[1] V. Raghavan and J. Li, “Evolution of physical-layer communications research in the post-5G era,” IEEE Access, vol. 7, pp. 10 392–10 401, 2019.
[2] ITU-R, “[IMT-2020.TECH PERF REQ] - minimum requirements related to technical performance for imt2020 radio interface(s),” Report ITU-R M.2410-0, Nov. 2017.
[3] P. Yang, Y. Xiao, M. Xiao, and S. Li, “6G wireless communications: Vision and potential tech- niques,” IEEE Network, vol. 33, no. 4, pp. 70–75, 2019.
[4] M. Z. Chowdhury, M. Shahjalal, S. Ahmed, and Y. M. Jang, “6G wireless communication systems: Applications, requirements, technologies, challenges, and research directions,” arXiv preprint arXiv:1909.11315, 2019.
[5] S. Dang, O. Amin, B. Shihada, and M.-S. Alouini, “From a human-centric perspective: What might 6G be?” arXiv preprint arXiv:1906.00741, 2019.
[6] C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J., vol. 27, pp. 379–423 and 623–656, July and Oct. 1948.
[7] D. J. Costello and G. D. Forney, “Channel coding: The road to channel capacity,” Proceedings of the IEEE, vol. 95, no. 6, pp. 1150–1177, 2007.
[8] R. W. Lucky, “Coding is dead,” IEEE Spectrum, Lucky Strikes Again: (Feats and Foibles of Engi- neers), pp. 243–245, 1993.
[9] M. J. Golay, “Notes on digital coding,” Proc. IRE, vol. 37, p. 657, 1949.
[10] D. E. Muller, “Application of boolean algebra to switching circuit design and to error detection,” IRE Trans. Electric Computers, no. 3, pp. 6–12, 1954.
[11] I. S. Reed, “A class of multiple-error-correcting codes and the decoding scheme,” IRE Trans. In- form. Theory, vol. 4, pp. 39–44, 1954.
[12] S. Kudekar, S. Kumar, M. Mondelli, H. D. Pfister, E. S¸ as¸olu, and R. L. Urbanke, “Reed–muller codes achieve capacity on erasure channels,” IEEE Trans. Inf. Theory, vol. 63, no. 7, pp. 4298– 4316, 2017.
[13] E. Prange, Cyclic Error-Correcting codes in two symbols. Air force Cambridge research center, 1957.
[14] A. Hocquenghem, “Codes correcteurs d’erreurs,” in Chiffres, pp. 147–156, 1959.
[15] R. C. Bose and D. K. Ray-Chaudhuri, “On a class of error correcting binary group codes,” Infor- mation and control, vol. 3, no. 1, pp. 68–79, 1960.
[16] I. S. Reed and G. Solomon, “Polynomial codes over certain finite fields,” Journal of the society for industrial and applied mathematics, vol. 8, no. 2, pp. 300–304, 1960.
[17] R. Singleton, “Maximum distance q-nary codes,” IEEE Trans. Inf. Theory, vol. 10, no. 2, pp. 116– 118, 1964.
[18] D. Gorenstein and N. Zierler, “A class of error correcting codes in pm symbols,” Journal of the Society of Industrial and Applied Mathematics, vol. 9, pp. 207–214, 1961.
[19] W. Peterson, “Encoding and error-correction procedures for the Bose-Chaudhuri codes,” IRE Trans- actions on Information Theory, vol. 6, no. 4, pp. 459–470, 1960.
[20] E. Berlekamp, Algebraic coding theory. World Scientific, 1968.
[21] ——, “On decoding binary Bose-Chadhuri-Hocquenghem codes,” IEEE Trans. Inf. Theory, vol. 11, no. 4, pp. 577–579, 1965.
[22] J. Massey, “Shift-register synthesis and BCH decoding,” IEEE Trans. Inf. Theory, vol. 15, no. 1, pp. 122–127, 1969.
[23] G. D. Forney, “On decoding BCH codes,” IEEE Trans. Inf. Theory, vol. 11, no. 4, pp. 549–557, 1965.
[24] ——, “Generalized minimum distance decoding,” IEEE Trans. Inf. Theory, vol. IT-12, no. 1, pp. 125–131, Jan. 1966.
[25] ——, “Burst-correcting codes for the classic bursty channel,” IEEE Transactions on Communica- tion Technology, vol. 19, no. 5, pp. 772–781, 1971.
[26] V. D. Goppa, “Codes associated with divisors,” Problemy Peredachi Informatsii, vol. 13, no. 1, pp. 33–39, 1977.
[27] M. A. Tsfasman, S. Vla˘dutx, and T. Zink, “Modular curves, shimura curves, and goppa codes, better than varshamov-gilbert bound,” Mathematische Nachrichten, vol. 109, no. 1, pp. 21–28, 1982.
[28] I. Blake, C. Heegard, T. Hoholdt, and V. Wei, “Algebraic-geometry codes,” IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2596–2618, 1998.
[29] P. Elias, “Coding for noisy channels,” IRE Conv. Rec., vol. pt. 4, pp. 37–46, 1955.
[30] J. M. Wozencraft and B. Reiffen, Sequential Decoding. MIT Press, 1961.
[31] R. Fano, “A heuristic discussion of probabilistic decoding,” IEEE Trans. Inf. Theory, vol. 9, no. 2, pp. 64–74, 1963.
[32] A. Viterbi, “Error bounds for convolutional codes and an asymptotically optimum decoding algo- rithm,” IEEE Trans. Inf. Theory, vol. 13, no. 2, pp. 260–269, 1967.
[33] G. D. Forney, “The viterbi algorithm: A personal history,” arXiv preprint cs/0504020, 2005.
[34] J. A. Heller, “Short constraint length convolutional codes,” Jet Prop. Lab., Space Prog. Summary, vol. III, no. 37-54, pp. 171–177, 1968.
[35] J. Heller and I. Jacobs, “Viterbi decoding for satellite and space communication,” IEEE Transac- tions on Communication Technology, vol. 19, no. 5, pp. 835–848, 1971.
[36] G. D. Forney, Concatenated Codes. MIT Press, 1966.
[37] E. Paaske, “Improved decoding for a concatenated coding system recommended by CCSDS,” IEEE Trans. Commun., vol. 38, no. 8, pp. 1138–1144, 1990.
[38] O. M. Collins and M. Hizlan, “Determinate state convolutional codes,” IEEE Trans. Commun., vol. 41, no. 12, pp. 1785–1794, 1993.
[39] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol. IT-20, pp. 284–287, Mar. 1974.
[40] S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun., vol. 49, no. 10, pp. 1727–1737, Oct. 2001.
[41] M. El-Hajjar and L. Hanzo, “Exit charts for system design and analysis,” IEEE Communications Surveys & Tutorials, vol. 16, no. 1, pp. 127–153, 2013.
[42] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near shannon limit error-correcting coding and decoding: Turbo-codes,” in Proc. 1993 IEEE International Conference on Communications (ICC), pp. 1064–1070, May 1993.
[43] C. Douillard, M. Je´ze´quel, C. Berrou, D. Electronique, A. Picart, P. Didier, and A. Glavieux, “Iterative correction of intersymbol interference: Turbo-equalization,” European transactions on telecommunications, vol. 6, no. 5, pp. 507–511, 1995.
[44] X. Wang and H. V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded cdma,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1046–1061, 1999.
[45] S. Ten Brink, G. Kramer, and A. Ashikhmin, “Design of low-density parity-check codes for mod- ulation and detection,” IEEE Trans. Commun., vol. 52, no. 4, pp. 670–678, 2004.
[46] R. G. Gallager, “Low-density parity-check codes,” IRE Trans. Info. Theory,, vol. 8, no. 1, pp. 21–28, Jan. 1962.
[47] D. J. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inf. The- ory, vol. 45, no. 2, pp. 399–431, Mar. 1999.
[48] M. Sipser and D. A. Spielman, “Expander codes,” IEEE Trans. Inf. Theory, vol. 42, no. 6, pp. 1710–1722, 1996.
[49] R. Tanner, “A recursive approach to low complexity codes,” IEEE Trans. Inf. Theory, vol. 27, no. 5, pp. 533–547, 1981.
[50] M. G. Luby, M. Mitzenmacher, M. A. Shokrollahi, and D. A. Spielman, “Improved low-density parity-check codes using irregular graphs,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 585–598, 2001.
[51] T. J. Richardson and R. L. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 599–618, Feb. 2001.
[52] S.-Y. Chung, T. J. Richardson, and R. L. Urbanke, “Analysis of sum-product decoding of low- density parity-check codes using a Gaussian approximation,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 657–670, Feb. 2001.
[53] S.-Y. Chung, G. D. Forney, T. J. Richardson, R. Urbanke et al., “On the design of low-density parity-check codes within 0.0045 db of the shannon limit,” IEEE Commun. Lett., vol. 5, no. 2, pp. 58–60, 2001.
[54] E. Arıkan, “Channel polarization: A method for constructing capacity-achieving codes for sym- metric binary-input memoryless channels,” IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 3051–3073, Jul. 2009.
[55] E. Arikan and E. Telatar, “On the rate of channel polarization,” in Proc. 2009 IEEE International Symposium on Information Theory (ISIT), pp. 1493–1495, 2009.
[56] T. Richardson and R. Urbanke, Modern Coding Theory. Cambridge University Press, 2008.
[57] I. Tal and A. Vardy, “List decoding of polar codes,” IEEE Trans. Inf. Theory, vol. 61, no. 5, pp. 2213–2226, May 2015.
[58] G. Liva, L. Gaudio, T. Ninacs, and T. Jerkovits, “Code design for short blocks: A survey,” arXiv:1610.00873, Oct. 2016.
[59] J. Van Wonterghem, A. Alloumf, J. J. Boutros, and M. Moeneclaey, “Performance comparison of short-length error-correcting codes,” in Proc. 2016 Symposium on Communications and Vehicular Technologies (SCVT), pp. 1–6, 2016.
[60] M. Shirvanimoghaddam, M. S. Mohammadi, R. Abbas, A. Minja, C. Yue, B. Matuz, G. Han, Z. Lin, W. Liu, Y. Li et al., “Short block-length codes for ultra-reliable low latency communications,” IEEE Communications Magazine, vol. 57, no. 2, pp. 130–137, 2018.
[61] G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inf. Theory, vol. 28, no. 1, pp. 55–67, Jan. 1982.
[62] L.-F. Wei, “Trellis-coded modulation with multidimensional constellations,” IEEE Trans. Inf. The- ory, vol. 33, no. 4, pp. 483–501, 1987.
[63] S. S. Pietrobon, R. H. Deng, A. Lafanechere, G. Ungerboeck, and D. J. Costello, “Trellis-coded multidimensional phase modulation,” IEEE Trans. Inf. Theory, vol. 36, no. 1, pp. 63–89, 1990.
[64] P. Robertson and T. Wo¨rz, “Bandwidth-efficient turbo trellis-coded modulation using punctured component codes,” IEEE J. Sel. Areas Commun., vol. 16, no. 2, pp. 206–218, Feb. 1998.
[65] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “Parallel concatenated trellis coded modula- tion,” in Proc. 1996 IEEE International Conference on Communications (ICC), vol. 2, pp. 974–978, Jun. 1996.
[66] H. Imai and S. Hirakawa, “A new multilevel coding method using error-correcting codes,” IEEE Trans. Inf. Theory, vol. 23, no. 3, pp. 371–377, May 1977.
[67] U. Wachsmann, R. F. Fischer, and J. B. Huber, “Multilevel codes: theoretical concepts and practical design rules,” IEEE Trans. Inf. Theory, vol. 45, no. 5, pp. 1361–1391, Jul. 1999.
[68] E. Zehavi, “8-PSK trellis codes for a Rayleigh channel,” IEEE Trans. Commun., vol. 40, no. 5, pp. 873–884, 1992.
[69] G. Caire, G. Taricco, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inf. Theory, vol. 44, no. 3, pp. 927–946, May 1998.
[70] L. Szczecinski and A. Alvarado, Bit-Interleaved Coded Modulation: Fundamentals, Analysis and Design. John Wiley & Sons, 2015.
[71] X. Li and J. A. Ritcey, “Bit-interleaved coded modulation with iterative decoding,” IEEE Commun. Lett., vol. 1, no. 6, pp. 169–171, 1997.
[72] U. Erez and R. Zamir, “Achieving 1/2 log (1+SNR) on the AWGN channel with lattice encoding and decoding,” IEEE Trans. Inf. Theory, vol. 50, no. 10, pp. 2293–2314, Oct. 2004.
[73] N. Di Pietro, J. J. Boutros, G. Ze´mor, and L. Brunel, “Integer low-density lattices based on con- struction A,” in Proc. 2012 IEEE Information Theory Workshop (ITW), pp. 422–426, 2012.
[74] N. Di Pietro, G. Ze´mor, and J. J. Boutros, “Lda lattices without dithering achieve capacity on the gaussian channel,” IEEE Trans. Inf. Theory, vol. 64, no. 3, pp. 1561–1594, 2017.
[75] E. S. Barnes and N. J. A. Sloane, “New lattice packings of spheres,” Canadian Journal of Mathe- matics, vol. XXXV, no. 1, pp. 117–130, 1983.
[76] A. Sakzad, M. Sadeghi, and D. Panario, “Turbo lattices: Construction and error decoding performance,” CoRR, vol. abs/1108.1873, 2011. [Online]. Available: http://arxiv.org/abs/1108. 1873
[77] A. Vem, Y.-C. Huang, K. R. Narayanan, and H. D. Pfister, “Multilevel lattices based on spatially- coupled LDPC codes with applications,” in Proc. 2014 IEEE International Symposium on Informa- tion Theory (ISIT), Jun. 2014.
[78] Y. Yan and C. Ling, “A construction of lattices from polar codes,” in Proceedings of the IEEE Information Theory Workshop, pp. 124–128, Lausanne, Switzerland, 2012.
[79] Y. Yan, L. Liu, C. Ling, and X. Wu, “Construction of capacity-achieving lattice codes: Polar lattices,” ArXiv e-prints, vol. abs/1411.0187, November 2014. [Online]. Available: http://arxiv.org/abs/1411.0187
[80] N. Sommer, M. Feder, and O. Shalvi, “Low-density lattice codes,” IEEE Trans. Inf. Theory, vol. 54, no. 4, pp. 1561–1585, Apr. 2008.
[81] O. Shalvi, N. Sommer, and M. Feder, “Signal codes: convolutional lattice codes,” IEEE Trans. Inf. Theory, vol. 57, no. 8, pp. 5203–5226, Aug. 2011.
[82] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups. Springer Science & Business Media, 2013, vol. 290.
[83] G. D. Forney, R. Gallager, G. Lang, F. Longstaff, and S. Qureshi, “Efficient modulation for band- limited channels,” IEEE J. Sel. Areas Commun., vol. 2, no. 5, pp. 632–647, Sep. 1984.
[84] F. R. Kschischang and S. Pasupathy, “Optimal nonuniform signaling for Gaussian channels,” IEEE Trans. Inf. Theory, vol. 39, no. 3, pp. 913–929, May 1993.
[85] G. D. Forney and G. Ungerboeck, “Modulation and coding for linear Gaussian channels,” IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2384–2415, Oct. 1998.
[86] G. D. Forney and L.-F. Wei, “Multidimensional constellations. I. Introduction, figures of merit, and generalized cross constellations,” IEEE J. Sel. Areas Commun., vol. 7, no. 6, pp. 877–892, Aug. 1989.
[87] G. D. Forney, “Multidimensional constellations. ii. voronoi constellations,” IEEE J. Sel. Areas Commun., vol. 7, no. 6, pp. 941–958, Aug. 1989.
[88] ——, “Trellis shaping,” IEEE Trans. Inf. Theory, vol. 38, no. 2, pp. 281–300, Mar. 1992.
[89] A. K. Khandani and P. Kabal, “Shaping multidimensional signal spaces. I. optimum shaping, shell mapping,” IEEE Trans. Inf. Theory, vol. 39, no. 6, pp. 1799–1808, Nov. 1993.
[90] F.-W. Sun and H. C. van Tilborg, “Approaching capacity by equiprobable signaling on the Gaussian channel,” IEEE Trans. Inf. Theory, vol. 39, no. 5, pp. 1714–1716, 1993.
[91] C. Fragouli, R. D. Wesel, D. Sommer, and G. Fettweis, “Turbo codes with non-uniform constella- tions,” in Proc. 2001 IEEE International Conference on Communications (ICC), vol. 1, pp. 70–73, 2001.
[92] N. Sommer, M. Feder, and O. Shalvi, “Shaping methods for low-density lattice codes,” in Proc. 2009 IEEE Information Theory Workshop (ITW), pp. 238–242, 2009.
[93] N. S. Ferdinand, B. M. Kurkoski, M. Nokleby, and B. Aazhang, “Low-dimensional shaping for high-dimensional lattice codes,” IEEE Trans. Wireless Commun., vol. 15, no. 11, pp. 7405–7418, Nov. 2016.
[94] F. Zhou and B. M. Kurkoski, “Shaping LDLC lattices using convolutional code lattices,” IEEE Commun. Lett., vol. 21, no. 4, pp. 730–733, Apr. 2017.
[95] P. Mitran and H. Ochiai, “Parallel concatenated convolutional lattice codes with constrained states,” IEEE Trans. Commun., vol. 63, no. 4, pp. 1081–1090, May 2015.
[96] S. Le Goff, A. Glavieux, and C. Berrou, “Turbo-codes and high spectral efficiency modulation,” in Proc. 1994 IEEE International Conference on Communications (ICC), pp. 645–649, May 1994.
[97] D. Raphaeli and A. Gurevitz, “Constellation shaping for pragmatic turbo-coded modulation with high spectral efficiency,” IEEE Trans. Commun., vol. 52, no. 3, pp. 341–345, 2004.
[98] A. R. Calderbank and L. H. Ozarow, “Nonequiprobable signaling on the Gaussian channel,” IEEE Trans. Inf. Theory, vol. 36, no. 4, pp. 726–740, Jul. 1990.
[99] A. K. Khandani and W. Tong, “Application of shaping technique with turbo coset codes,” IEEE Trans. Veh. Technol., vol. 56, no. 6, pp. 3770–3779, Nov. 2007.
[100] S. Y. Le Goff, B. K. Khoo, C. C. Tsimenidis, and B. S. Sharif, “Constellation shaping for bandwidth-efficient turbo-coded modulation with iterative receiver,” IEEE Trans. Wireless Com- mun., vol. 6, no. 6, Jun. 2007.
[101] M. C. Valenti and X. Xiang, “Constellation shaping for bit-interleaved LDPC coded APSK,” IEEE Trans. Commun., vol. 60, no. 10, pp. 2960–2970, Oct. 2012.
[102] Y.-C. Tsai, H.-H. Chung, and M.-C. Lin, “Scrambling-based shaping for turbo coded modulation,” IEEE Trans. Commun., vol. 58, no. 11, pp. 3148–3153, Nov. 2010.
[103] E. Agrell and A. Alvarado, “Signal shaping for BICM at low SNR,” IEEE Trans. Inf. Theory, vol. 59, no. 4, pp. 2396–2410, 2013.
[104] G. Bo¨cherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matched low-density parity- check coded modulation,” IEEE Trans. Commun., vol. 63, no. 12, pp. 4651–4665, Dec. 2015.
[105] F. Steiner and G. Bo¨cherer, “Comparison of geometric and probabilistic shaping with application to ATSC 3.0,” in 11th International ITG Conference on Systems, Communications and Coding (SCC), pp. 1–6, 2017.
[106] G. D. Forney, M. D. Trott, and S.-Y. Chung, “Sphere-bound-achieving coset codes and multilevel coset codes,” IEEE Trans. Inf. Theory, vol. 46, no. 3, pp. 820–850, May 2000.
[107] T. Matsumine and H. Ochiai, “A new turbo coded modulation approach exploiting non-binary field,” in Proc. 2018 IEEE Radio and Wireless Symposium (RWS), pp. 72–75, Jan. 2018.
[108] ——, “A design of non-binary turbo codes over finite fields based on gaussian approximation and union bounds,” in Proc. 2018 IEEE Vehicular Technology Conference (VTC-Spring), Jun. 2018.
[109] ——, “Capacity-approaching non-binary turbo codes: A hybrid design based on exit charts and union bounds,” IEEE Access, vol. 6, pp. 70 952–70 963, 2018.
[110] D. J. MacKay and R. M. Neal, “Near shannon limit performance of low density parity check codes,” Electronics letters, vol. 32, no. 18, pp. 1645–1646, Aug. 1996.
[111] H. C. Davey and D. J. MacKay, “Low density parity check codes over GF(q),” IEEE Commun. Lett., vol. 2, no. 6, pp. 165–167, Jun. 1998.
[112] J. Berkmann, “Symbol-by-symbol map decoding of nonbinary codes,” Proc. ITG-Fachtagung Codierung fu¨r Quelle, Kanal und Ubertragung, 1998.
[113] D. J. MacKay and M. C. Davey, “Evaluation of Gallager codes for short block length and high rate applications,” in Proc. IMA Workshop Codes, Syst, Graphical Models, 1999.
[114] H. Song and J. Cruz, “Reduced-complexity decoding of Q-ary LDPC codes for magnetic record- ing,” IEEE Trans. Magn., vol. 39, no. 2, pp. 1081–1087, Mar. 2003.
[115] S. Benedetto and G. Montorsi, “Unveiling turbo codes: Some results on parallel concatenated coding schemes,” IEEE Trans. Inf. Theory, vol. 42, no. 2, pp. 409–428, Mar. 1996.
[116] M. Luby, M. Mitzenmacher, A. Shokrollah, and D. Spielman, “Analysis of low density codes and improved designs using irregular graphs,” in Proc. the 30th Annual ACM Symp. Theory of Comput- ing (STOC), pp. 249–258, 1998.
[117] H. El Gamal and A. R. Hammons, “Analyzing the turbo decoder using the Gaussian approxima- tion,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 671–686, Feb. 2001.
[118] G. Li, I. J. Fair, and W. A. Krzymien, “Density evolution for nonbinary LDPC codes under Gaussian approximation,” IEEE Trans. Inf. Theory, vol. 55, no. 3, pp. 997–1015, Mar. 2009.
[119] D. Divsalar, S. Dolinar, and F. Pollara, “Iterative turbo decoder analysis based on density evolution,” IEEE J. Sel. Areas Commun., vol. 19, no. 5, pp. 891–907, May 2001.
[120] A. Bennatan and D. Burshtein, “Design and analysis of nonbinary LDPC codes for arbitrary discrete-memoryless channels,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 549–583, Jan. 2006.
[121] J. Kliewer, S. X. Ng, and L. Hanzo, “Efficient computation of EXIT functions for nonbinary itera- tive decoding,” IEEE Trans. Commun., vol. 54, no. 12, pp. 2133–2136, Dec. 2006.
[122] S. X. Ng, O. R. Alamri, Y. Li, J. Kliewer, and L. Hanzo, “Near-capacity turbo trellis coded modu- lation design based on EXIT charts and union bounds,” IEEE Trans. Commun., vol. 56, no. 12, pp. 2030–2039, Dec. 2008.
[123] V. Savin, “Split-extended LDPC codes for coded cooperation,” in Proc. 2010 International Sympo- sium on Information Theory and its Applications (ISITA), pp. 151–156, 2010.
[124] B. M. Kurkoski, K. Yamaguchi, and K. Kobayashi, “Single-gaussian messages and noise thresholds for decoding low-density lattice codes,” in Proc. 2009 IEEE International Symposium on Informa- tion Theory (ISIT), pp. 734–738, 2009.
[125] M. Takai and K. Ishibashi, “Repeat-accumulate signal codes,” IEEE Trans. Commun., vol. 67, no. 4, pp. 2607–2619, 2019.
[126] A. C. Reid, T. A. Gulliver, and D. P. Taylor, “Rate-1/2 component codes for nonbinary turbo codes,” IEEE Trans. Commun., vol. 53, no. 9, pp. 1417–1422, Sep. 2005.
[127] J. Berkmann, “On turbo decoding of nonbinary codes,” IEEE Commun. Lett., vol. 2, no. 4, pp. 94–96, Apr. 1998.
[128] M. Ferrari and S. Bellini, “Rate variable, multi-binary turbo codes with controlled error-floor,” IEEE Trans. Commun., vol. 57, no. 5, 2009.
[129] C. Douillard and C. Berrou, “Turbo codes with rate-m/(m + 1) constituent convolutional codes,” IEEE Trans. Commun., vol. 53, no. 10, pp. 1630–1638, Oct. 2005.
[130] G. S. White, D. Costello et al., “Construction and performance of q-ary turbo codes for use with M - ary modulation techniques,” in Proc. 2000 IEEE International Symposium on Information Theory (ISIT), pp. 220–220, 2000.
[131] J. A. Briffa and H. G. Schaathun, “Non-binary turbo codes and applications,” in Proc. 2008 5th International Symposium on Turbo Codes and Related Topics (ISTC), pp. 294–298, 2008.
[132] H. Balta, C. Douillard, and R. Lucaciu, “Multi-non-binary turbo codes,” EURASIP J. Wireless Commun. Netw. https://doi.org/10.1186/1687-1499-2013-279, Dec. 2013.
[133] G. Liva, E. Paolini, B. Matuz, S. Scalise, and M. Chiani, “Short turbo codes over high order fields,” IEEE Trans. Commun., vol. 61, no. 6, pp. 2201–2211, Jun. 2013.
[134] F. Steiner, G. Bocherer, and G. Liva, “Protograph-based LDPC code design for bit-metric decod- ing,” in Proc. 2015 IEEE International Symposium on Information Theory (ISIT), Jun. 2015.
[135] F. Steiner, G. Liva, and G. Bcherer, “Ultra-sparse non-binary LDPC codes for probabilistic ampli- tude shaping,” arXiv:1708.05558, Aug. 2017.
[136] T. Prinz, P. Yuan, G. Bocherer, F. Steiner, O. Iscan, R. Bohnke, and W. Xu, “Polar coded proba- bilistic amplitude shaping for short packets,” in Proc. 2017 IEEE 18th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Jul. 2017.
[137] O. ˙Is¸can, R. Bo¨hnke, and W. Xu, “Shaped polar codes for higher order modulation,” IEEE Commun. Lett., vol. 22, no. 2, pp. 252–255, Feb. 2018.
[138] N. S. Loghin, J. Zo¨llner, B. Mouhouche, D. Ansorregui, J. Kim, and S.-I. Park, “Non-uniform constellations for ATSC 3.0,” IEEE Trans. Broadcast., vol. 62, no. 1, pp. 197–203, Mar. 2016.
[139] M. El-Hajjar and L. Hanzo, “EXIT charts for system design and analysis,” IEEE Communications Surveys & Tutorials, vol. 16, no. 1, pp. 127–153, 2014.
[140] J. Hou, P. H. Siegel, L. B. Milstein, and H. D. Pfister, “Capacity-approaching bandwidth-efficient coded modulation schemes based on low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 49, no. 9, pp. 2141–2155, 2003.
[141] S. Benedetto, R. Garello, and G. Montorsi, “A search for good convolutional codes to be used in the construction of turbo codes,” IEEE Trans. Commun., vol. 46, no. 9, pp. 1101–1105, Sep. 1998.
[142] X.-Y. Hu, E. Eleftheriou, and D.-M. Arnold, “Regular and irregular progressive edge-growth Tan- ner graphs,” IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 386–398, Jan. 2005.
[143] C. Poulliat, M. Fossorier, and D. Declercq, “Design of regular (2, dc)-LDPC codes over GF(q) using their binary images,” IEEE Trans. Commun., vol. 56, no. 10, pp. 1626–1635, Oct. 2008.
[144] R. G. Gallager, Information Theory and Reliable Communication. Wiley, 1968.
[145] T. Matsumine, B. M. Kurkoski, and H. Ochiai, “Construction d lattice decoding and its application to bch code lattices,” in Proc. 2018 IEEE Global Communications Conference (GLOBECOM), pp. 1–6, 2018.
[146] W. Kositwattanarerk and F. Oggier, “Connections between construction d and related constructions of lattices,” arXiv:1308.6175, Aug. 2013.
[147] M. P. C. Fossorier and S. Lin, “Soft-decision decoding of linear block codes based on ordered statistics,” IEEE Trans. Inf. Theory, vol. 41, no. 5, pp. 1379–1396, Sep. 1995.
[148] P. R. Branco da Silva and D. Silva, “Multilevel LDPC lattices with efficient encoding and decoding and a generalization of construction D’,” ArXiv e-prints, vol. abs/1712.08201, December 2017.
[149] A. Sakzad, M. Sadeghi, and D. Panario, “Construction of turbo lattices,” in Proceedings 48th An- nual Allerton Conference on Communication, Control, and Computing, pp. 14–21, Monticello, IL, USA, September 2010.
[150] R. H. Morelos-Zaragoza, The Art of Error Correcting Coding. John Wiley & Sons, 2006.
[151] A. Ingber, R. Zamir, and M. Feder, “Finite-dimensional infinite constellations.” IEEE Trans. Inf. Theory, vol. 59, no. 3, pp. 1630–1656, 2013.
[152] D. Agrawal and A. Vardy, “Generalized minimum distance decoding in Euclidean space: Perfor- mance analysis,” IEEE Trans. Inf. Theory, vol. 46, no. 1, pp. 60–83, 2000.
[153] O. F. Acikel and W. E. Ryan, “Punctured turbo-codes for BPSK/QPSK channels,” IEEE Trans. Commun., vol. 47, no. 9, pp. 1315–1323, Sep. 1999.
[154] T. Matsumine and H. Ochiai, “Triple parallel concatenated trellis coded modulation,” in Proc. 2017 IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), pp. 1–5, 2017.
[155] C. Fragouli and R. D. Wesel, “Turbo-encoder design for symbol-interleaved parallel concatenated trellis-coded modulation,” IEEE Trans. Commun., vol. 49, no. 3, pp. 425–435, Mar. 2001.
[156] D. Divsalar and F. Pollara, “Multiple turbo codes for deep-space communications,” JPL TDA Progr. Rep., pp. 66–77, May 1995.
[157] M. Breiling, “A logarithmic upper bound on the minimum distance of turbo codes,” IEEE Trans. Inf. Theory, vol. 50, no. 8, pp. 1692–1710, Aug. 2004.
[158] N. Kahale and R. Urbanke, “On the minimum distance of parallel and serially concatenated codes,” in Proc. 1998 IEEE International Symposium on Information Theory (ISIT), p. 31, 1998.
[159] S. ten Brink, “Convergence of multidimensional iterative decoding schemes,” in Proc. 35th Asilo- mar Conf. Signals, Systems and Computers (ACSSC), vol. 1, pp. 270–274, 2001.
[160] K. S. Arkoudogiannis, C. E. Dimakis, and K. V. Koutsouvelis, “Turbo trellis-coded modulation: A weight spectrum view at the odd-even constraint,” in Proc. 9th International Symposium on Turbo Codes and Iterative Information Processing (ISTC), pp. 76–80, 2016.
[161] Y.-J. Wu and H. Ogiwara, “Symbol-interleaver design for turbo trellis-coded modulation,” IEEE Commun. Lett., vol. 8, no. 10, pp. 632–634, Oct. 2004.
[162] H. Chen and A. Haimovich, “EXIT charts for turbo trellis-coded modulation,” IEEE Commun. Lett., vol. 8, no. 11, pp. 668–670, Nov. 2004.
[163] C. He, M. Lentmaier, D. J. Costello, and K. S. Zigangirov, “Joint permutor analysis and design for multiple turbo codes,” IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 4068–4083, Sep. 2006.
[164] E. Boutillon and D. Gnaedig, “Maximum spread of D-dimensional multiple turbo codes,” IEEE Trans. Commun., vol. 53, no. 8, pp. 1237–1242, Aug. 2005.
[165] J. Han and O. Y. Takeshita, “On the decoding structure for multiple turbo codes,” in Proc. 2001 IEEE International Symposium on Information Theory (ISIT), p. 98, 2001.
[166] T. Matsumine and H. Ochiai, “A serial concatenation of binary-input nonbinary-output convolu- tional code and recursive convolutional lattice code,” IEEE Access, vol. 6, pp. 24 809–24 817, 2018.
[167] M. Nakamura and H. Torii, “Ternary phase shift keying and its performance,” in Proc. 2002 IEEE International International Symposium on Wireless Personal Multimedia Communications (WPMC), pp. 1284–1288, Oct. 2002.
[168] M. Tanahashi and H. Ochiai, “A multilevel coded modulation approach for hexagonal signal con- stellation,” IEEE Trans. Wireless Commun., vol. 8, no. 10, pp. 4993–4997, Oct. 2009.
[169] M. Abdelaziz and T. A. Gulliver, “Ternary convolutional codes for ternary phase shift keying,” IEEE Commun. Lett., vol. 20, no. 9, pp. 1709–1712, Aug. 2016.
[170] M. Bingeman and A. K. Khandani, “Symbol-based turbo codes,” IEEE Commun. Lett., vol. 3, no. 10, pp. 285–287, Oct. 1999.
[171] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “Serial concatenation of interleaved codes: Performance analysis, design, and iterative decoding,” IEEE Trans. Inf. Theory, vol. 44, no. 3, pp. 909–926, May 1998.
[172] B. Scanavino, G. Montorsi, and S. Benedetto, “Convergence properties of iterative decoders work- ing at bit and symbol level,” in Proc. 2001 IEEE Global Communications Conference (GLOBE- COM), vol. 2, pp. 1037–1041, 2001.
[173] H. Ochiai, “Exact and approximate distributions of instantaneous power for pulse-shaped single- carrier signals,” IEEE Trans. Wireless Commun., vol. 10, no. 2, pp. 682–692, Feb. 2011.
[174] ——, “An analysis of band-limited communication systems from amplifier efficiency and distortion perspective,” IEEE Trans. Commun., vol. 61, no. 4, pp. 1460–1472, Apr. 2013.
[175] D. Divsalar and E. Pollara, “Turbo codes for PCS applications,” in Proc. 1995 IEEE International Conference on Communications (ICC), vol. 1, pp. 54–59, Jun. 1995.
[176] T. Matsumine, T. Koike-Akino, D. S. Millar, K. Kojima, and K. Parsons, “Short lattice-based shap- ing approach exploiting non-binary coded modulation,” in Proc. 45th European Conference on Optical Communication (ECOC), pp. 1–3, 2019.
[177] J. Conway and N. Sloane, “Fast quantizing and decoding and algorithms for lattice quantizers and codes,” IEEE Trans. Inf. Theory, vol. 28, no. 2, pp. 227–232, 1982.
[178] B. M. Kurkoski, “Encoding and indexing of lattice codes,” IEEE Trans. Inf. Theory, vol. 64, no. 9, pp. 6320–6332, 2018.
[179] J. Conway and N. Sloane, “Voronoi regions of lattices, second moments of polytopes, and quanti- zation,” IEEE Trans. Inf. Theory, vol. 28, no. 2, pp. 211–226, 1982.
[180] T. Matsumine, T. Koike-Akino, D. S. Millar, K. Kojima, and K. Parsons, “Polar-coded modula- tion for joint channel coding and probabilistic shaping,” in Proc. Optical Fiber Communication Conference (OFC), p. M4B.2, 2019.
[181] P. Trifonov and V. Miloslavskaya, “Polar subcodes,” IEEE J. Sel. Areas Commun., vol. 34, no. 2, pp. 254–266, 2016.
[182] P. Schulte and G. Bo¨cherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory, vol. 62, no. 1, pp. 430–434, 2016.
[183] T. Fehenberger, D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, “Multiset-partition dis- tribution matching,” IEEE Trans. Commun., vol. 67, no. 3, pp. 1885–1893, Nov. 2018.
[184] D. S. Millar, T. Fehenberger, T. Koike-Akino, K. Kojima, and K. Parsons, “Distribution matching for high spectral efficiency optical communication with multiset partitions,” J. Lightw. Technol., vol. 37, no. 2, pp. 517–523, Mar. 2019.
[185] A. Amari, S. Goossens, Y. C. Gultekin, O. Vassilieva, I. Kim, T. Ikeuchi, C. Okonkwo, F. M. Willems, and A. Alvarado, “Introducing enumerative sphere shaping for optical communication systems with short blocklengths,” arXiv preprint arXiv:1904.06601, 2019.
[186] P. Schulte and F. Steiner, “Divergence-optimal fixed-to-fixed length distribution matching with shell mapping,” IEEE Wireless Commun. Lett., vol. 8, no. 2, pp. 620–623, 2019.
[187] J. Honda and H. Yamamoto, “Polar coding without alphabet extension for asymmetric models,” IEEE Trans. Inf. Theory, vol. 59, no. 12, pp. 7829–7838, 2013.
[188] M. Mondelli, S. H. Hassani, and R. L. Urbanke, “How to achieve the capacity of asymmetric channels,” IEEE Trans. Inf. Theory, vol. 64, no. 5, pp. 3371–3393, 2018.
[189] Y. Polyanskiy, H. V. Poor, and S. Verdu´, “Channel coding rate in the finite blocklength regime,” IEEE Trans. Inf. Theory, vol. 56, no. 5, pp. 2307–2359, May 2010.
[190] E. Agrell, J. Lassing, E. G. Strom, and T. Ottosson, “On the optimality of the binary reflected Gray code,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3170–3182, 2004.
[191] O. ˙Is¸can, R. Bo¨hnke, and W. Xu, “Probabilistically shaped multi-level coding with polar codes for fading channels,” in 2018 IEEE Globecom Workshops (GC Wkshps), pp. 1–5, 2018.
[192] Y. C. Gultekin, W. v. Houtum, A. Koppelaar, and F. M. Willems, “Partial enumerative sphere shap- ing,” arXiv preprint arXiv:1904.04528v2, 2019.