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On a Diophantine equation related to perfect codes


Author: Ronald Alter
Journal: Math. Comp. 25 (1971), 621-624
DOI: https://doi.org/10.1090/S0025-5718-1971-0307795-6
MathSciNet review: 0307795
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Abstract | References | Additional Information

Abstract: A necessary condition for the existence of perfect double Hamming-error-correcting codes on q symbols, for q a prime power, is that the Diophantine equation

$\displaystyle \left( {\begin{array}{*{20}{c}} n \\ 0 \\ \end{array} } \right) +... ...ft( {\begin{array}{*{20}{c}} n \\ 2 \\ \end{array} } \right){(q - 1)^2} = {q^k}$

have a nontrivial solution in positive integers. In this paper this equation is considered for all q, and by applying Newton's method for approximating the roots of a polynomial, it is established that it has no nontrivial solutions for all n, odd k, and q of the form $ q = 2{s^2}$.

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

  • [1] Elwyn R. Berlekamp, Algebraic coding theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1968. MR 0238597
  • [2] R. Alter, Perfect Double Hamming-Error-Correcting Codes on Q-Symbols, Proc. Third Annual Princeton Conf. on Information Sciences and Systems, 1969, pp. 547-550.
  • [3] J. H. van Lint, On the nonexistence of perfect 2- and 3-Hamming-error-correcting codes over 𝐺𝐹(𝑞), Information and Control 16 (1970), 396–401. MR 0270824


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1971-0307795-6
Keywords: Perfect double Hamming-error-correcting codes, Newton approximation method, exponential Diophantine equation, Thue-Siégel-Roth Theorem
Article copyright: © Copyright 1971 American Mathematical Society