Summary: ON THE SEMIPRIMITIVITY OF CYCLIC CODES
YVES AUBRY AND PHILIPPE LANGEVIN
Abstract. 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.
Irreducible cyclic codes are extensively studied in the literature.
They can be defined by three parameters p, m and v and are de-
noted c(p, m, v) (see section 2 for a precise definition). Such codes
with only few different (Hamming) weights are highly appreciated, es-
pecially those with exactly two non-zero weights, called two-weight
codes. The classification of two-weight codes is a classical problem in
coding theory (see ); it is still an open problem but recent progress
has been made. An infinite family, namely the semiprimitive codes (i.e.