All two-weight irreducible cyclic codes? (Q1604413)

The authors obtain necessary and sufficient conditions for an irreducible cyclic code to have at most two weights, and it is done without evaluating the corresponding Gauss sums in McEliece's identity. Their approach enables a unified explanation for all known two-weight irreducible codes, and it gives four new sporadic examples. The authors conjecture that there are no other two-weight irreducible cyclic codes and give a partial proof of this conjecture conditionally on the generalized Riemann hypothesis.