Abstract：This paper describes a class of rate-compatible nonbinary low-density parity-check (LDPC) codes based on an improved extension method. The check symbols for lower code rates involve not only the codewords of the highest-rate code but also the check symbols of moderate-rate codes. The construction process optimizes the masking matrix and the base matrix with random replacement of nonbinary elements. This paper also describes an algebraic method to design the parity matrices to further reduce the design complexity. The codes are not only rate compatible, but also have a quasi-cyclic structure which will benefit hardware implementations of the encoder and decoder. Numerical tests show that the codes can achieve good performance within a wide range of code rates in both the waterfall region and in the error-floor region.
[1] DAVEY M C, MACKAY D J C. Low-density parity check codes over GF(q)[J]. IEEE Communications Letters, 1998, 2(6):165-167. [2] POULLIAT C, FOSSORIER M, DECLERCQ D. Design of regular (2, dc)-LDPC codes over GF(q) using their binary images[J]. IEEE Transactions on Communications, 2008, 56(10):1626-1635. [3] DOLECEK L, DIVSALAR D, SUN Y Z, et al. Non-binary protograph-based LDPC codes:Enumerators, analysis, and designs[J]. IEEE Transactions on Information Theory, 2014, 60(7):3913-3941. [4] LIN W, BAI B M, LI Y, et al. Design of q-ary irregular repeat-accumulate codes[C]//Proceedings of 2009 International Conference on Advanced Information Networking and Applications. Bradford, UK:IEEE, 2009:201-206. [5] ZENG L Q, LAN L, TAI Y Y, et al. Transactions papers-constructions of nonbinary quasi-cyclic LDPC codes:A finite field approach[J]. IEEE Transactions on Communications, 2008, 56(4):545-554. [6] ZENG L Q, LAN L, TAI Y Y, et al. Construction of nonbinary cyclic, quasi-cyclic and regular LDPC codes:A finite geometry approach[J]. IEEE Transactions on Communications, 2008, 56(3):378-387. [7] ZHOU B, KANG J Y, TAI Y Y, et al. High performance non-binary quasi-cyclic LDPC codes on Euclidean geometries LDPC codes on Euclidean geometries[J]. IEEE Transactions on Communications, 2009, 57(5):1298-1311. [8] LI J E, LIU K K, LIN S, et al. A matrix-theoretic approach to the construction of non-binary quasi-cyclic LDPC codes[J]. IEEE Transactions on Communications, 2015, 63(4):1057-1068. [9] Ryan W E, Lin S. 信道编码:经典与现代[M]. 白宝明, 马啸, 译. 北京:电子工业出版社, 2017. RYAN W E, LIN S. Channel codes:Classical and modern[M]. BAI B M, MA X, Trans. Beijing:Publishing House of Electronics Industry, 2017. (in Chinese) [10] ISCAN O, LENTNER D, XU W. A comparison of channel coding schemes for 5G short message transmission[C]//Proceedings of 2016 IEEE Globecom Workshops. Washington DC, USA:IEEE, 2016:1-6. [11] SYBIS M, WESOLOWSKI K, JAYASINGHE K, et al. Channel coding for ultra-reliable low-latency communication in 5G systems[C]//Proceedings of the 84th Vehicular Technology Conference. Montreal, QC, Canada:IEEE, 2016:1-5. [12] DURISI G, KOCH T, POPOVSKI P. Toward massive, ultrareliable, and low-latency wireless communication with short packets[J]. Proceedings of the IEEE, 2016, 104(9):1711-1726. [13] WILLIAMSON A R, CHEN T Y, WESEL R D. Variable-length convolutional coding for short blocklengths with decision feedback[J]. IEEE Transactions on Communications, 2015, 63(7):2389-2403. [14] ÖSTMAN J, DURISI G, STR? M E G, et al. Low-latency ultra-reliable 5G communications:Finite-blocklength bounds and coding schemes[C]//Proceedings of the 11th International ITG Conference on Systems, Communications and Coding. Hamburg, Germany:IEEE, 2017. [15] HUANG J, ZHOU W, ZHOU S L. Structured nonbinary rate-compatible low-density parity-check codes[J]. IEEE Communications Letters, 2011, 15(9):998-1000. [16] ZHOU L, BAI B M, XU M. Design of nonbinary rate-compatible LDPC codes utilizing bit-wise shortening method[J]. IEEE Communications Letters, 2010, 14(10):963-965. [17] KASAI K, DECLERCQ D, POULLIAT C, et al. Multiplicatively repeated nonbinary LDPC codes[J]. IEEE Transactions on Information Theory, 2011, 57(10):6788-6795. [18] ZHU M, WEI Y, BAI B M, et al. Nonbinary Kite codes:A family of nonbinary rate-compatible LDPC codes[C]//Proceedings of 2015 IEEE International Symposium on Information Theory. Hong Kong, China:IEEE, 2015:1094-1098. [19] CHANG B Y, DOLECEK L, DIVSALAR D. EXIT chart analysis and design of non-binary protograph-based LDPC codes[C]//Proceedings of 2011 Military Communications Conference. Baltimore, MD, USA:IEEE, 2011:566-571. [20] KARIMI M, BANIHASHEMI A H. Counting short cycles of quasi cyclic protograph LDPC codes[J]. IEEE Communications Letters, 2012, 16(3):400-403. [21] 3GPP. 3GPP TSG RAN WG1 Meeting NR Ad-Hoc#2.. https://www.3GPP.org. [22] 3GPP. 3GPP TSG RAN WG1 Meeting #88.. https://www.3GPP.org. [23] TAL I, VARDY A. List decoding of Polar codes[J]. IEEE Transactions on Information Theory, 2015, 61(5):2213-2226.
2018, Vol. 58 
