Cyclotomic trace codes (Q2005571)

\textit{C. Ding} [IEEE Trans. Inf. Theory 61, No. 6, 3265--3275 (2015; Zbl 1359.94685)] gave a construction of codes using the set of all nonzero squares in the field of order \(p^m\), \(p\) an odd prime. The authors generalize this construction in two ways using a defining set as the collection of non-zero \(s\)-th powers \((s \geq 2)\) and using fields of arbitrary finite order. They use this construction to obtain optimal and near optimal codes as well as projective codes over the field of order \(4\) that give optimal or near optimal quantum codes. When \(s\) is relatively prime to \(p^m-1\), the resulting codes are one wight codes and when \(s\) is not relatively prime to \(p^m-1\), some two weight codes are constructed that give rise to strongly regular graphs and block designs.