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Perfect multiple error-correcting arithmetic codes

Author: Daniel M. Gordon
Journal: Math. Comp. 49 (1987), 621-633
MSC: Primary 11T71; Secondary 94B40
MathSciNet review: 906195
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Abstract: An arithmetic code is a subgroup of $ {{\mathbf{Z}}_{{r^n} \pm 1}}$, with the arithmetic distance $ d(x,y) = {\min _t}x - y \equiv \Sigma _{i = 1}^t{c_i}{r^{n(i)}}\;(\bmod {r^n} \pm 1)$, for $ \vert{c_i}\vert < r$, $ n(i) \geqslant 0$ for $ 1 \leqslant i \leqslant t$. A perfect e-error-correcting code is one from which all $ x \in {{\mathbf{Z}}_{{r^n} \pm 1}}$, are within distance e of exactly one codeword. Necessary and sufficient (assuming the Generalized Riemann Hypothesis) conditions for the existence of infinitely many perfect single error-correcting codes for a given r are known. In this paper some conditions for the existence of perfect multiple error-correcting codes are given, as well as the results of a computer search for examples.

References [Enhancements On Off] (What's this?)

  • [1] W. E. Clark & J. J. Liang, "On arithmetic weight for a general radix representation of integers," IEEE Trans. Inform. Theory, v. 19, 1973, pp. 823-826. MR 0396060 (52:16850)
  • [2] W. E. Clark & J. J. Liang, "On modular weight and cyclic nonadjacent forms for arithmetic codes," IEEE Trans. Inform. Theory, v. 20, 1974, pp. 767-770. MR 0359983 (50:12433)
  • [3] H. Halberstam & H. E. Richert, Sieve Methods, Academic Press, New York, 1974. MR 0424730 (54:12689)
  • [4] D. E. Knuth, The Art of Computer Programming, Vol. 2, 2nd ed., Addison-Wesley, Reading, Mass., 1981. MR 633878 (83i:68003)
  • [5] H. W. Lenstra, Jr., Perfect Arithmetic Codes, Séminaire Delange-Pisot-Poitou (Théorie des Nombres, 1977/78).
  • [6] J. H. van Lint, Introduction to Coding Theory, Springer-Verlag, New York, 1982.
  • [7] H. Riesel, Prime Numbers and Computer Methods for Factorization, Birkhäuser, Boston, 1985. MR 897531 (88k:11002)
  • [8] D. Shanks, Solved and Unsolved Problems in Number Theory, 3rd ed., Chelsea, New York, 1985. MR 798284 (86j:11001)

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Article copyright: © Copyright 1987 American Mathematical Society

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