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Linear complementary pairs (LCPs) of codes have been studied since they were introduced in the context of discussing mitigation measures against possible hardware attacks to integrated circuits. Since the security parameters for LCPs of codes are defined from the (Hamming) distance and the dual distance of the codes in the pair, and the additional algebraic structure of skew constacyclic codes provides tools for studying the the dual and the distance of a code, we study the properties of LCPs of skew constacyclic codes. As a result, we give a characterization for those pairs, as well as multiple results that lead to constructing pairs with designed security parameters. We extend skew BCH codes to a constacyclic context and show that an LCP of codes can be immediately constructed from a skew BCH constacyclic code. Additionally, we describe a Hamming weight-preserving automorphism group in the set of skew constacyclic codes, which can be used for constructing LCPs of codes., Comment: 25 pages, 0 figures

A class of linear codes that extends classic Goppa codes to a non-commutative context is defined. An efficient decoding algorithm, based on the solution of a non-commutative key equation, is designed. We show how the parameters of these codes, when the alphabet is a finite field, may be adjusted to propose a McEliece-type cryptosystem.

The tensor product of one code endowed with the Hamming metric and one endowed with the rank metric is analyzed. This gives a code which naturally inherits the sum-rank metric. Specializing to the product of a cyclic code and a skew-cyclic code, the resulting code turns out to belong to the recently introduced family of cyclic-skew-cyclic. A group theoretical description of these codes is given, after investigating the semilinear isometries in the sum-rank metric. Finally, a generalization of the Roos and the Hartmann-Tzeng bounds for the sum rank-metric is established, as well as a new lower bound on the minimum distance of one of the two codes constituting the product code., Comment: 18 pages

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., Comment: 15 pages

Finding the minimum distance of linear codes is an NP-hard problem. Traditionally, this computation has been addressed by means of the design of algorithms that find, by a clever exhaustive search, a linear combination of some generating matrix rows that provides a codeword with minimum weight. Therefore, as the dimension of the code or the size of the underlying finite field increase, so it does exponentially the run time. In this work, we prove that, given a generating matrix, there exists a column permutation which leads to a reduced row echelon form containing a row whose weight is the code distance. This result enables the use of permutations as representation scheme, in contrast to the usual discrete representation, which makes the search of the optimum polynomial time dependent from the base field. In particular, we have implemented genetic and CHC algorithms using this representation as a proof of concept. Experimental results have been carried out employing codes over fields with two and eight elements, which suggests that evolutionary algorithms with our proposed permutation encoding are competitive with regard to existing methods in the literature. As a by-product, we have found and amended some inaccuracies in the MAGMA Computational Algebra System concerning the stored distances of some linear codes.

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., Comment: 15 pages

Differential Convolutional Codes with designed Hamming distance are defined, and an algebraic decoding algorithm, inspired by Peterson-Gorenstein-Zierler's algorithm, is designed for them., Comment: Revised version: some typos have been fixed and references updated

We give necessary and sufficient conditions on an Ore extension $A[x;\sigma,\delta]$, where $A$ is a finite dimensional algebra over a field $\mathbb{F}$, for being a Frobenius extension over the ring of commutative polynomials $\mathbb{F}[x]$. As a consequence, as the title of this paper highlights, we provide a negative answer to a problem stated by Caenepeel and Kadison.

With a small suitable modification, dropping the projectivity condition, we extend the notion of a Frobenius algebra to grant that a Frobenius algebra over a Frobenius commutative ring is itself a Frobenius ring. The modification introduced here also allows Frobenius finite rings to be precisely those rings which are Frobenius finite algebras over their characteristic subrings. From the perspective of linear codes, our work expands one's options to construct new finite Frobenius rings from old ones. We close with a discussion of generalized versions of the MacWilliams identities that may be obtained in this context.

We design a heuristic method, a genetic algorithm, for the computation of an upper bound of the minimum distance of a linear code over a finite field. By the use of the row reduced echelon form, we obtain a permutation encoding of the problem, so that its space of solutions does not depend on the size of the base field or the dimension of the code. Actually, the efficiency of our method only grows non-polynomially with respect to the length of the code.

In this paper a framework to study the dual of skew cyclic codes is proposed. The transposed Hamming ring extensions are based in the existence of an anti-isomorphism of algebras between skew polynomial rings. Our construction is applied to left ideal convolutional codes, skew constacyclic codes and skew Reed-Solomon code, showing that the dual of these codes belong to the same class.

In this paper we deal with the problem of extending Zadeh's operators on fuzzy sets (FSs) to interval-valued (IVFSs), set-valued (SVFSs) and type-2 (T2FSs) fuzzy sets. Namely, it is known that seeing FSs as SVFSs, or T2FSs, whose membership degrees are singletons is not order-preserving. We then describe a family of lattice embeddings from FSs to SVFSs. Alternatively, if the former singleton viewpoint is required, we reformulate the intersection on hesitant fuzzy sets and introduce what we have called closed-valued fuzzy sets. This new type of fuzzy sets extends standard union and intersection on FSs. In addition, it allows handling together membership degrees of different nature as, for instance, closed intervals and finite sets. Finally, all these constructions are viewed as T2FSs forming a chain of lattices.

The use of skew polynomial rings allows to endow linear codes with cyclic structures which are not cyclic in the classical (commutative) sense. Whenever these skew cyclic structures are carefully chosen, some control over the Hamming distance is gained, and it is possible to design efficient decoding algorithms. In this paper, we give a version of the Hartmann-Tzeng bound that works for a wide class of skew cyclic codes. We also provide a practical method for constructing them with designed distance. For skew BCH codes, which are covered by our constructions, we discuss decoding algorithms. Detailed examples illustrate both the theory as the constructive methods it supports.

We design a non-commutative version of the Peterson-Gorenstein-Zierler decoding algorithm for a class of codes that we call skew RS codes. These codes are left ideals of a quotient of a skew polynomial ring, which endow them of a sort of non-commutative cyclic structure. Since we work over an arbitrary field, our techniques may be applied both to linear block codes and convolutional codes. In particular, our decoding algorithm applies for block codes beyond the classical cyclic case.

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 that the effective factorization of Ore polynomials over $\mathbb{F}_q(t)$ is still an open problem. This is so because the known algorithm in [1] presents two gaps, and therefore it does not cover all the examples. We amend one of the gaps, and we discuss what kind of partial factorizations can be then computed by using [1].

This paper investigates the application of the theoretical algebraic notion of a separable ring extension, in the realm of cyclic convolutional codes or, more generally, ideal codes. We work under very mild conditions, that cover all previously known as well as new non trivial examples. It is proved that ideal codes are direct summands as left ideals of the underlying non-commutative algebra, in analogy with cyclic block codes. This implies, in particular, that they are generated by an idempotent element. Hence, by using a suitable separability element, we design an efficient algorithm for computing one of such idempotents.

We develop a fast algorithm for computing the bound of an Ore polynomial over a skew field, under mild conditions. As an application, we state a criterion for deciding whether a bounded Ore polynomial is irreducible, and we discuss a factorization algorithm. The asymptotic time complexity in the degree of the given Ore polynomial is studied. In the class of Ore polynomials over a finite field, our algorithm is an alternative to Giesbretch's one that reduces the complexity in the degree of the polynomial.

In this paper, a refinement of the weight distribution in an MDS code is computed. Concretely, the number of codewords with a fixed amount of nonzero bits in both information and redundancy parts is obtained. This refinement improves the theoretical approximation of the information-bit and -symbol error rate, in terms of the channel bit-error rate, in a block transmission through a discrete memoryless channel. Since a bounded distance reproducing encoder is assumed, the computation of the here-called false positive (a decoding failure with no information-symbol error) is provided. As a consequence, a new performance analysis of an MDS code is proposed.

In this paper we introduce prime fuzzy ideals over a noncommutative ring. This notion of primeness is equivalent to level cuts being crisp prime ideals. It also generalizes the one provided by Kumbhojkar and Bapat in [Not-so-fuzzy fuzzy ideals, Fuzzy Sets and Systems 37 (1990), 237--243], which lacks this equivalence in a noncommutative setting. Semiprime fuzzy ideals over a noncommutative ring are also defined and characterized as intersection of primes. This allows us to introduce the fuzzy prime radical and contribute to establish the basis of a Fuzzy Noncommutative Ring Theory., Comment: Section 1 has been rewritten, corrected typos, added references

While semisimple artinian rings and semisimple coalgebras over a field can be described in terms of matrices (either matrix ring over division rings or comatrix coalgebras over the ground field), semisimple corings seem to have a more intrincated structure in general. It turns out that some well-known properties of semisimple rings or coalgebras, which are immediately deduced from the aforementioned structure, are not evident over a (left) semi-simple coring. For instance, it is not evident that the notion of semi-simple coring is left-right symmetric. To be precise, if every left comodule decomposes a a direct sum of simple comodules, do the right comodules have such a decomposition? In other words, is every left semi-simple coring a right semi-simple coring? We develope the basic essentials for a theory of semi-simple corings, giving a positive answer for the last question, as well as some information about the structure of semi-simple corings.

In this paper we use refiltering methods to prove that certain types of multi-filtered algebras are Auslander-regular and Cohen-Macaulay. This is applied to obtain that the quantized enveloping algebra associated to a Cartan matrix is Auslander-regular and Cohen-Macaulay., Comment: To appear in Algebras and Representation Theory

Let $A$ be an algebra over a commutative ring $R$. If $R$ is noetherian and $A^\circ$ is pure in $R^A$, then the categories of rational left $A$-modules and right $A^\circ$-comodules are isomorphic. In the Hopf algebra case, we can also strengthen the Blattner-Montgomery duality theorem. Finally, we give sufficient conditions to get the purity of $A^\circ$ in $R^A$., Comment: AMSLaTeX with xypic