  In this article, we present a new construction of codes from algebraic
curves. Given a curve over a non-prime finite field, the obtained codes are
defined over a subfield. We call them Cartier Codes since their construction
involves the Cartier operator. This new class of codes can be regarded as a
natural geometric generalisation of classical Goppa codes. In particular, we
prove that a well-known property satisfied by classical Goppa codes extends
naturally to Cartier codes. We prove general lower bounds for the dimension and
the minimum distance of these codes and compare our construction with a
classical one: the subfield subcodes of Algebraic Geometry codes. We prove that
every Cartier code is contained in a subfield subcode of an Algebraic Geometry
code and that the two constructions have similar asymptotic performances.
  We also show that some known results on subfield subcodes of Algebraic
Geometry codes can be proved nicely by using properties of the Cartier operator
and that some known bounds on the dimension of subfield subcodes of Algebraic
Geometry codes can be improved thanks to Cartier codes and the Cartier
operator.
