Skew-Polynomial Rings and Skew-Cyclic Codes

Gluesing-Luerssen, Heide

doi:10.48550/arxiv.1902.03516

Cited by 3 publications

(3 citation statements)

References 25 publications

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“…In this section we recall some basic notions on skew polynomial rings and skew cyclic codes. The interested reader is referred to the recent survey of Gluesing-Luerssen [21].…”

Section: Skew Cyclic Codes and Skew Reed-solomon Codesmentioning

confidence: 99%

Roos bound for skew cyclic codes in Hamming and rank metric

Alfarano,

Lobillo,

Neri

2020

Preprint

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In this paper, a Roos like bound on the minimum distance for skew cyclic codes over a general field is provided. The result holds in the Hamming metric and in the rank metric. The proofs involve arithmetic properties of skew polynomials and an analysis of the rank of parity-check matrices. For the rank metric case, a way to arithmetically construct codes with a prescribed minimum rank distance, using the skew Roos bound, is also given. Moreover, some examples of MDS codes and MRD codes over finite fields are built, using the skew Roos bound.

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“…In this section we recall some basic notions on skew polynomial rings and skew cyclic codes. The interested reader is referred to the recent survey of Gluesing-Luerssen [21].…”

Section: Skew Cyclic Codes and Skew Reed-solomon Codesmentioning

confidence: 99%

Roos bound for skew cyclic codes in Hamming and rank metric

Alfarano,

Lobillo,

Neri

2020

Preprint

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“…The novelty of the paper is concentrated in the third section, more exactly, the new results are Definition 3.3 (inspired in the notion of algebraic set given in Section 5 of [7]), Theorem 3.4, Corollary 3.5, Lemma 3.11 and Theorem 3.12 (main theorem). Some examples that illustrate the main theorem are presented in Example 3.…”

Section: Introductionmentioning

confidence: 99%

Algebraic sets, ideals of points and the Hilbert's Nullstellensatz theorem for skew PBW extensions

Lezama

2021

Preprint

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In this paper we define the algebraic sets and the ideal of points for bijective skew P BW extensions with coefficients in left Noetherian domains. Some properties of affine algebraic sets of commutative algebraic geometry will be extended, in particular, a Zariski topology will be constructed. Assuming additionally that the extension is quasi-commutative with polynomial center and the ring of coefficients is an algebraically closed field, we will prove an adapted version of the Hilbert's Nullstellensatz theorem that covers the classical one. The Gröbner bases of skew P BW extensions will be used for defining the algebraic sets and for proving the main theorem. Many key algebras and rings coming from mathematical physics and non-commutative algebraic geometry are skew P BW extensions.

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“…(iii) The previous example can be generalized in the following way (see [5], Lemma 4.1, or also [32], Proposition 3.3.15): Let q ∈ F − {0} and A := F q [x 1 , . .…”

mentioning

confidence: 99%

Noncommutative Coding Theory and Algebraic Sets for Skew PBW Extensions

2024

J Math Techniques Comput Math

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The classical commutative coding theory has been recently extended to noncommutative rings of polynomial type. There are many interesting works in coding theory over single Ore extensions. In this review article we present the most relevant algebraic tools and properties of single Ore extensions used in noncommutative coding theory. The last section represents the novelty of the paper. We will discuss the algebraic sets arising in noncommutative coding theory but for skew PBW extensions. These extensions conform a general class of noncommutative rings of polynomial type and cover several algebras arising in physics and noncommutative algebraic geometry, in particular, they cover the Ore extensions of endomorphism injective type and the polynomials rings over fields.

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Skew-Polynomial Rings and Skew-Cyclic Codes

Cited by 3 publications

References 25 publications

Roos bound for skew cyclic codes in Hamming and rank metric

Roos bound for skew cyclic codes in Hamming and rank metric

Algebraic sets, ideals of points and the Hilbert's Nullstellensatz theorem for skew PBW extensions

Noncommutative Coding Theory and Algebraic Sets for Skew PBW Extensions

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