Convolutional codes with additional algebraic structure

López-Permouth, Sergio R.; Szabo, Steve

doi:10.1016/j.jpaa.2012.09.017

Cited by 22 publications

(34 citation statements)

References 9 publications

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“…If such separability element is computed, then it allows us to calculate a generating idempotent of any ideal code of the extension [5,6]. Additionally, in this case, we give a positive answer to the question posed in [13]: is any ideal code a direct summand, as a left ideal, of the working algebra A[z; σ]?…”

Section: Algorithm 1 Separable Automorphismmentioning

confidence: 98%

“…The elements of A[z; σ] are polynomials in z with coefficients on the right, and the multiplication is derived from the rule az = zσ(a) for all a ∈ A. The action given by left multiplication of F[z] on A[z; σ] makes A[z; σ] a free F[z]-module of rank m. Recall from [13] that an ideal code is a left ideal I ≤ A[z; σ] such that I is a direct summand of A[z; σ] as an F[z]-module. So, ideal codes are convolutional codes.…”

mentioning

confidence: 99%

“…So, ideal codes are convolutional codes. This additional algebraic structure has been studied from the perspective of cyclic convolutional codes in [3,13,6], where the separability hypothesis on the ground algebra A plays, implicitly in [3,13], a prominent role.…”

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confidence: 99%

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Separable Automorphisms on Matrix Algebras over Finite Field Extensions

Gómez-Torrecillas

Lobillo

Navarro

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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Let F ⊆ K an extension of finite fields and A = Mn(K) be the ring of square matrices of order n over K viewed as an algebra over F. Given an F-automorphism σ on A, the Ore extension A[z; σ] may be used to built certain convolutional codes, namely, the ideal codes. We provide an algorithm to decide if the automorphism σ on A is a separable returning the corresponding separability element p. In this case p is also a separability element for the extension F[z] ⊆ A[z; σ], and as a consequence ideal codes are generated by idempotents in A[z; σ], which can be computed applying previous algorithms of the authors.

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Section: Algorithm 1 Separable Automorphismmentioning

confidence: 98%

mentioning

confidence: 99%

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Separable Automorphisms on Matrix Algebras over Finite Field Extensions

Gómez-Torrecillas

Lobillo

Navarro

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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“…Remark 10. If A = FG is the group algebra of a finite group G, and θ is the involution defined on G as θ(g) = g −1 , then Proposition 9 gives [25,Proposition 4.18] in the case where the skew derivation used in [25] is zero. Remark 12.…”

Section: 1mentioning

confidence: 99%

Dual skew codes from annihilators: Transpose Hamming ring extensions

Gómez-Torrecillas¹,

Lobillo²,

Navarro³

2019

Contemporary Mathematics

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“…The usual applications to physics are focused in algebras over complex or real numbers. On the other hand, in [15], Piret used skew polynomials over finite fields to introduce Cyclic Convolutional Codes, whose algebraic structure is deeply studied in [6,7], and extended in [13]. This connection with Coding Theory suggests that new algorithms over skew polynomials should be developed.…”

Section: Introductionmentioning

confidence: 99%

On isomorphisms of modules over non-commutative PID

Gómez-Torrecillas

Lobillo

Navarro

2014

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation

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Let R be an Ore extension of a skew-field. A basic computational problem is to decide effectively whether two given Ore polynomials f, g ∈ R (of the same degree) are similar, that is, if there exists an isomorphism of left R-modules between R/Rf and R/Rg. Since these modules are of finite length, we consider the more general problem of deciding when two given left R-modules of finite length are isomorphic. We show that if R is free of finite rank as a module over its center C, then this problem can be reduced to check the existence of an isomorphism of C-modules. This method works for a large class of left R-modules of finite length. Our result is proven in the realm of non-commutative principal ideal domains, and generalizes a result by Jacobson for some Ore extensions of a skew field by an automorphism. As a consequence, we propose an algorithm to check whether two given left R-modules of finite length are isomorphic by associating a matrix with coefficients in C to each of the modules, and checking if the corresponding rational canonical forms are equal. Our method is illustrated with examples of computations for Ore extensions of finite fields, and of the Hamilton quaternions.

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Convolutional codes with additional algebraic structure

Cited by 22 publications

References 9 publications

Separable Automorphisms on Matrix Algebras over Finite Field Extensions

Separable Automorphisms on Matrix Algebras over Finite Field Extensions

Dual skew codes from annihilators: Transpose Hamming ring extensions

On isomorphisms of modules over non-commutative PID

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