Codes and the Cartier operator

Codes and the Cartier Operator

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-kn...

Linear Recurrences with Polynomial Coefficients and Computation of the Cartier-Manin Operator on Hyperelliptic Curves

We improve an algorithm originally due to Chudnovsky and Chudnovsky for computing one selected term in a linear recurrent sequence with polynomial coefficients. Using baby-steps / giant-steps techniques, the nth term in such a sequence can be computed in time proportional to √ n, instead of n for a naive approach. As an intermediate result, we give a fast algorithm for computing the values take...

Linear Recurrences with Polynomial Coefficients and Application to Integer Factorization and Cartier-Manin Operator

We study the complexity of computing one or several terms (not necessarily consecutive) in a recurrence with polynomial coefficients. As applications, we improve the best currently known upper bounds for factoring integers deterministically, and for computing the Cartier-Manin operator of hyperelliptic curves.

Codes and Vertex Operator Algebras

The Rank of the Cartier Operator on Cyclic Covers of the Projective Line

We give a lower bound on the rank of the Cartier operator of Jacobian varieties of hyperelliptic and superelliptic curves in terms of their genus.