Skew-cyclic codes

We generalize the notion of cyclic codes by using generator polynomials 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 properties. We give many examples of codes which improve the previously best known linear codes

A general framework for Noetherian well ordered polynomial reductions

Polynomial reduction is one of the main tools in computational algebra with innumerable applications in many areas, both pure and applied. Since many years both the theory and an efficient design of the related algorithm have been solidly established. This paper presents a general definition of polynomial reduction structure, studies its features and highlights the aspects needed in order to grant and to efficiently test the main properties (noetherianity, confluence, ideal membership). The most significant aspect of this analysis is a negative reappraisal of the role of the notion of term order which is usually considered a central and crucial tool in the theory. In fact, as it was already established in the computer science context in relation with termination of algorithms, most of the properties can be obtained simply considering a well-founded ordering, while the classical requirement that it be preserved by multiplication is irrelevant. The last part of the paper shows how the polynomial basis concepts present in literature are interpreted in our language and their properties are consequences of the general results established in the first part of the paper.Comment: 36 pages. New title and substantial improvements to the presentation according to the comments of the reviewer

Zeros with multiplicity, Hasse derivatives and linear factors of general skew polynomials

In this work, multiplicities of zeros of general skew polynomials are studied. Two distinct definitions are considered: First, $a$ is said to be a zero of $F$ of multiplicity $r$ if $(x-a)^r$ divides $F$ on the right; second, $a$ is said to be a zero of $F$ of multiplicity $r$ if some skew polynomial $P = (x-a_r) \cdots (x-a_2) (x-a_1)$, having $a_1 = a$ as its only right zero, divides $F$ on the right. The first notion was considered earlier, while the second one was recently introduced by Bolotnikov over the quaternions. Neither of these two notions implies the other for general skew polynomials. We show that, in the first case, Lam and Leroy's concept of P-independence does not behave naturally, whereas a union theorem, as stated by Lam, still holds. In contrast, we show that P-independence for the second notion of multiplicities behaves naturally. As a consequence, we provide extensions of classical commutative results to general skew polynomials. These include the upper bound on the number of (P-independent) zeros (counting multiplicities) of a skew polynomial by its degree, and the equivalence of P-independence, the solvability of Hermite interpolation and the invertibility of confluent Vandermonde matrices (for which we introduce skew polynomial Hasse derivatives). We provide characterizations of skew polynomials of the form $P = (x-a_r) \cdots (x-a_2) (x-a_1)$ having a single right zero, assuming conjugacy classes are algebraic. Based on these, we explicitly describe such skew polynomials in particular cases of interest, recovering Bolotnikov's results over the quaternions as a particular case