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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Perfect multiple error-correcting arithmetic codes
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by Daniel M. Gordon PDF
Math. Comp. 49 (1987), 621-633 Request permission

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 $|{c_i}| < r$, $n(i) \geqslant 0$ 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
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Math. Comp. 49 (1987), 621-633
  • MSC: Primary 11T71; Secondary 94B40
  • DOI: https://doi.org/10.1090/S0025-5718-1987-0906195-7
  • MathSciNet review: 906195