Equidistant Arithmetic Codes and Character Sums Original Research Article

A cyclic arithmetic code is a subgroup of Z/(rn − 1)Z, where the weight of a word x is the minimal number of nonzero coefficients in the representation x ≡ ∑n − 1i = 0ciri with ci < r for all i. A code is called equidistant if all nonzero codewords have the same weight. In this paper necessary conditions for the existence of equidistant codes are given. By relating these conditions to character sums on certain intervals, it is shown that for r = 2, 3 no new equidistant codes exist, and several infinite families of equidistant codes are given.