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Cyclic Convolutional Codes over Separable Extensions
Abstract
We show that, under mild conditions of separability, an ideal code, as defined in Lopez-Permouth and Szabo (J Pure Appl Algebra 217(5):958–972, 2013), is a direct summand of an Ore extension and, consequently, it is generated by an idempotent element. We also design an algorithm for computing one of these idempotents.
... This in particular would imply that the code is generated by an idempotent. A sufficient condition appears in [4,5], where it is shown that, if there is a certain separability element for the ring extension F[z] ⊆ A[z, σ], then every ideal code is a direct summand as a left ideal, and an algorithm to compute explicitly the generating idempotent of each ideal code is designed. ...
... In this section we consider the following problem: When is every ideal code a split ideal code? We will give a sufficient condition, in terms of the non singular matrix U , based on the general results from [4,5]. ...
... If the extension F[z] ⊆ A[z; σ U ] is separable, then every ideal code of A[z; σ U ] is a split ideal code. With this motivation, the separability of these ring extensions has been investigated in [4,5]. We collect some results of [4,5] in the next proposition. ...
Let Mn(����) be the algebra of n _ n matrices over the _nite _eld ����. In this paper we prove that the dual code of each ideal convolutional code in the skew-polynomial ring Mn(����)[z;σU] which is a direct summand as a left ideal, is also an ideal convolutional code over Mn(����)[z;σUT] and a direct summand as a left ideal. Moreover we provide an algorithm to decide if _U is a separable automorphism and returns the corresponding separability element, when pertinent.
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.
In this paper we deal with the theory of rough ideals started in B. Davvaz, Roughness in rings, Information Sciences 164 (1–4) (2004) 147–163. We show that the approximation spaces built from an equivalence relation compatible with the ring structure, i.e. associated to a two-sided ideal, are too naive in order to develop practical applications. We propose the use of certain crisp equivalence relations obtained from fuzzy ideals. These relations make available more flexible approximation spaces since they are enriched with a wider class of rough ideals. Furthermore, these are fully compatible with the notion of primeness (semiprimeness). The theory is illustrated by several examples of interest in Engineering and Mathematics.
Maximum-distance separable (MDS) convolutional codes are
characterized through the property that the free distance attains the
generalized singleton bound. The existence of MDS convolutional codes
was established by two of the authors by using methods from algebraic
geometry. This correspondence provides an elementary construction of MDS
convolutional codes for each rate k/n and each degree δ. The
construction is based on a well-known connection between quasi-cyclic
codes and convolutional codes
Convolutional codes have appeared in the literature endowed with sufficient additional algebraic structure to be considered as (left) ideals of a (code-ambient) automorphism-twisted polynomial ring with coefficients in a (word-ambient) semisimple finite group ring. In this paper we extend the present scope of the theory by considering a code-ambient twisted polynomial ring having, in addition to an automorphism σσ, the action of a σσ-derivation δδ. In addition, we develop the basic theory without any specific restrictions for the semisimple finite word-ambient ring. This second element therefore extends even the original notions of both cyclic and group convolutional codes considered thus far in the literature. Among other results, in this paper we develop a matrix-based approach to the study of our extended notion of group convolutional codes (and therefore of cyclic convolutional codes as well), inspired by the use of circulant matrices by Gluesing-Luerssen and Schmale, and then use it to extend to this level the results on the existence of dual codes that were originally established by those authors for cyclic codes (in the narrower sense without a σσ-derivation). Various examples illustrate the potential value of extending the search for good convolutional codes in this direction.
We investigate the notion of cyclicity for convolutional codes as it has been introduced by Piret and Roos. Codes of this type are described as submodules of F[z]
n
with some additional generalized cyclic structure but also as specific left ideals in a skew polynomial ring. Extending a result of Piret, we show in a purely algebraic setting that these ideals are always principal. This leads to the notion of a generator polynomial just like for cyclic block codes. Similarly a parity check polynomial can be introduced by considering the right annihilator ideal. An algorithmic procedure is developed which produces unique reduced generator and parity check polynomials. We also show how basic code properties and a minimal generator matrix can be read off from these objects. A close link between polynomial and vector description of the codes is provided by certain generalized circulant matrices.
Algebraic convolutional coding theory is considered. It is shown that any convolutional code has a canonical direct decomposition into subcodes and that this decomposition leads in a natural way to a minimal encoder. Considering cyclic convolutional codes, as defined by Piret, an easy application of the general theory yields a canonical direct decomposition into cyclic subcodes, and at the same time a canonical minimal encoder for such codes. A list of pairs (n,k) admitting completely proper cyclic (n, k) -convolutional codes is included.
The encoded sequences of an (n,k) convolutional code are treated as sequences of polynomials in the ring of polynomials modulo X^{n} - 1 . Any such sequence can then be written as a power series in two variables w(X,D) , where the polynomial coefficient of D^{j} is the "word" at time unit j in the sequence. Necessary and sufficient conditions on the ring "multiplication" for the set of such sequences so that the set becomes alinear associative algebra are derived. Cyclic convolutional codes (CCC's)are then defined to be left ideals in this algebra. A canonical decomposition of a CCC into minimal ideals is given which illuminates the cyclic structure. As an application of the ideas in the paper, a number of CCC's with large free distance are constructed.