On the weight distribution of a class of cyclic codes

Let q = p<sup>n</sup> with n = 2m and p be a prime. Let 0 ≤ k ≤ n − 1, k ≠ m. Assume k/(m,k) and m/(m,k) are both odd. In this paper we will study the following exponential sums of the equation where Tr<inf>1</inf><sup>n</sup> : F<inf>q</inf> − F<inf>p</inf> and Tr<inf>1</inf><sup>m</sup> : F<inf>p</inf><sup>m</sup> → F<inf>p</inf> are the canonical trace mappings and ζ<inf>p</inf> = e 2πi/p is a primitive p-th root of unity. As an application, we will determine the weight distribution of the cyclic codes C over F<inf>p</inf> with parity-check polynomial h<inf>1</inf>(x)h<inf>2</inf>(x) where h<inf>1</inf>(x) and h<inf>2</inf>(x) are the minimal polynomials of πequation and πequation over F<inf>p</inf> respectively for a primitive element π of F<inf>q</inf>.

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