HIGH - RATE BINOMIAL CONVOLUTIONAL CODES by Ph . PIRET

A construction is described of high-rate RS-like convolutional codes, and a lower bound is given on their free distance. These codes can be used to construct asymptotically good binary convolutional codes. 1. Introduetion I~ a preceding report 4), a class of RS-like convolutional codes was introduced. These codes were obtained only for length n, rate R = kln and constraint length 'll related by the inequality \ Rn 'V~ n + I, (1) and their free distance was lower-bounded as follows: df~'V(n-k+I). (2) Our aim in this paper will be to obtain codes with a slightly weaker bound on df, but for which n, k and 'Vwould no more be so drastically bounded as in (I). After recalling the formalism used to represent the RS-like codes, and some useful arguments for computing df> we shall present in sec. 3 a construction of high-rate RS-like convolutional codes, and a lower bound on their free distance. These codes are then used to obtain long, asymptotically good, binary convolutional codes, of arbitrary rate, by use of some arguments developed by Justesen 3). 2. Representation of the RS-like codes Let q be a power of 2, and consider a linear convolutional code of length n, rate kln and constraint length 'Von GF(q). lts generator matrix can be given the form (3) where we suppose that all G, generate cyclic RS block codes 2). We shall consider an n-tuple with polynomial representation xn_ 1 gr(X)= ---, r = 0, 1, ... , n-I, X(/.r (4) where oe,primitive of order n, is in GF(q) but in no proper subfield. The k HIGH-RATE BINOMIAL CqNVOLUTIONAL CODES 447 n-tuples of each Gj are chosen in this set so that each G, can be represented by a column of k integers referring to the indices of the chosen n-tuples. 'using this convention, we represent the generator matrix G by a rectangular array of integers gij (0 ~ i ::::;; k I, 0 ~ j ~ v I). The space of the admitted sequences is then generated by the rows of a matrix r that we now describe. Consider therefore the matrix G, represented by v columns of gij, and the , operator T" that shifts G, s columns to the right. The matrix T', is then given by ,