Non-commutative convolutional codes over the infinite dihedral group

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 noncommutative polynomial algebra F2{X,Y }/(X − 1, Y 2 − 1). We study this algebra and derive a criterion for an element to be regular. Such an element defines a transfer function. Moreover, we show how encoding over D∞ can be represented by two classical convolutions over Z, alternating according to the parity of the index of the input bits. Furthermore, we adapt the Viterbi algorithm to decode these codes using two different trellis. Finally, we show that these codes have performances similar to classic convolutional codes. Unfortunately, they are not more resistant to code recognition. However, under certain conditions, we get more optimal codes in terms of free distance than conventional convolutional codes, which can be a great asset for non-generic dedicated proprietary transmitter/receiver.