Ruud Pellikaan

The minimum distance of codes in an array coming from telescopic semigroups

The concept of an error-correcting array gives a new bound on the minimum distance of linear codes and a decoding algorithm which decodes up to half this bound. This gives a unified point of view which explains several improvements on the minimum distance of algebraic-geometric codes. Moreover, it is explained in terms of linear algebra and the theory of semigroups only.

Generalized Hamming weights of q-ary Reed-Muller codes

The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as well as the q-ary Reed-Muller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain condition.

On the efficient decoding of algebraic-geometric codes

This talk is intended to give a survey on the existing literature on the decoding of algebraic-geometric codes. Although the motivation originally was to find an efficient decoding algorithm for algebraic-geometric codes, the latest results give algorithms which can be explained purely in terms of linear algebra. We will treat the following subjects: 1. The decoding problem 2. Decoding by error location 3. Decoding by error location of algebraic-geometric codes 4. Majority coset decoding 5. Decoding algebraic-geometric codes by solving the key equation 6. Improvements of the complexity

On Weierstrass semigroups and the redundancy of improved geometric Goppa codes

Improved geometric Goppa codes have a smaller redundancy and the same bound on the minimum distance as ordinary algebraic-geometry codes. For an asymptotically good sequence of function fields we give a formula for the redundancy.

Equality of geometric Goppa codes and equivalence of divisors

We give necessary and sufficient conditions for two geometric Goppa codes CL(D,G) and CL(D,H) to be the same. As an application we characterize self-dual geometric Goppa codes.

Doing More with Fewer Bits

We present a variant of the Diffie-Hellman scheme in which the number of bits exchanged is one third of what is used in the classical Diffie-Hellman scheme, while the offered security against attacks known today is the same. We also give applications for this variant and conjecture a extension of this variant further reducing the size of sent information.

On the unique representation of very strong algebraic geometry codes

This paper addresses the question of retrieving the triple

Order Functions and Evaluation Codes

{(\mathcal X,\mathcal P, E)}