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Algebraic Coding 1993: Algebraic Coding pp 141-153 | Cite as

Spectral-null codes and null spaces of Hadamard submatrices

  • Ron M. Roth
Sequences
Part of the Lecture Notes in Computer Science book series (LNCS, volume 781)

Abstract

Codes C(m,r) of length 2m over {1,−1} are defined as null spaces of certain submatrices of Hadamard matrices. It is shown that the codewords of C(m, r) all have an rth order spectral null at zero frequency. Establishing the connection between C(m, r) and the parity-check matrix of Reed-Muller codes, the minimum distance of C(m, r) is obtained along with upper bounds on the redundancy of C(m, r).

Keywords

Minimum Distance Null Space Decode Algorithm Integer Vector Hadamard Matrice 
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.
    S. Al-Bassam, B. Bose, On balanced codes, IEEE Trans. Inform. Theory, IT-36 (1990), 406–408.Google Scholar
  2. 2.
    N. Alon, E.E. Bergmann, D. Coppersmith, A.M. Odlyzko, Balancing sets of vectors, IEEE Trans. Inform. Theory, IT-34 (1988), 128–130.Google Scholar
  3. 3.
    A.M. Barg, Incomplete sums, DC-constrained codes, and codes that maintain synchronization, Designs, Codes, and Cryptography, 3 (1993), 105–116.Google Scholar
  4. 4.
    A.M. Barg, S.N. Lytsin, DC-constrained codes from Hadamard matrices, IEEE Trans. Inform. Theory, IT-37 (1991), 801–807.Google Scholar
  5. 5.
    M. Blaum, A (16,9,6,5,4) error-correcting DC-free block code, IEEE Trans. Inform. Theory, IT-34 (1988), 138–141.MathSciNetGoogle Scholar
  6. 6.
    E. Eleftheriou, R. Cideciyan, On codes satisfying M th order running digital sum constraints, IEEE Trans. Inform. Theory, IT-37 (1991), 1294–1313.MathSciNetGoogle Scholar
  7. 7.
    T. Etzion, Constructions of error-correcting DC-free block codes, IEEE Trans. Inform. Theory, IT-36 (1990), 899–905.Google Scholar
  8. 8.
    H.C. Ferreira, Lower bounds on the minimum Humming distance achievable with runlength constrained or DC-free block codes and the synthesis of a (16,8), Dmin=4, DC-free block code, IEEE Trans. Magn., MAG-20 (1984), 881–883.Google Scholar
  9. 9.
    W.H. Gottschalk, G.A. Hedlung, Topological Dynamics, Colloquium Publications of the AMS, 36, American Math. Society, Providence, Rhode Island, 1955.Google Scholar
  10. 10.
    H.D.L. Hollman, K.A.S. Immink, Performance of efficient balanced codes, IEEE Trans. Inform. Theory, IT-37 (1991), 913–918.Google Scholar
  11. 11.
    L.K. Hua, Introduction to Number Theory, Springer, Berlin, 1982.Google Scholar
  12. 12.
    K.A.S. Immink, Coding Techniques for Digital Recorders, Prentice-Hall, London, 1991.Google Scholar
  13. 13.
    K.A.S. Immink, G. Beenker, Binary transmission codes with higher order spectral zeros at zero frequency, IEEE Trans. Inform. Theory, IT-33 (1987), 452–454.Google Scholar
  14. 14.
    R. Karabed, P.H. Siegel, Matched spectral-null codes for partial-response channels, IEEE Trans. Inform. Theory, IT-37 (1991), 818–855.Google Scholar
  15. 15.
    D.E. Knuth, Efficient balanced codes, IEEE Trans. Inform. Theory, IT-32 (1986), 51–53.Google Scholar
  16. 16.
    F.J. MacWilliams, N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977.Google Scholar
  17. 17.
    C.M. Monti, G.L. Pierobon, Codes with a multiple spectral null at zero frequency, IEEE Trans. Inform. Theory, IT-35 (1989), 463–472.Google Scholar
  18. 18.
    R.M. Roth, G.M. Benedek, Interpolation and approximation of sparse multi-variate polynomials over GF(2), SIAM J. Comput., 20 (1991), 291–314.Google Scholar
  19. 19.
    R.M. Roth, P.H. Siegel, A. Vardy, High-order spectral-null codes: Constructions and bounds, submitted to IEEE Trans. Inform. Theory.Google Scholar
  20. 20.
    H. van Tilborg, M. Blaum, On error-correcting balanced codes, IEEE Trans. Inform. Theory, IT-35 (1989), 1091–1095.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Ron M. Roth
    • 1
  1. 1.Computer Science DepartmentTechnion - Israel Institute of TechnologyHaifaIsrael

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