Two-weight cyclic codes constructed as the direct sum of two one-weight cyclic codes
A new family of cyclic codes with only two weights is presented; these codes are constructed as the direct sum of two one-weight cyclic codes of the same length and dimension; the weight enumerator polynomials of these codes are given and a characterization of those which are projective. The proof uses results on characters and Gaussian sums. The two-weight cyclic codes presented by \textit{T. Helleseth} [IEEE Trans. Inf. Theory 22, 631--632 (1976; Zbl 0336.94009)] and \textit{G. Vega} and \textit{J. Wolfmann} [Des. Codes Cryptography 42, No. 3, 327--334 (2007; Zbl 1132.94011)] are special kinds of the new family. By means of some examples (over the fields \(\mathbb F_2\), \(\mathbb F_{16}\), \(\mathbb F_9\)), the authors outline a method for the systematic search for the two-weight cyclic codes constructed as the direct sum of two on-weight cylic codes.
scientific article; zbMATH DE number 5316928
scientific article