quotient rings of skew polynomial rings by a two sided ideals are studied. As an application, skew constacyclic… (More)

In analogy to cyclic codes, we study linear codes over finite fields obtained from left ideals in a quotient ring of a (non commutative) skew polynomial ring. The paper shows how existence and properties of such codes are linked to arithmetic properties of skew polynomials. This class of codes is a generalization of the θ-cyclic codes discussed in [1].… (More)

In previous works we considered codes defined as ideals of quotients of non com-mutative 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… (More)

In [4], starting from an automorphism θ of a finite field F q and a skew polynomial ring R = F q [X; θ], module θ-codes are defined as left R-submodules of R/Rf where f ∈ R. In [4] it is conjectured that an Euclidean self-dual module θ-code is a θ-constacyclic code and a proof is given in the special case when the order of θ divides the length of the code.… (More)

The aim of t.lIis paper is to flccide whether a liuear tliffer-fmtial equation with polponIia1 coefficients df~pcudiug on piLI?ll~lf~tCrS has got. polymrlial solut.ioIIs. 3lore pref:isclv we want to fwust.rufX il finit.f~ set 2' of nef:fwarp ad suffi&nt dgelxilif' iUld arithmetic: codit icms sudl tllilt t,lwrc is a poly-IIoIiiinl solution if and ouly if… (More)

In this work the definition of codes as modules over skew polynomial rings of automorphism type is generalized to skew polynomial rings, whose multiplication is defined using an automorphism and a derivation. This produces a more general class of codes which, in some cases, produce better distance bounds than module skew codes constructed only with an… (More)

The construction of cyclic codes can be generalized to so called module θ-cyclic codes using noncommutative polynomials. The product of the generator polynomial g of a self-dual module θ-cyclic code and its " skew reciprocal polynomial " is known to be a noncommu-tative polynomial of the form X n − a, reducing the problem of the computation of all such… (More)

In this paper we introduce a new key exchange algorithm (Diffie-Hellman like) based on so called (non-commutative) skew poly-nomials. The algorithm performs only polynomial multiplications in a special small field and is very efficient. The security of the scheme can be interpretated in terms of solving binary quadratic equations or exhaustive search of a… (More)

Next-generation sequencing (NGS) allows faster acquisition of metagenomic data, but complete exploration of complex ecosystems is hindered by the extraordinary diversity of microorganisms. To reduce the environmental complexity, we created an innovative solution hybrid selection (SHS) method that is combined with NGS to characterize large DNA fragments… (More)