Parametric Solvable Polynomial Rings and Applications

Kredel, Heinz

doi:10.1007/978-3-319-24021-3_21

Cited by 14 publications

(11 citation statements)

References 14 publications

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“…Skew-polynomial rings and their ideals have been already used in other applications and when generalizing other classical notions like Gröbner bases [3] to a non-commutative setting, e.g. see [14,16,31,32,34,35,44,45,58], where they appear as examples of solvable polynomial rings, operator theory [26], and other codes, in particular (cyclic) convolutional codes and MDS codes cf. [21,22,24,25,[38][39][40].…”

Section: Nonassociative Algebra S F = S[t; σ δ]/S[t; σ δ] F Is Defimentioning

confidence: 99%

How to obtain lattices from $$(f,\sigma ,\delta )$$ ( f , σ , δ ) -codes via a generalization of Construction A

Pumpluen

2017

AAECC

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We show how cyclic ( f, σ, δ)-codes over finite rings canonically induce a Zlattice in 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 coding. Previous results by Ducoat and Oggier are obtained as special cases.

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Section: Nonassociative Algebra S F = S[t; σ δ]/S[t; σ δ] F Is Defimentioning

confidence: 99%

How to obtain lattices from $$(f,\sigma ,\delta )$$ ( f , σ , δ ) -codes via a generalization of Construction A

Pumpluen

2017

AAECC

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“…Note that ⋄ is commutative when A is a twisted monoid ring R[S] over a commutative ring R and a commutative monoid S, as polynomial rings, solvable polynomial rings [15,16], [26,IV.49.5], multivariate Ore extensions [30,31,8,9] . .…”

Section: Definition 14mentioning

confidence: 99%

Buchberger–Weispfenning theory for effective associative rings

Ceria¹,

Mora²

2017

Journal of Symbolic Computation

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“…Further we plan to implement comprehensive Gröbner bases for parametric solvable polynomial rings [13]. There are also many opportunities for utilizing parallelism, see [3] and [10] for a start.…”

Section: Examples and Gröbner Bases Over Regular Ringsmentioning

confidence: 99%

Comprehensive Gröbner Bases in a Java Computer Algebra System

Kredel

2014

Computer Mathematics

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We present an implementation of the algorithms for computing comprehensive Gröbner bases in a Java computer algebra system (JAS). Contrary to approaches to implement comprehensive Gröbner bases with minimal requirements to the computer algebra system, we aim to provide and utilize all necessary algebraic structures occurring in the algorithm. In the implementation of a condition we aim at the maximal semantic exploitation of the occurring algebraic structures: the set of equations equal zero are implemented as an ideal (with Gröbner base computation) and the set of inequalities are implemented as a multiplicative set which is simplified to polynomials of minimal degrees using, for example, square-free decomposition. With our approach we can also make the transition of a comprehensive Gröbner system to a polynomial ring over a (commutative, finite, von Neuman) regular coefficient ring and test or compute Gröbner bases in such polynomial rings.

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Parametric Solvable Polynomial Rings and Applications

Cited by 14 publications

References 14 publications

How to obtain lattices from $$(f,\sigma ,\delta )$$ ( f , σ , δ ) -codes via a generalization of Construction A

How to obtain lattices from $$(f,\sigma ,\delta )$$ ( f , σ , δ ) -codes via a generalization of Construction A

Buchberger–Weispfenning theory for effective associative rings

Comprehensive Gröbner Bases in a Java Computer Algebra System

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