Quotients of orders in algebras obtained from skew polynomials with applications to coding theory
Susanne Pumpluen
备注：
The title changed from "Quotients of orders in algebras obtained from skew polynomials and possible applications" to "Quotients of orders in algebras obtained from skew polynomials with applications to coding theory". This version contains some minor corrections of the previous one, mostly typos
We describe families of nonassociative finite unital rings that occur as quotients of natural nonassociative orders in generalized nonassociative cyclic division algebras over number fields. These natural orders have already been used to systematically construct fully diverse fast-decodable space-time block codes. We show how the quotients of natural orders can be employed for coset coding. Previous results by Oggier and Sethuraman involving quotients of orders in associative cyclic division algebras are obtained as special cases.
Differential graded algebra techniques have played a crucial role in the development of homological algebra, especially in the study of homological properties of commutative rings carried out by Serre, Tate, Gulliksen, Avramov, and others. In this article, we extend the construction of the Koszul complex and acyclic closure to a more general setting. As an application of our constructions, we shine some light on the structure of the Ext algebra of quotients of skew polynomial rings by ideals generated by normal elements. As a consequence, we give a presentation of the Ext algebra when the elem…
Automorphic Transcendence Degree, Growth, and Skew Normalization
Faculty of Advanced Education, Ho Chi Minh City University of Technology and Engineering（胡志明市工程技术大学高级教育学院）；Department of Computer Science, University of Bath（巴斯大学计算机系）；Analytical and Algebraic Methods in Optimization Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University（顿铎成大学数学与统计学院优化分析代数方法研究组）
Let $D$ be a finite-dimensional division algebra over its center and $R=D[t;σ,δ]$ a skew polynomial ring. Under certain assumptions on $δ$ and $σ$, the ring of central quotients $D(t;σ,δ) = \{f/g \,|\, f \in D[t;σ,δ], g \in C(D[t;σ,δ])\}$ of $D[t;σ,δ]$ is a central simple algebra with reduced norm $N$. We calculate the norm $N(f)$ for some skew polynomials $f\in R$ and investigate when and how the reducibility of $N(f)$ reflects the reducibility of $f$.
This paper constitutes a first attempt to do analysis with skew polynomials. Precisely, our main objective is to develop a theory of residues for skew rational functions (which are, by definition, the quotients of two skew polynomials). We prove in particular a skew analogue of the residue formula and a skew analogue of the classical formula of change of variables for residues.
Noether's normalization in iterated skew polynomial rings
Faculty of Advanced Education, Ho Chi Minh City University of Technology and Engineering（胡志明市理工大学与工程学院先进教育学院）；Analytical and Algebraic Methods in Optimization Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University（同大大学数学与统计学院分析与代数优化方法研究组）
本文在更一般的设定下研究非交换多项式环的诺特正则化引理，扩展了Elad Paran和Thieu N. Vo的研究，并引入了除环上的组合 Nullstellensatz 新版本。
The concepts of evaluation and interpolation are extended from univariate skew polynomials to multivariate skew polynomials, with coefficients over division rings. Iterated skew polynomial rings are in general not suitable for this purpose. Instead, multivariate skew polynomial rings are constructed in this work as follows: First, free multivariate skew polynomial rings are defined, where multiplication is additive on degrees and restricts to concatenation for monomials. This allows to define the evaluation of any skew polynomial at any point by unique remainder division. Multivariate skew pol…
In this work, free multivariate skew polynomial rings are considered, together with their quotients over ideals of skew polynomials that vanish at every point (which includes minimal multivariate skew polynomial rings). We provide a full classification of such multivariate skew polynomial rings (free or not) over finite fields. To that end, we first show that all ring morphisms from the field to the ring of square matrices are diagonalizable, and that the corresponding derivations are all inner derivations. Secondly, we show that all such multivariate skew polynomial rings over finite fields a…
Symmetric rings were introduced by Lambek to extend usual commutative ideal theory in noncommutative rings. In this paper, we study symmetric rings over which Ore extensions are symmetric. A ring R is called strongly σ-symmetric if the skew polynomial ring R[x;σ] is symmetric. We consider some properties and extensions of strongly σ-symmetric rings. Then we show the relationship between strongly σ-symmetric rings and other classes of rings. We next argue the polynomial extensions over strongly σ-symmetric rings. Moreover, we prove that if R is a σ-rigid ring, then R[x]/(x_n) is a strongly σ-sy…
Families of codes such as group codes, constacyclic and skew cyclic codes, some of which independently suggested in the literature, turn out to be special instances of the general family of crossed product codes. Hamming-metric is a main feature of ambient code spaces which is used to evaluate the efficiency of their various codes. This note aims at classifying the ambient spaces of crossed products up to Hamming-isometry. We establish a criterion for two crossed products of a group G over a base ring R to be isometric in terms of a certain G-automorphism action on the second cohomology of G w…
We provide an algebraic description for sum-rank metric codes, as quotient space of a skew polynomial ring. This approach generalizes at the same time the skew group algebra setting for rank-metric codes and the polynomial setting for codes in the Hamming metric. This allows to construct twisted linearized Reed-Solomon codes, a new family of maximum sum-rank distance codes extending at the same time Sheekey's twisted Gabidulin codes in the rank metric and twisted Reed-Solomon codes in the Hamming metric. Furthermore, we provide an analogue in the sum-rank metric of Trombetti-Zhou construction,…
The question of whether a noncommutative graded quotient singularity $A^G$ is isolated depends on a subtle invariant of the $G$-action on $A$, called the pertinency. We prove a partial dichotomy theorem for isolatedness, which applies to a family of noncommutative quotient singularities arising from a graded cyclic action on the $(-1)$-skew polynomial ring. Our results generalize and extend some results of Bao, He and the third-named author and results of Gaddis, Kirkman, Moore and Won.
We consider the algorithmic problem of computing a primitive idempotent of a central simple algebra over the field of rational functions over a finite field. The algebra is given by a set of structure constants. The problem is reduced to the computation of a division algebra Brauer equivalent to the central simple algebra. This division algebra is constructed as a cyclic algebra, once the Hasse invariants have been computed. We give an application to skew constacyclic convolutional codes.
We propose a decoding algorithm for a class of convolutional codes called skew BCH convolutional codes. These are convolutional codes of designed Hamming distance endowed with a cyclic structure yielding a left ideal of a non-commutative ring (a quotient of a skew polynomial ring). In this setting, right and left division algorithms exist, so our algorithm follows the guidelines of the Sugiyama's procedure for finding the error locator and error evaluator polynomials for BCH block codes.
We show how cyclic $(f,σ,δ)$-codes over finite rings canonically induce a $\mathbb{Z}$-lattice in $\mathbb{R}^N$ by using certain quotients of orders in nonassociative division algebras defined using the skew polynomial $f$. This construction generalizes the one using certain $σ$-constacyclic codes by Ducoat and Oggier, which used quotients of orders in non-commutative associative division algebras defined by $f$, and can be viewed as a generalization of the classical Construction A for lattices from linear codes. It has the potential to be applied to coset coding, in particular to wire-tap co…
Let $K$ be a field and $σ$ an automorphism of $K$ of order $n$.Employing a nonassociative algebra, we study the eigenspace of a bounded skew polynomial $f\in K[t;σ]$. We mainly treat the case that $K/F$ is a cyclic field extension of degree $n$ with Galois group generated by $σ$. We obtain lower bounds on the dimension of the eigenspace, and compute it in special cases as a quotient algebra. Conditions under which a monic polynomial $f\in F[t]\subset K[t;σ]$ is reducible are obtained in special cases.
We consider a skew-symmetric $n$-ary bracket on the polynomial algebra $K[x_1,\ldots,x_n,x_{n+1}]$ ($n\geq 2$) over a field $K$ of characteristic zero defined by $\{a_1,\ldots,a_n\}=J(a_1,\ldots,a_n,C)$, where $C$ is a fixed element of $K[x_1,\ldots,x_n,x_{n+1}]$ and $J$ is the Jacobian. If $n=2$ then this bracket is a Poisson bracket and if $n\geq 3$ then it is an $n$-Lie-Poisson bracket on $K[x_1,\ldots,x_n,x_{n+1}]$. We describe the center of the corresponding $n$-Lie-Poisson algebra and show that the quotient algebra $K[x_1,\ldots,x_n,x_{n+1}]/(C-λ)$, where $(C-λ)$ is the ideal generated b…
An $(n,r,h,a,q)$-Local Reconstruction Code (LRC) is a linear code over $\mathbb{F}_q$ of length $n$, whose codeword symbols are partitioned into $n/r$ local groups each of size $r$. Each local group satisfies `$a$' local parity checks to recover from `$a$' erasures in that local group and there are further $h$ global parity checks to provide fault tolerance from more global erasure patterns. Such an LRC is Maximally Recoverable (MR), if it offers the best blend of locality and global erasure resilience -- namely it can correct all erasure patterns whose recovery is information-theoretically fe…
