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Dual codes of systematic group codes over abelian groups

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Abstract

For systematic codes over finite fields the following result is well known: If [I¦P] is the generator matrix then the generator matrix of its dual code is [ −P tr¦I]. The main result is a generalization of this for systematic group codes over finite abelian groups. It is shown that given the endomorphisms which characterize a group code over an abelian group, the endomorphisms which characterize its dual code are identified easily. The self-dual codes are also characterized. It is shown that there are self-dual and MDS group codes over elementary abelian groups which can not be obtained as linear codes over finite fields.

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References

  1. Biglieri E., Elia M.: Construction of linear block codes over groups. IEEE International Symposium on Information Theory, San Antonio, Texas, 1993

    Google Scholar 

  2. Conway J. H., Sloane N. J. A.: Sphere Packings, Lattices and Groups. Springer Berlin, Heidelberg New York 1988

    MATH  Google Scholar 

  3. Delsarte P.: Bounds for unrestricted codes by linear programming. Philips Research Dev. J.,27, 272–289 (1972)

    MATH  MathSciNet  Google Scholar 

  4. Forney G. D.: Geometrically Uniform Codes. IEEE Transactions on Information Theory, Vol. IT-37,5, 1241–1260 (1991)

    Article  MathSciNet  Google Scholar 

  5. Forney G. D. Jr.: On the Hamming distance property of group codes. IEEE Trans. Information Theory, Vol. IT-38,6, 1797–1801 (1992)

    Article  MathSciNet  Google Scholar 

  6. Hall M., Jr.: The Theory of Groups. MacMillan 1964

  7. Loeliger H. A., Mittelholzer T.: Linear codes over groups and new Slepian-type signal sets. Proc. 1991 IEEE International Symposium on Information Theory, Budapest, Hungary, June 24–28 (1991)

  8. Loeliger H. A.: On Euclidean Space Group Codes, Ph. D. Dissertation, Swiss Federal Institute of Technology, Zurich, Switzerland (1992)

    Google Scholar 

  9. Loeliger H. A.: Signal sets matched to groups. IEEE Trans. Information Theory, Vol. IT-37,6, 1675–1682(1991)

    Article  MathSciNet  Google Scholar 

  10. MacWilliams F. J., Sloane N. J. A.: The Theory of Error Correcting Codes, North-Holland (1977)

  11. Massey J. L., Mittelholzer T.: Systematicity and rotational invariance of convolutional codes over rings, Proc. 2nd Int. Workshop on Algebraic and Combinatorial Coding Theory, Leningrad Russia, Sept. 16–22 (1990)

  12. Massey J. L., Mittelholzer T., Riedel T., Vollenweider M.: Ring Convolutional codes for Phase Modulation, Int. Symp. Inform. Theory, San Diego, CA, Jan. 14–19 (1990).

  13. McDonald B. R.: Finite rings with identity, Marcel Dekker, New York (1974)

    MATH  Google Scholar 

  14. Sundar Rajan B., Siddiqi M. U.: Transform domain characterization of cyclic codes over Zm. AAECC 5(5), 261–275 (1994)

    Article  MATH  Google Scholar 

  15. Van der Warden: Modern Algebra II, Fredrick Ungar, New York

  16. Vardy A.: The Nordstrom-Robinson Code: Representation over GF(4) and Efficient Decoding. IEEE Trans. IT, 48(5) 1686–1693 (1994)

    Article  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Electrical Engineering, Indian Institute of Technology, Delhi, Hauz Khas, 110016, New Delhi, India

    A. A. Zain & B. Sundar Rajan

Authors
  1. A. A. Zain
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  2. B. Sundar Rajan
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Part of this correspondence was presented in 1994 International Symposium on Information Theory, Trondheim, Norway, June 1994

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Zain, A.A., Rajan, B.S. Dual codes of systematic group codes over abelian groups. AAECC 8, 71–83 (1997). https://doi.org/10.1007/s002000050054

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  • DOI: https://doi.org/10.1007/s002000050054

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