Keywords: series connection, parallel connection, input-state-output representation, convolutional code, concatenated code, free distance.

Reliable and efficient communication is becoming an increasingly indispensable tool of the modern world. The construction of error correcting codes, along with efficient decoding algorithms, is the goal of modern coding theory. Error correcting codes generally fall into two categories: block codes and convolutional codes. The class of convolutional codes generalizes the class of linear block codes in a natural way. A major parameter of a convolutional code is its free distance since it determines the decoding capability of a code under maximum likelihood decoding. This motivates the search for convolutional codes with a specified rate and degree that have maximum free distance. Good convoltional codes are often looked for through computer searches. Several authors have extended constructions known for block codes to convolutional codes. However, Forney [3,4], in his goal to find a class of codes whose probability of error decreased exponentially with code length, while decoding complexity increased only algebraically, arrived at a solution consisting of the multilevel coding structure known as concatenated code. It consists of a cascade of an inner an outer a code.

On the other hand, taking into account that a convolutional code is essentially a linear system defined over a finite field, Rosenthal Schumacher and York [5] showed that the input-state-output representation commonly used in system theory are very useful for the construction of convolutional code with a designed free distance. This construction requires that a controllability matrix associated with the input-state-output system was the parity-check matrix of a Reed-Solomon code. Rosenthal and York [8] using a new parity-check matrix introduced another class of convolutional codes with a designed free distance. Moreover, Rosenthal and Smarandache [6,7] introduced two new classes of convolutional codes with designed lower free distance and a state space approach for constructing maximum distance separable convolutional codes. Recently, Climent, Herranz and Perea [2] introduced new concatenated models and obtained some interesting results. They have based on the fact that if we take into account that a multi-variable systems can be considered to be made up of several connected simpler subsystems then the output of subsystems act as input to others directly or by means of simple algebraic operations. Forney's idea of a concatenated code [3,4], and the input-state-output representation of a convolutional code [5] are used here. In this paper, following the line of Climent, Herranz and Perea [2], as well as, two of the classical connection models in system theory (series and parallel) new models of series and parallel concatenated convolutional codes are introduced. We present conditions to obtain a minimal and noncatastrophic representation of these models. Finally, we present a lower bound of the free distance of the last parallel model.

1

J. Climent, V. Herranz, and C. Perea New convolutional codes from old convolutional codes, Proceedings Sisteen International Symposium on: Mathematical Theory of Networks and Systems. 1-14, 2004.

2

J. Climent, V. Herranz, and C. Perea A first approximation of concatenated convolutional codes from system viewpoint, Submitted 2006. doi:10.1016/j.laa.2007.03.017

3

G.D. Forney, Jr."Concatenated Codes", Cambridge, MA: M.I.T.Press, 1966.

4

G.D. Forney, Jr., Convolutional codes I: Algebraic structure, IEEE Trans. Inform. Theory, IT-16, 720-738, 1970. doi:10.1109/TIT.1970.1054541

5

J. Rosenthal, J.M. Schumacher and E.V. York. On behaviors and convolutional codes. IEEE Transactions on Information Theory, 42(6) 1881-1891, 1996. doi:10.1109/18.556682

6

J. Rosenthal and R. Smarandache. Construction of Convolutional Codes using Methods from Linear Systems Theory Proceedings of the 35th Allerton Conference on Communication, Control and Computing, 953-960, 1997.

7

J. Rosenthal and R. Smarandache. A state space approach for constructing MDS rate convolutional codes. Proceedings of the 1998 IEEE Information Theory Workshop on Information Theory, 116-117, Killarney, Kerry, Ireland, 1998.

8

J. Rosenthal; E. V. York, BCH Convolutional Codes, IEEE. Transactions on Information Theory 45(6), 1833-1844, 1999. doi:10.1109/18.782104

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