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. INTRODUCTIONThe main reason why cyclic block codes are useful is that it is possible to exploit the ring structure of their word-ambient space to get a better control of the parameters of the code, and to design efficient decoding algorithms. A classical example is the procedure developed by Sugiyama, Kasahara, Hirasawa and Namekawa [18] for nearest neighbor decoding of BCH codes. Commonly known as Sugiyama Algorithm, it is a variation of the decoding scheme proposed by Peterson [13] and, Gorenstein and Zierler [6], which computes the error positions of a received polynomial by a clever use of the extended Euclidean algorithm.When dealing with convolutional codes, the Viterbi algorithm is, by far, the most often used for decoding convolutional codes over binary symmetric or additive white Gaussian noise channels. It makes use of the trellis structure of these codes in order to find the shortest path and return a maximum-likelihood estimation by means of hard and soft decission schemes. Tomás, Rosenthal and Smarandache [19] use large finite windows in the infinite sliding generating matrix associated to convolutional codes to design a decoding algorithm over the erasure channel. It is known that endowing convolutional codes with a cyclic structure requires of a non-commutative multiplication [14]. However, the different proposals of cyclic convolutional codes in the literature seem to have failed to take advantage of their algebraic structure for finding efficient and practical decoding algorithms, aiming to provide an alternative to the Viterbi algorithm. This is probably due to the fact that the non-commutative polynomial rings used in this classical approach [4,14,15] are more complicated than expected. In particular, no Euclidean division algorithm is available here. In [5] we proposed a simpler approach that follows the idea of Piret [14] and Roos [15] of using a non-commutative multiplication, but implements it with a different algebraic construction. Thus, a skew cyclic convolutional code (SCCC) becomes a left ideal, whose generator is expressed in a adequate way, of a suitable factor ring of a skew polynomial ring with coefficients in a rational function field. This paper is the natural continuation of [5]. We mainly attempt to show more solid evidences of the great potential of this notion, and provide a decoding algorithm for a class of SCCCs. Hopefully, this could lay the foundations of a practical alternative of the Viterbi algorithm. By analogy with BCH codes, we call these codes skew BCH convolutional codes. The sim...

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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.

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. 2010 Mathematics Subject Classification. Primary .

Abstract. 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 [16], 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.