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Algebraic and Geometric Coding Theory

  • Gregory S. Chirikjian
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

Coding theory is concerned with methods for “packaging” and “unpackaging” messages in order that the most information can be reliably send over a communication channel. In this chapter, a greater emphasis is given to the roles of geometry and group theory in communication problems than is usually the case in presentations of this subject. Geometry and group theory enter in problems of communication in a surprising number of different ways. These include the use of finite groups and sphere packings in highdimensional spaces for the design of error-correcting codes (such as those due to Golay and Hamming). These codes facilitate the efficient and robust transmission of information. Additionally, Lie groups enter in certain decoding problems related to determining the state of various motion sensors.

Keywords

Code Theory Gray Code Sphere Packing Circle Packing Perfect Code 
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.

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References

  1. 1.
    Ahlswede, R., “Group codes do not achieve Shannon’s channel capacity for general discrete channels,” Ann. Math. Statist., 42(1), pp. 224–240, 1971.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Ash, A., Gross, R., Fearless Symmetry: Exposing the Hidden Patterns of Numbers, Princeton University Press, Princeton, NJ, 2006.Google Scholar
  3. 3.
    Berlekamp, E.R., Algebraic Coding Theory, Aegean Park Press, Laguna Hills, CA, 1984.Google Scholar
  4. 4.
    Chirikjian, G.S., Stein, D., “Kinematic design and commutation of a spherical stepper motor,” IEEE/ASME Trans. Mechatron., 4(4), pp. 342–353, 1999.CrossRefGoogle Scholar
  5. 5.
    Chirikjian, G.S., Kim, P.T., Koo, J.Y., Lee, C.H., “Rotational matching problems,” Int. J. Comput. Intell. Applic., 4(4), pp. 401–416, 2004.CrossRefGoogle Scholar
  6. 6.
    Chung, S.-Y., Forney, G.D., Jr., Richardson, T.J., Urbanke, R., “On the design of lowdensity parity check codes within 0.0045 dB of the Shannon limit,” IEEE Commun. Lett., 5, pp. 58–60, 2001.CrossRefGoogle Scholar
  7. 7.
    Conway, J.H., Sloane, N.J.A., Sphere Packings, Lattices and Groups, 3rd ed., Springer, New York, 1999.Google Scholar
  8. 8.
    Coxeter, H.S.M., “The problem of packing a number of equal nonoverlapping circles on a sphere,” Trans. NY Acad. Sci., 24, pp. 320–331, 1962.Google Scholar
  9. 9.
    Dai, W., Liu, Y., Rider, B., “Volume growth and general rate quantization on Grassmann manifolds,” IEEE Global Telecommunications Conference (GLOBECOM), 2007.Google Scholar
  10. 10.
    Dai, W., Liu, Y., Rider, B., “Quantization bounds on Grassmann manifolds and applications to MIMO communications,” IEEE Trans. Inform. Theory, 54(3), pp. 1108–1123, 2008.MathSciNetCrossRefGoogle Scholar
  11. 11.
    de Bruijn, N.G. “A combinatorial problem,” Koninklijke Nederlandse Akademie Wetenschappen, 49, pp. 758–764, 1946.Google Scholar
  12. 12.
    Delsarte, P., Goethals, J.M., Seidel, J.J., “Spherical codes and designs,” Geom. Dedicata, 6, pp. 363–388, 1977.MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Fuertes, J.M., Balle, B., Ventura, E., “Absolute-type shaft encoding using LFSR sequences with a prescribed length,” IEEE Trans. Instrum. Measur., 57(5), pp. 915–922, 2008.CrossRefGoogle Scholar
  14. 14.
    Golay, M.J.E., “Notes on digital coding,” Proc. IRE, 37, p. 657, 1949.Google Scholar
  15. 15.
    Gray, F., “Pulse code communication,” US Patent 2632058, March 1958.Google Scholar
  16. 16.
    Gross, D., Audenaert, K., Eisert, J., “Evenly distributed unitaries: On the structure of unitary designs,” J. Math. Phys., 48, p. 052104, 2007.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hamming, R.W., Coding and Information Theory, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1986.Google Scholar
  18. 18.
    Henkel, O., “Sphere-packing bounds in the Grassmann and Stiefel manifolds,” IEEE Trans. Inform. Theory, 51(10), pp. 3445–3456, 2005.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hochwald, B.M., Marzetta, T.L., “Unitary space-time modulation for multiple-antenna communication in Rayleigh flat-fading,” IEEE Trans. Inform. Theory, 46, pp. 543–564, 2000.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Huffman, W.C., Pless, V., Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.Google Scholar
  21. 21.
    Jain, A.K., Mustafa, T., Zhou, Y., Burdette, C., Chirikjian, G.S., Fichtinger, G., “FTRAC—A robust fluoroscope tracking fiducial,” Med. Phys., 32(10), pp. 3185–3198, 2005.CrossRefGoogle Scholar
  22. 22.
    Jain, A., Zhou, Y., Mustufa, T., Burdette, E.C., Chirikjian, G.S., Fichtinger, G., “Matching and reconstruction of brachytherapy seeds using the Hungarian algorithm (MARSHAL),” Med. Phys., 32(11), pp. 3475–3492, 2005.CrossRefGoogle Scholar
  23. 23.
    Jing, Y., Hassibi, B., “Design of fully-diverse multiple-antenna codes based on Sp(2),” IEEE Trans. Inform. Theory, 50(11), pp. 2639–2656, 2004.MathSciNetCrossRefGoogle Scholar
  24. 24.
    Jing, Y., Hassibi, B., “Three-transmit-antenna space-time codes based on SU(3),” IEEE Trans. Signal Process., 53(10), pp. 3688–3702, 2005.MathSciNetCrossRefGoogle Scholar
  25. 25.
    Karpovsky, M.G., “Fast Fourier transforms on finite non-Abelian groups,” IEEE Trans. Computers, 26(10), pp. 1028–1030, 1977.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Karpovsky, M.G., Stankovic, R.S., Astola, J.T., Spectral Logic and Its Applications for the Design of Digital Devices, Wiley-Interscience, New York, 2008.Google Scholar
  27. 27.
    Lee, S., Fichtinger, G., Chirikjian, G.S., “Novel algorithms for robust registration of fiducials in CT and MRI,” Med. Phys., 29(8), pp. 1881–1891, 2002.CrossRefGoogle Scholar
  28. 28.
    Leech, J., “Some sphere packings in higher space,” Can. J. Math., 16, pp. 657–682, 1964.MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Moon, T.K., Error Correction Coding: Mathematical Methods and Algorithms, John Wiley and Sons, New York, 2005.Google Scholar
  30. 30.
    Nebe, G., Rains, E.M., Sloane, N.J.A., Self-Dual Codes and Invariant Theory, Algorithms and Computation in Mathematics Vol. 17, Springer, New York, 2006.Google Scholar
  31. 31.
    Neutsch, W., Coordinates, Walter de Gruyter and Co., Berlin, 1996.Google Scholar
  32. 32.
    Pless, V., Introduction to the Theory of Error-Correcting Codes, 3rd ed., John Wiley and Sons, New York, 1998.Google Scholar
  33. 33.
    Rogers, C.A., “The packing of equal spheres,” Proc. LondonMath. Soc., Ser. 3, 8, pp. 609– 620, 1958.Google Scholar
  34. 34.
    Rogers, C.A., Packing and Covering, Cambridge University Press, Cambridge, 1964.Google Scholar
  35. 35.
    Scheinerman, E., Chirikjian, G.S., Stein, D., “Encoders for spherical motion using discrete sensors,” in Algorithmic and Computational Robotics: New Directions 2000 WAFR (B. Donald, K. Lynch, D. Rus, eds.), A.K. Peters, Natick, Mass., 2001.Google Scholar
  36. 36.
    Scheinerman, E.R., “Determining planar location via complement-free de Brujin sequences using discrete optical sensors,” IEEE Trans. Robot. Autom., 17(6), pp. 883–889, 2001.CrossRefGoogle Scholar
  37. 37.
    Sloane, N.J.A., The Theory of Error-Correcting Codes, North-Holland Mathematical Library, Elsevier, Amsterdam, 1977.Google Scholar
  38. 38.
    Stein, D., Scheinerman, E.R., Chirikjian, G.S., “Mathematical models of binary sphericalmotion encoders,” IEEE-ASME Trans. Mechatron., 8(2), pp. 234–244, 2003.CrossRefGoogle Scholar
  39. 39.
    Stephenson, K., Introduction to Circle Packing, the Theory of Discrete Analytic Functions, Cambridge University Press, Cambridge, 2005.Google Scholar
  40. 40.
    Tarokh, V., Seshadri, N., Calderbank, A.R., “Space-time codes for high data rate wireless communication: Performance criterion and code construction,” IEEE Trans. Inform. Theory, 44(2), pp. 744–765, 1998.MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Thompson, T.M., From Error-Correcting Codes through Sphere Packings to Simple Groups, The Carus Mathematical Monographs No. 21, Mathematical Association of America, Washington, DC, 1983.Google Scholar
  42. 42.
    Zheng, L., Tse, D.N.C., “Communication on the Grassmann manifold: A geometric approach to the noncoherent multiple-antenna channel,” IEEE Trans. Inform. Theory, 48(2), pp. 359–383, 2002.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringThe Johns Hopkins UniversityBaltimoreUSA

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