Diophantine Equations and Cyclic Codes

Abstract

We consider a polynomial equation of the type

$${a_1}x_1^{{d_1}} + \cdots + {a_s}x_s^{{d_s}} = b,$$

(2.1)

, where s (≥ 2), d,..., d _s are positive integers, b∈ F _q , a ₁, ... , a _s ∈ F ^x _q and q = p ^r with a prime p. Such an equation is called a diagonal equation. By the number N of solutions of this equation (2.1) in F ^s _q we mean the number of s-tuples (γ₁...γ _s for which ${a_1}\gamma _1^{{d_1}} + \cdots + {a_s}\gamma _s^{{d_s}} = b$.