\(\mathbb F_q\)-linear cyclic codes over \(\mathbb F_{q^m}\): DFT approach
Codes over \(\mathbb F_{q^m}\) that are closed under addition and multiplication with elements from \(\mathbb F_q\) are called \(\mathbb F_q\)-linear codes over \(\mathbb F_{q^m}\). A transform domain characterization is obtained of \(\mathbb F_q\)-linear cyclic codes using the discrete Fourier transform over an extension field of \(\mathbb F_{q^m}\). A transform domain condition is given for two vectors to be \(\mathbb F_q\)-dual of each other. This is used to prove the nonexistence of certain self-dual \(\mathbb F_q\)-linear cyclic codes and equivalently the nonexistence of the corresponding self-dual quasi-cyclic codes. The transform domain characterization of \(\mathbb F_q\)-linear cyclic codes is used to derive minimum distance bound for the corresponding quasi-cyclic codes.
scientific article; zbMATH DE number 2139211
scientific article