On the semiprimitivity of cyclic codes

Overview & Synopsis

We prove, without assuming the Generalized Riemann Hypothesis, but with at most one exception, that an irreducible cyclic code c(p,m,v) with v prime and p of index 2 modulo v is a two-weight code if and only if it is a semiprimitive code or it is one of the six sporadic known codes. The result is proved without any exception for index-two irreducible cyclic c(p,m,v) codes with v prime not congruent to 3 modulo 8. Finally, we prove that these two results hold true in fact for irreducible cyclic code c(p,m,v) such that there is three p-cyclotomic cosets modulo v.