Nonuniform Fuchsian codes for noisy channels

We develop a new transmission scheme for additive white Gaussian noisy (AWGN) channels based on Fuchsian groups from rational quaternion algebras. The structure of the proposed Fuchsian codes is nonlinear and nonuniform, hence conventional decoding methods based on linearity and symmetry do not apply. Previously, only brute force decoding methods with complexity that is linear in the code size exist for general nonuniform codes. However, the properly discontinuous character of the action of the Fuchsian groups on the complex upper half-plane translates into decoding complexity that is logarithmic in the code size via a recently introduced point reduction algorithm

Basic concepts in quantum computation

Section headings: 1 Qubits, gates and networks 2 Quantum arithmetic and function evaluations 3 Algorithms and their complexity 4 From interferometers to computers 5 The first quantum algorithms 6 Quantum search 7 Optimal phase estimation 8 Periodicity and quantum factoring 9 Cryptography 10 Conditional quantum dynamics 11 Decoherence and recoherence 12 Concluding remarksComment: 37 pages, lectures given at les Houches Summer School on "Coherent Matter Waves", July-August 199

Modular forms and elliptic curves over the field of fifth roots of unity

Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F.Comment: Added appendix by Mark Watkins, who found an elliptic curve missing from our tabl

Pseudorandomness and Dynamics of Fermat Quotients

We obtain some theoretic and experimental results concerning various properties (the number of fixed points, image distribution, cycle lengths) of the dynamical system naturally associated with Fermat quotients acting on the set $\{0, ..., p-1\}$. We also consider pseudorandom properties of Fermat quotients such as joint distribution and linear complexity