New Results on the Minimum Distance of Repeat Multiple Accumulate Codes

In this paper we consider the ensemble of codes formed by a serial concatenation of a repetition code with multiple accumulators through uniform random interleavers. Based on finite length weight enumerators for these codes, asymptotic expressions for the minimum distance and an arbitrary number of accumulators larger than one are derived. In accordance with earlier results in the literature, we first show that the minimum distance of RA codes can grow, at best, sublinearly with the block length. Then, for RAA codes and rates of1/3 or smaller, it is proved that these codes exhibit linear distance growth with block length, where the gap to the Gilbert-Varshamov bound can be made arbitrarily small by increasing the number of accumulators beyond two. In order to address rates larger than1/3, random puncturing of a low-rate mother code is introduced. We show that in this case the resulting ensemble of RAA codes asymptotically achieves linear distance growth close to the Gilbert-Varshamov bound. This holds even for very high rate codes. I. I NTRODUCTION Since the invention of turbo codes. iterative decoding schemes with even better performance than turbo codes have been designed. Among these are serially concatenated codes , where the simplest code is the repeat-accumulate (RA) code [1] which consists only of a repetition code, an interleaver , and an accumulator. Such a code has the advantage of very low decoding complexity compared to serially concatenated cod e constructions with convolutional constituent codes. Anot her benefit of RA codes, compared to powerful code constructions such as LDPC codes, is the extremely low encoding complexity ofO(1), whereas LDPC codes have an encoding complexity of O(g), with g much smaller than the block length [2]. This makes RA codes well suited in powerlimited environments, for example, for physical layer erro r correction in battery powered sensor network nodes or in spa ce communications. In this paper we address multiple serially concatenated RA codes, where design guidelines for more general double seri ally concatenated codes have been given in [3]. In particula r, we focus on the serial concatenation of an outer repetition c ode with two or more accumulators connected through random interleavers. The resulting code ensemble is then analyzed using the uniform interleaver approach [4] by averaging ove r all possible interleavers. Our work is mainly motivated by [5], where a similar setup was considered. There, the author s This work was supported in part by NSF Grant CCF05-15012, NAS A Grant NNX07AK53G, and the University of Notre Dame Faculty Researc h Program. show that for an asymptotically large number of accumulator s and uniform interleavers there are codes in the ensemble whose minimum distance grows linearly with block length and which achieve the Gilbert-Varshamov-Bound (GVB). They also provide numerical calculations of the minimum distanc e for different numbers of accumulators and finite block lengt hs, but they do not make any statement regarding the asymptotic minimum distance growth rate for the practically more relev ant case of a finite (small) number of accumulators. For the case of two accumulators, in [6] it is shown that there exist RAA ensembles with linear distance growth, but besides the existence proof no growth rate is given. In the following, we try to fill this gap and present an analysis of the asymptotic minimum distance growth rate for RA and repeat multiple accumulate codes. The main result of the paper is that for RAA codes and rates equal to 1/3 or smaller, we show that these codes exhibit linear distance growth with block lengt h, where the gap to the GVB can be made arbitrarily small by increasing the number of accumulators beyond two. In addition, we consider random puncturing at the output of the last accumulator. We show that in this case the resulting ensemble of RAA codes achieves linear distance growth close to the GVB for any code rate smaller than one. The paper is organized as follows. In Section II we consider the ensemble average weight spectrum of RA codes and show that the minimum distance cannot grow linearly with block length. Section III addresses the case of repeat and multipl e accumulate codes, and random puncturing and its effect on th e minimum distance is discussed in Section IV. II. ENSEMBLE AVERAGE WEIGHT SPECTRUM FORRA CODES In this section we briefly address the minimum distance of RA codes, and we show that these codes cannot achieve linear distance growth with block length, i.e., they are asymptoti cally bad. Related results have already been established in [6], [7], and [8], where lower and upper bounds on minimum distance for more general serially concatenated codes have been derived. In order to introduce the notation and for tuto rial reasons we restate these results, where our derivation is ba sed on the uniform interleaver approach. The RA encoder is depicted in Fig. 1. The binary input sequenceu has lengthK and weight w, R denotes the repetition code of rateR = 1/q, which leads to a code π1 weightqw weightw u v