Abstract
In this paper we discuss the weight distribution of prime cyclic arithmetic codes. This is equivalent to the following number-theoretic problem: A norm | | dependent on a positive integer r is defined on Z p = {0, 1, …, p − 1} as follows: Let 〈 r〉 denote the subgroup of the group of non-zero elements of Z p generated by r. Let | x| be the number of elements of the coset 〈 r〉 x which lie in the interval M (p,r)= p r+t + 1 p r+1 + 2, … rp r+1 where the bracket denotes the greatest integer function. In coding theory | x| is called the weight of x. We study the deviation Δ( p, r) of the weight of x in Z p from the average weight of the non-zero elements of Z p . Several bounds are found for Δ( p, r), and using elementary facts concerning quadratic residues some new conditions are found which imply that Δ( p, r) = 0.
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