On a Diophantine equation related to perfect codes

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