Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras

The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be published in Lecture Notes in Computer Scienc

Solving Shift Register Problems over Skew Polynomial Rings using Module Minimisation

For many algebraic codes the main part of decoding can be reduced to a shift register synthesis problem. In this paper we present an approach for solving generalised shift register problems over skew polynomial rings which occur in error and erasure decoding of $\ell$-Interleaved Gabidulin codes. The algorithm is based on module minimisation and has time complexity $O(\ell \mu^2)$ where $\mu$ measures the size of the input problem.Comment: 10 pages, submitted to WCC 201

Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes

Let S be a unital ring, S[t; σ, δ] a skew polynomial ring where σ is an injective endomorphism and δ a left σ -derivation, and suppose f ε S[t; σ, δ] has degree m and an invertible leading coefficient. Using right division by f to define the multiplication, we obtain unital nonassociative algebras Sf on the set of skew polynomials in S[t; σ, δ] of degree less than m. We study the structure of these algebras. When S is a Galois ring and f base irreducible, these algebras yield families of finite unital nonassociative rings A, whose set of (left or right) zero divisors has the form pA for some prime p. For reducible f, the Sf can be employed both to design linear (f, σ, δ)-codes over unital rings and to study their behaviour

Semifields from skew polynomial rings

Skew polynomial rings were used to construct finite semifields by Petit in 1966, following from a construction of Ore and Jacobson of associative division algebras. In 1989 Jha and Johnson constructed the so-called cyclic semifields, obtained using irreducible semilinear transformations. In this work we show that these two constructions in fact lead to isotopic semifields, show how the skew polynomial construction can be used to calculate the nuclei more easily, and provide an upper bound for the number of isotopism classes, improving the bounds obtained by Kantor and Liebler in 2008 and implicitly in recent work by Dempwolff