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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)


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.

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PII: S 0025-5718(1987)0906195-7
Article copyright: © Copyright 1987 American Mathematical Society