One dimensional Convolutional Goppa Codes over the projective line

We give a general method to construct MDS one-dimensional convolutional codes. Our method generalizes previous constructions [5]. Moreover we give a classification of one-dimensional Convolutional Goppa Codes and propose a characterization of MDS codes of this type. Introduction One of the main problems in coding theory is the construction of codes with a large distance, such as so-called MDS codes. The aim of this paper is to give a very general method to construct onedimensional MDS convolutional codes using the techniques developed in our previous papers [1, 2, 3]. In Section 1 we give a general introduction to convolutional codes, reformulated in terms that enables a good understanding of the choices of the generator matrices and the submodules generated by them. The treatment is fairly selfcontained, with only a few references for proofs of certain statements. Moreover, a characterization of one-dimensional MDS convolutional codes in terms of their associated block linear codes is given (Theorem 1.11). In Section 2 we describe the notion of Convolutional Goppa Code, introduced in [1] and [2] and we recall the construction of convolutional Goppa codes over the projective line. We use this construction in Section 3 to give families of examples of one-dimensional convolutional Goppa codes; moreover, we prove, using Theorem 1.11, that they are MDS. This is the main result in this paper. The examples constructed in [5] are particular cases of ours. Finally, we give in Section 3.1 a classification of one-dimensional convolutional Goppa codes defined over the projective line, which could give rise to a characterization of MDS convolutional Goppa codes of dimension one. 1. Convolutional codes Given a finite field Fq, representing the symbols in which an information word u ∈ Fkq is written, each k × n matrix of rank k with entries in Fq defines an injective linear map F k q G −→ Fnq u 7→ x = uG , This work was partially supported by the research contracts MTM2009-11393 of the Spanish Ministry for Science and Innovatión. J.A. Domı́nguez Pérez, J.M. Muñoz Porras and G. Serrano Sotelo are in the Department of Mathematics, University of Salamanca, Plaza de la Merced 1-4, 37008 Salamanca, Spain (email: [email protected], [email protected], and [email protected]). 1 2 J.A. DOMÍNGUEZ PÉREZ, J.M. MUÑOZ PORRAS AND G. SERRANO SOTELO whose image subspace is the linear code C = ImG ⊆ Fnq of length n, dimension k, and rate k/n. G is called a generator matrix of the code, and G ′ is another generator matrix of the code if there exists an element B ∈ GL(k,Fq) such that G ′ = B · G. In practical applications, the codification process is not limited to a single word, but to a sequence of information words depending on time, ut ∈ F k q , t ≥ 0, which after the codification are transformed into the sequence of codified words xt = utG and xt at the instant t depends only on the information word ut at the same instant t. The basic idea of convolutional codification is to allow xt to depend not only on ut but also on ut−1, . . . , ut−m for some positive integer m, which is the memory of the code. If one denotes a sequence of words as a polynomial vector u(z) =