Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



On a Diophantine equation related to perfect codes

Author: Ronald Alter
Journal: Math. Comp. 25 (1971), 621-624
MathSciNet review: 0307795
Full-text PDF

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] E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York, 1968, pp. 302-309. MR 38 #6873. MR 0238597 (38:6873)
  • [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 $ {\text{GF}}(q)$,'' Information and Control, v. 16, 1970, pp. 396-401. MR 0270824 (42:5710)

Additional Information

Keywords: Perfect double Hamming-error-correcting codes, Newton approximation method, exponential Diophantine equation, Thue-Siégel-Roth Theorem
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society