Title: Non-commutative convolutional codes over the infinite dihedral group

Authors: Marion Candau; Roland Gautier; Johannes Huisman

Addresses: Laboratoire de Mathématiques de Bretagne Atlantique, UMR CNRS 6205, Université de Bretagne Occidentale, 6 avenue Victor Le Gorgeu, CS 93837, 29238 Brest, Cedex 3, France ' Lab-STICC, UMR CNRS 6285, Université de Bretagne Occidentale, 6 avenue Victor Le Gorgeu, CS 93837, 29238 Brest, Cedex 3, France ' Laboratoire de Mathématiques de Bretagne Atlantique, UMR CNRS 6205, Université de Bretagne Occidentale, 6 avenue Victor Le Gorgeu, CS 93837, 29238 Brest, cedex 3, France

Abstract: Classic convolutional codes are defined as the convolution of a message and a transfer function over ℤ. In this paper, we study convolutional codes over the infinite dihedral group D. The goal of this study is to design convolutional codes with good and interesting properties and intended to be more resistant to code recognition. Convolution of two functions on D corresponds to the product of two polynomials in the non-commutative polynomial algebra 𝔽2{X,Y}/{X² − 1, Y² − 1}. We show how encoding over D can be represented by two classical convolutions over ℤ. Furthermore, we adapt the Viterbi algorithm to decode these codes using two different trellises. Finally, we show that these codes have performances similar to classic convolutional codes, but are not more resistant to code recognition. However, we get more optimal codes in terms of free distance than conventional.

Keywords: convolutional codes; coding theory; non-commutative polynomial algebra; non-commutative groups; infinite dihedral groups; transfer function; adapted Viterbi algorithm; free distance; information theory; code recognition.

DOI: 10.1504/IJICOT.2015.068694

International Journal of Information and Coding Theory, 2015 Vol.3 No.1, pp.1 - 14

Available online: 06 Apr 2015

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