4096 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 10, OCTOBER 2013 Low-Rate Non-Binary LDPC Codes for Coherent and Blockwise Non-Coherent AWGN Channels Bal´ azs Matuz, Student Member, IEEE, Gianluigi Liva, Member, IEEE, Enrico Paolini, Member, IEEE, Marco Chiani, Fellow, IEEE, and Gerhard Bauch, Senior Member, IEEE Abstract—Low-rate non-binary low-density parity-check (LDPC) codes for coherent and blockwise non-coherent additive white Gaussian noise (AWGN) channels are developed. The proposed construction is based on the concatenation of non- binary outer LDPC codes with inner binary codes. In case the binary codes are chosen to be Hadamard or Reed-Muller (RM) codes, the complexity of the decoding scheme is considerably reduced. An asymptotic analysis of the concatenation with help of composite capacity considerations and density evolution (DE) is provided, from which guidelines on the choice of both inner and outer codes are devised. Finite length designs presented in this work confirm the excellent performance of the proposed codes. Index Terms—Blockwise non-coherent AWGN channel, code design, coherent AWGN channel, concatenated codes, low-rate non-binary LDPC/turbo codes. I. I NTRODUCTION L OW-RATE codes have been widely considered in the context of spread spectrum communications [1], [2]. In that context, some of the most successful and powerful coding schemes are based on Hadamard-Walsh sequences either for orthogonal modulation [2], [3] or as component codes for concatenated schemes [4]–[7]. For instance, a low-rate coding scheme consisting of the concatenation of an outer rate-1/3 convolutional code and an inner Hadamard code, yielding an overall code rate of 1/32, was selected for the uplink of the Interim Standard 95 (IS-95)(A) standard [2], [3]. Originally, coding and spreading have been treated sep- arately in code division multiple access systems, although spreading may also be considered as a special case of channel coding. Results in literature [5] suggest that the optimal Manuscript received April 30, 2013; revised July 9, 2013. The editor coordinating the review of this paper and approving it for publication was Dr. A. Graell i Amat. B. Matuz and G. Liva are with the Institute of Communications and Navi- gation, German Aerospace Center (DLR), Oberpfaffenhofen, 82234 Weßling, Germany (e-mail: {balazs.matuz, gianluigi.liva}@dlr.de). E. Paolini and M. Chiani are with CNIT, DEI, University of Bologna, via Venezia 52, 47521 Cesena (FC), Italy (e-mail: {e.paolini, marco.chiani}@unibo.it). G. Bauch is with Hamburg University of Technology (TUHH), Institute of Communications, Eißendorfer Straße 40, 21073 Hamburg, Germany (e-mail: bauch@tuhh.de). The research leading to these results has received funding in part from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement n. 288502 CONCERTO. This work was presented in part at the IEEE International Symposium on Turbo Codes and Iterative Information Processing, Gothenburg, Sweden, August 2012, and in part at the IEEE Symposium on Source and Channel Coding, Munich, Germany, January 2013. Digital Object Identifier 10.1109/TCOMM.2013.090513.130320 multiple access channel capacity is achievable by choosing an appropriate set of low-rate channel codes. Practical approaches of combining channel coding and spreading by making use of low-rate coding schemes have been presented, for instance, in [8], [9]. Further applications of low-rate coding schemes lie in the field of satellite and deep-space communications. In the former context, interactive satellite return links turn out to be a suitable target, due to low transmit powers and moderate to low data rates. Concerning deep-space communications low- rate long turbo codes are applied for telemetry, while short low-rate coding schemes are currently under investigation for telecommand links where it is crucial to guarantee the data integrity [10]. Iteratively-decodable codes able to approach the Shannon limit at low code rates have been introduced in the past [6], [7], [11]–[13]. However, most of them suffer either from high error floors [6], [7] or from visible losses in comparison with the sphere packing bound (SPB) [14] when the code dimension k is within few hundreds of bits [13]. It is well established that non-binary LDPC codes represent an excellent solution in the short/moderate block length regime [15]–[17] with efficient practical implementations [18]–[20]. Low-rate LDPC codes over high-order finite fields F q (i.e., for field orders 64 ≤ q ≤ 256) possessing iterative (IT) decoding thresholds close to the AWGN channel capacity have been recently proposed in [21]. The codes of [21], also referred to as multiplicatively repeated (MR)-LDPC codes, rely on the repetition of the codeword symbols and their multiplication by non-zero coefficients of F q . This construction provides a large flexibility in terms of code rate, and in fact, as discussed later, it may be interpreted as the serial concatenation of an outer non-binary LDPC code and random-like binary inner codes of dimension p, with p = log 2 q. With this regard, the analysis of concatenated schemes provided next is applicable to MR-LDPC codes. In this work, a novel low-rate coding scheme is proposed. It is based on the concatenation of non-binary LDPC outer codes [22] and binary algebraic inner codes, where the inner code dimension matches the LDPC code field order. Likewise, some of the proposed LDPC constructions coincide with the non-binary turbo codes of [17]. On the one hand, for a proper inner code selection, the proposed scheme has significant complexity advantages over competing non-binary schemes with similar performance [21], while on the other hand it shows large gains w.r.t. conventional binary schemes. For instance, improvements of 2 dB or more w.r.t. the IS-95(A) standard can be observed. 0090-6778/13$31.00 c  2013 IEEE