Matroidal structure of skew polynomial rings with application to network coding

Liu, Siyu; Manganiello, Felice; Kschischang, Frank R.

doi:10.1016/j.ffa.2017.04.007

Cited by 6 publications

(13 citation statements)

References 18 publications

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“…In [19,Section 4], the equivalence between the matroid of P-independent subsets of one conjugacy class and linearly independent sets of points in the projective space P K (F) is given for finite fields. We give a similar result in the language of lattices.…”

Section: Inner Derivationsmentioning

confidence: 99%

Skew and linearized Reed–Solomon codes and maximum sum rank distance codes over any division ring

2018

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Reed-Solomon codes and Gabidulin codes have maximum Hamming distance and maximum rank distance, respectively. A general construction using skew polynomials, called skew Reed-Solomon codes, has already been introduced in the literature. In this work, we introduce a linearized version of such codes, called linearized Reed-Solomon codes. We prove that they have maximum sum-rank distance. Such distance is of interest in multishot network coding or in singleshot multi-network coding. To prove our result, we introduce new metrics defined by skew polynomials, which we call skew metrics, we prove that skew Reed-Solomon codes have maximum skew distance, and then we translate this scenario to linearized Reed-Solomon codes and the sum-rank metric. The theories of Reed-Solomon codes and Gabidulin codes are particular cases of our theory, and the sum-rank metric extends both the Hamming and rank metrics. We develop our theory over any division ring (commutative or non-commutative field). We also consider non-zero derivations, which give new maximum rank distance codes over infinite fields not considered before.1 Interestingly, the term "sum" carries the Hamming part of the metric, and the term "rank" carries the rank part. We do not know if this popular terminology was intentional in this sense.2 One can define the right notion of erasures characterizing the sum-rank metric, and then use the bound by degrees on the sum-dimensions of zero sets of linear operator polynomials [15, Theorem 2.1].

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Section: Inner Derivationsmentioning

confidence: 99%

Skew and linearized Reed–Solomon codes and maximum sum rank distance codes over any division ring

2018

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“…This defines an equivalence relation on F q m , where two elements a, b ∈ K are σ-conjugates if there is some c ∈ K * such that a c = b. As seen in [15], 0 is conjugate only with itself, and the size of the remaining conjugacy classes is m , with q − 1 distinct conjugacy classes.…”

Section: Background On Skew Polynomial Ringsmentioning

confidence: 99%

“…We prove some important incremental results that allow us to construct another isomorphism. The first is largely a rephrasing of Lemma 1 from [15]:…”

Section: Matroid Isomorphismmentioning

confidence: 99%

“…For the other direction, the proof is essentially the same as in the right case. The key note is that d i=1 f i x q m−1 −1 is an (m − 1)-linearized polynomial, and so by Theorem 4 of [15], there are also only q d solutions to (5), so once we find d F q -linearly independent b i 's, we must have that all roots are a (q m−1 − 1)th power of some F q -linear combination of them. Proof.…”

Section: Matroid Isomorphismmentioning

confidence: 99%

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Matroidal root structure of skew polynomials over finite fields

Baumbaugh

Manganiello

2019

Journal of Discrete Mathematical Sciences and Cryptography

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A skew polynomial ring R = K[x; σ, δ] is a ring of polynomials with non-commutative multiplication. This creates a difference between left and right divisibility, and thus a concept of left and right evaluations and roots. A polynomial in such a ring may have more roots than its degree, which leads to the concepts of closures and independent sets of roots. There is also a structure of conjugacy classes on the roots. In R = Fqm [x, σ], this leads to the matroids Mr and M l of right independent and left independent sets, respectively. The matroids Mr and M l are isomorphic via the extension of the map φ : [1] → [1] defined by φ(a) = a m , where i = q i−1 −1 q−1 is a notation introduced to simplify the exponents in evaluation polynomials. Additionally, extending the field of coefficients of R results in a new skew polynomial ring S of which R is a subring, and if the extension is taken to include roots of an evaluation polynomial of f (x) (which does not depend on which side roots are being considered on), then all roots of f (x) in S are in the same conjugacy class.

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“…Matroid theory [28][29][30] is a generalization of graph and linear algebra theories. It has been used in information coding [31] and cryptology [32]. Recently, the combination between rough sets and matroids has attracted many interesting research.…”

Section: Introductionmentioning

confidence: 99%

Four Operators of Rough Sets Generalized to Matroids and a Matroidal Method for Attribute Reduction

Wang

Zhang

2018

Symmetry

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Rough sets provide a useful tool for data preprocessing during data mining. However, many algorithms related to some problems in rough sets, such as attribute reduction, are greedy ones. Matroids propose a good platform for greedy algorithms. Therefore, it is important to study the combination between rough sets and matroids. In this paper, we investigate rough sets and matroids through their operators, and provide a matroidal method for attribute reduction in information systems. Firstly, we generalize four operators of rough sets to four operators of matroids through the interior, closure, exterior and boundary axioms, respectively. Thus, there are four matroids induced by these four operators of rough sets. Then, we find that these four matroids are the same one, which implies the relationship about operators between rough sets and matroids. Secondly, a relationship about operations between matroids and rough sets is presented according to the induced matroid. Finally, the girth function of matroids is used to compute attribute reduction in information systems.

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Matroidal structure of skew polynomial rings with application to network coding

Abstract: Over a finite field F q m , the evaluation of skew polynomials is intimately related to the evaluation of linearized

Cited by 6 publications

References 18 publications

Skew and linearized Reed–Solomon codes and maximum sum rank distance codes over any division ring

Skew and linearized Reed–Solomon codes and maximum sum rank distance codes over any division ring

Matroidal root structure of skew polynomials over finite fields

Four Operators of Rough Sets Generalized to Matroids and a Matroidal Method for Attribute Reduction

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