J. A. Dominguez Perez, J. M. Muñoz Porras, G. Serrano Sotelo
A new kind of Convolutional Codes generalizing Goppa Codes is proposed. This provides a systematic method for constructing convolutional codes with prefixed properties. In particular, examples of Maximum-Distance Separable (MDS) convolutional codes are obtained.
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We define Convolutional Goppa Codes over algebraic curves and construct their corresponding dual codes. Examples over the projective line and over elliptic curves are described, obtaining in particular some Maximum-Distance Separable (MDS) convolutional codes.
We show that the free distance, as a function on a space parameterizing a family of convolutional codes, is a lower-semicontinuous function and that, therefore, the property of being Maximum Distance Separable (MDS) is an open condition. For a class of convolutional codes, an algorithm is offered to compute the free distance. The behaviour of the free distance by enlargements of the alphabet and by increasing the length is also studied. As an application, the algebraic equations characterizing the subfamily of MDS codes is explicitly computed for families of 1-dimensional convolutional Goppa c…
Multidimensional convolutional codes generalize (one dimensional) convolutional codes and they correspond under a natural duality to multidimensional systems widely studied in the systems literature.
Decoding of convolutional codes poses a significant challenge for coding theory. Classical methods, based on e.g. Viterbi decoding, suffer from being computationally expensive and are restricted therefore to codes of small complexity. Based on analogies with model predictive optimal control, we propose a new iterative method for convolutional decoding that is cheaper to implement than established algorithms, while still offering significant error correction capabilities. The algorithm is particularly well-suited for decoding special types of convolutional codes, such as e.g. cyclic convolution…
A MacWilliams Identity for convolutional codes will be established. It makes use of the weight adjacency matrices of the code and its dual, based on state space realizations (the controller canonical form) of the codes in question. The MacWilliams Identity applies to various notions of duality appearing in the literature on convolutional coding theory.
Detailed information about the weight distribution of a convolutional code is given by the adjacency matrix of the state diagram associated with a controller canonical form of the code. We will show that this matrix is an invariant of the code. Moreover, it will be proven that codes with the same adjacency matrix have the same dimension and the same Forney indices and finally that for one-dimensional binary convolutional codes the adjacency matrix determines the code uniquely up to monomial equivalence.
An iterative decoding algorithm for convolutional codes is presented. It successively processes $N$ consecutive blocks of the received word in order to decode the first block. A bound is presented showing which error configurations can be corrected. The algorithm can be efficiently used on a particular class of convolutional codes, known as doubly cyclic convolutional codes. Due to their highly algebraic structure those codes are well suited for the algorithm and the main step of the procedure can be carried out using Reed-Solomon decoding. Examples illustrate the decoding and a comparison wit…
It is known that maximum distance separable and maximum distance profile convolutional codes exist over large enough finite fields of any characteristic for all parameters $(n,k,δ)$. It has been conjectured that the same is true for convolutional codes that are strongly maximum distance separable. Using methods from linear systems theory, we resolve this conjecture by showing that, over a large enough finite field of any characteristic, codes which are simultaneously maximum distance profile and strongly maximum distance separable exist for all parameters $(n,k,δ)$.
MDS convolutional codes have the property that their free distance is maximal among all codes of the same rate and the same degree. In this paper we introduce a class of MDS convolutional codes whose column distances reach the generalized Singleton bound at the earliest possible instant. We call these codes strongly MDS convolutional codes. It is shown that these codes can decode a maximum number of errors per time interval when compared with other convolutional codes of the same rate and degree. These codes have also a maximum or near maximum distance profile. A code has a maximum distance pr…
After a discussion of the Griesmer and Heller bound for the distance of a convolutional code we present several codes with various parameters, over various fields, and meeting the given distance bounds. Moreover, the Griesmer bound is used for deriving a lower bound for the field size of an MDS convolutional code and examples are presented showing that, in most cases, the lower bound is tight. Most of the examples in this paper are cyclic convolutional codes in a generalized sense as it has been introduced in the seventies. A brief introduction to this promising type of cyclicity is given at t…
In this paper we extend the relation between convolutional codes and linear systems over finite fields to certain commutative rings through first order representations . We introduce the definition of rings with representations as those for which these representations always exist, and we show that finite products of finite fields belong to this class. We develop the input/state/output representations for convolutional codes over these rings, and we show how to use them to construct observable convolutional codes as in the classical case.
Maximum distance profile codes are characterized by the property that two trajectories which start at the same state and proceed to a different state will have the maximum possible distance from each other relative to any other convolutional code of the same rate and degree. In this paper we use methods from systems theory to characterize maximum distance profile codes algebraically. Tha main result shows that maximum distance profile codes form a generic set inside the variety which parametrizes the set of convolutional codes of a fixed rate and a fixed degree.
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