Abstract
There is a relative lack of basic and fundamental chapters that can serve as a starting point for researchers in the field of using algebraic geometry theory in forward error correction and especially in BTCs. Even the algebraic geometry approach found to be efficient in dealing with binary and non-binary fields. So this chapter will concentrate on the construction and decoding aspects of AG codes to build up a sound knowledge to start developing the new BTC and IBTC.
Keywords
- Forward Error Correction
- Turbo Code
- Decode Algorithm
- Additive White Gaussian Noise Channel
- Iterative Decode
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Berlekamp ER (1972) A survey of algebraic coding theory: lectures held at the department for automation and information. In: Courses and lectures—international centre for mechanical sciences. Springer, Berlin. http://books.google.co.uk/books?id=i-RQAAAAMAAJ
Goppa VD (1981) Codes on algebraic curves. Soviet Math Dokl 24:75–91
Tsfasman MA, Vladut SG, Zink T (1982) Modular curves, Shimura curves and Goppa codes, better than Varshamov-Gilbert bound. Math Nachtrichten 109:21–28
Justesen J, Larsen K, Jensen H, Havemose A, Hoholdt T (1989) Construction and decoding of a class of algebraic geometry codes. IEEE Trans Inf Theory 35(4):811–821. doi:10.1109/18.32157
Skorobogatov AN, Vladut SG (1990) On the decoding of algebraic-geometric codes. IEEE Trans Inf Theory 36(5):1051–1060. doi:10.1109/18.57204
Justesen J, Larsen K, Jensen H, Hoholdt T (1992) Fast decoding of codes from algebraic plane curves. IEEE Trans Inf Theory 38(1):111–119. doi:10.1109/18.108255
Sakata S (1988) Finding a minimal set of linear recurring relations capable of generating a given finite two-dimensional array. J Symbolic Comput 5(3):321–337. doi:10.1016/S0747-7171(88)80033-6. http://www.sciencedirect.com/science/article/pii/S0747717188800336
Massey J (1969) Shift-register synthesis and bch decoding. IEEE Trans Inf Theory 15(1):122–127. doi:10.1109/TIT.1969.1054260
Berlekamp ER (1984) Algebraic coding theory. No. M-6. Aegean Park Press, California. http://books.google.co.uk/books?id=leSbQgAACAAJ
Feng GL, Rao TRN (1993) Decoding algebraic-geometric codes up to the designed minimum distance. IEEE Trans Inf Theory 39(1):37–45. doi:10.1109/18.179340
Duursma IM (1993) Majority coset decoding. IEEE Trans Inf Theory 39(3):1067–1070. doi:10.1109/18.256518
Feng GL, Rao TRN (1994) A simple approach for construction of algebraic-geometric codes from affine plane curves. IEEE Trans Inf Theory 40(4):1003–1012. doi:10.1109/18.335972
Feng GL, Rao TRN (1995) Improved geometric Goppa codes. i. basic theory. IEEE Trans Inf Theory 41(6):1678–1693. doi:10.1109/18.476241
Yaghoobian T, Blake I (1994) Reed-solomon codes and their applications, Chap. 13. IEEE Press, Piscataway, USA, pp 293–314
Sakata S, Justesen J, Madelung Y, Jensen H, Hoholdt T (1995) Fast decoding of algebraic-geometric codes up to the designed minimum distance. IEEE Trans Inf Theory 41(6):1672–1677. doi:10.1109/18.476240
Sakata S (1990) Extension of the Berlekamp-Massey algorithm to N dimensions. Inf Comput 84(2):207–239. doi:10.1016/0890-5401(90)90039-K. http://dx.doi.org/10.1016/0890-5401(90)90039-K
Heegard C, Little J, Saints K (1995) Systematic encoding via Grobner bases for a class of algebraic-geometric Goppa codes. IEEE Trans Inf Theory 41(6):1752–1761. doi:10.1109/18.476247
Blake I, Heegard C, Hoholdt T, Wei V (1998) Algebraic-geometry codes. IEEE Trans Inf Theory 44(6):2596–2618. doi:10.1109/18.720550
Xing C, Niederreiter H, Lam KY (1999) Constructions of algebraic-geometry codes. IEEE Trans Inf Theory 45(4):1186–1193. doi:10.1109/18.761259
Liu CW (1999) Determination of error values for decoding Hermitian codes with the inverse affine Fourier transform. IEICE Trans Fundam Electron Commun Comput Sci 82(10):2302–2305. http://ci.nii.ac.jp/naid/110003208168/en/
Johnston M, Carrasco R, Burrows BL (2004) Design of algebraic-geometric codes over fading channels. Electron Lett 40(21):1355–1356. doi:10.1049/el:20045392
Johnston M, Carrasco RA (2005) Construction and performance of algebraic-geometric codes over awgn and fading channels. IEE Proc Commun 152(5):713–722. doi:10.1049/ip-com:20045153
Berrou C, Glavieux A, Thitimajshima P (1993) Near shannon limit error-correcting coding and decoding: Turbo-codes. 1. In: IEEE ICC’93, vol 2, pp 1064–1070. doi:10.1109/ICC.1993.397441
Pyndiah R, Glavieux A, Picart A, Jacq S (1994) Near optimum decoding of product codes. In: IEEE GLOBECOM ’94, pp 339–343. doi:10.1109/GLOCOM.1994.513494
Picart A, Pyndiah R (1999) Adapted iterative decoding of product codes. In: IEEE GLOBECOM ’99, vol 5, pp 2357–2362. doi:10.1109/GLOCOM.1999.831724
Hirst SA, Honary B, Markarian G (2001) Fast chase algorithm with an application in turbo decoding. IEEE Trans Commun 49(10):1693–1699. doi:10.1109/26.957387
Martin P, Taylor D (2002) Distance based adaptive scaling in suboptimal iterative decoding. IEEE Trans Commun 50(6):869–871. doi:10.1109/TCOMM.2002.1010602
Berrou C, Jezequel M, Douillard C, Kerouedan S (2001) The advantages of non-binary turbo codes. In: 2001 IEEE information theory workshop, pp 61–63. doi:10.1109/ITW.2001.955136
Aitsab O, Pyndiah R (1996) Performance of reed-solomon block turbo code. In: IEEE GLOBECOM ’96, vol 1, pp 121–125. doi:10.1109/GLOCOM.1996.594345
Sweeney P, Wesemeyer S (2000) Iterative soft-decision decoding of linear block codes. IEE Proc Commun 147(3):133–136. doi:10.1049/ip-com:20000300
Zhou R, Picart A, Pyndiah R, Goalie A (2004a) Reliable transmission with low complexity reed-solomon block turbo codes. In: 1st international symposium on wireless communication systems, pp 193–197. doi:10.1109/ISWCS.2004.1407236
Zhou R, Picart A, Pyndiah R, Goalic A (2004b) Potential applications of low complexity non-binary high code rate block turbo codes. In: IEEE MILCOM 2004, vol 3, pp 1694–1699. doi:10.1109/MILCOM.2004.1495192
Diatta De Geest D, Geller B (2004) Reed-solomon turbo codes for high data rate transmission. In: IEEE 59th VTC 2004, vol 2, pp 1023–1027. doi:10.1109/VETECS.2004.1388986
Piriou E, Jego C, Adde P, Le Bidan R, Jezequel M (2006) Efficient architecture for reed-solomon block turbo code. In: IEEE ISCAS 2006, p 4. doi:10.1109/ISCAS.2006.1693426
Frey B, Mackay D (1999) Irregular turbo codes. In: 37th allerton conference on communication, control and computing. Allerton House, Illinois
Frey B, Mackay D (2000) Irregular turbo-like codes. In: 2nd international symposium on turbo codes and related topics. Brest, France, pp 67–72
Richardson TJ, Shokrollahi MA, Urbanke RL (2001) Design of capacity-approaching irregular low-density parity-check codes. IEEE Trans Inf Theory 47(2):619–637. doi:10.1109/18.910578
Sawaya HE, Boutros JJ (2003) Irregular turbo codes with symbol-based iterative decoding. In: 3rd international symposium on turbo codes and related topics. Brest, France
Sholiyi A (2011) Irregular block turbo codes for communication systems. Ph.D. thesis, Swansea University, Swansea, UK
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2014 The Author(s)
About this chapter
Cite this chapter
Alzubi, J.A., Alzubi, O.A., Chen, T.M. (2014). Literature Review. In: Forward Error Correction Based On Algebraic-Geometric Theory. SpringerBriefs in Electrical and Computer Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-08293-6_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-08293-6_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08292-9
Online ISBN: 978-3-319-08293-6
eBook Packages: EngineeringEngineering (R0)