A critical review and some remarks about one- and two-weight irreducible cyclic codes

A lot of work has already been done in one- and two-weight irreducible cyclic codes. In particular, in the remarkable work of Schmidt and White [8], all one- and two-weight irreducible cyclic codes were characterized, and it was conjectured that all of them are either subfield codes, or semiprimitive two-weight codes, or they belong to an exceptional set of eleven codes. In addition, the authors give, indirectly, a characterization of all semiprimitive two-weight irreducible cyclic codes over any finite field. The purpose of this work is to stress the importance of such characterization, and also, to present some remarks about one- and two-weight irreducible cyclic codes. This characterization and our remarks are important because through them we show that most of the recent results in one- and two-weight irreducible cyclic codes are, in one way or another, consequences of the work of Schmidt and White. For example, we will show that all new results regarding the two-weight irreducible cyclic codes, recently presented by C. Ding and J. Yang in [4], can be viewed as mere instances of such characterization. Moreover, through this characterization we generalize the results about projective two-weight irreducible cyclic codes that were presented by J. Wolfmann in [11]. Our results will also be useful to clarify that the family of two-weight irreducible cyclic codes presented by Rao and Pinnawala, in [7], is already included in the paper of Schmidt and White through an equivalence notion, and therefore, a major revision of the Schmidt and White conjecture is not needed.