In this work, we study a class of generalized quasi-cyclic (GQC) codes called skew GQC codes. By the factorization theory of ideals, we give the Chinese Remainder Theorem over the skew polynomial ring, which leads to a canonical decomposition of skew GQC codes. We also focus on some characteristics of skew GQC codes in details. For a 1-generator skew GQC code, we define the parity-check polynom...

We study two register arithmetic computation and skew arithmetic circuits. Our main results are the following: • For commutative computations, we show that an exponential circuit size lower bound for a model of 2-register straight-line programs (SLPs) which is a universal model of computation (unlike width-2 algebraic branching programs that are not universal [AW11]). • For noncommutative compu...

In this paper, we study convolutional codes with a specific cyclic structure. By definition, these codes are left ideals in a certain skew polynomial ring. Using that the skew polynomial ring is isomorphic to a matrix ring we can describe the algebraic parameters of the codes in a more accessible way. We show that the existence of such codes with given algebraic parameters can be reduced to the...

Given an iterated skew polynomial ring C[y1; τ1, δ1] . . . [yn; τn, δn] over a complete local ring C with maximal ideal m, we prove, under suitable assumptions, that the completion at the ideal m+ 〈y1, y2, . . . , yn〉 is an iterated skew power series ring. When C is a field, this completion is a local, noetherian, Auslander regular domain with Krull, classical Krull and global dimension all equ...

In this paper convolutional codes with cyclic structure will be investigated. These codes can be understood as left principal ideals in a suitable skew-polynomial ring. It has been shown in [4] that only certain combinations of the algebraic parameters (field size, length, dimension, and Forney indices) can occur for such cyclic codes. We will investigate whether all these combinations can inde...

In this paper convolutional codes with cyclic structure will be investigated. These codes can be understood as left principal ideals in a suitable skew-polynomial ring. It has been shown in [3] that only certain combinations of the parameters (field size, length, dimension, and Forney indices) can occur for cyclic codes. We will investigate whether all these combinations can indeed be realized ...

Abstract Let D be a finite-dimensional division algebra over its center and R = [ t ; ? , ? ] skew polynomial ring. Under certain assumptions on the ring of central quotients ( ) { f / g | ? ], C ])} is simple with reduced norm N . We calculate for some polynomials investigate when how reducibility reflects