(2,1)-separating systems beyond the probabilistic bound

Building on previous results of Xing, we give new lower bounds on the rate of intersecting codes over large alphabets. The proof is constructive, and uses algebraic geometry, although nothing beyond the basic theory of linear systems on curves. Then, using these new bounds within a concatenation argument, we construct binary (2,1)-separating systems of asymptotic rate exceeding the one given by the probabilistic method, which was the best lower bound available up to now. This answers (negatively) the question of whether this probabilistic bound was exact, which has remained open for more than 30 years. (By the way, we also give a formulation of the separation property in terms of metric convexity, which may be an inspirational source for new research problems.)Comment: Version 7 is a shortened version, so that numbering should match with the journal version (to appear soon). Material on convexity and separation in discrete and continuous spaces has been removed. Readers interested in this material should consult version 6 instea

Mod-2 dihedral Galois representations of prime conductor

For all odd primes N up to 500000, we compute the action of the Hecke operator T_2 on the space S_2(Gamma_0(N), Q) and determine whether or not the reduction mod 2 (with respect to a suitable basis) has 0 and/or 1 as eigenvalues. We then partially explain the results in terms of class field theory and modular mod-2 Galois representations. As a byproduct, we obtain some nonexistence results on elliptic curves and modular forms with certain mod-2 reductions, extending prior results of Setzer, Hadano, and Kida

Mod-2 dihedral Galois representations of prime conductor

For all odd primes N up to 500000, we compute the action of the Hecke operator T_2 on the space S_2(Gamma_0(N), Q) and determine whether or not the reduction mod 2 (with respect to a suitable basis) has 0 and/or 1 as eigenvalues. We then partially explain the results in terms of class field theory and modular mod-2 Galois representations. As a byproduct, we obtain some nonexistence results on elliptic curves and modular forms with certain mod-2 reductions, extending prior results of Setzer, Hadano, and Kida.Comment: 16 pages; v2: final submitted versio

Torsion Limits and Riemann-Roch Systems for Function Fields and Applications

The Ihara limit (or -constant) $A(q)$ has been a central problem of study in the asymptotic theory of global function fields (or equivalently, algebraic curves over finite fields). It addresses global function fields with many rational points and, so far, most applications of this theory do not require additional properties. Motivated by recent applications, we require global function fields with the additional property that their zero class divisor groups contain at most a small number of $d$-torsion points. We capture this by the torsion limit, a new asymptotic quantity for global function fields. It seems that it is even harder to determine values of this new quantity than the Ihara constant. Nevertheless, some non-trivial lower- and upper bounds are derived. Apart from this new asymptotic quantity and bounds on it, we also introduce Riemann-Roch systems of equations. It turns out that this type of equation system plays an important role in the study of several other problems in areas such as coding theory, arithmetic secret sharing and multiplication complexity of finite fields etc. Finally, we show how our new asymptotic quantity, our bounds on it and Riemann-Roch systems can be used to improve results in these areas.Comment: Accepted for publication in IEEE Transactions on Information Theory. This is an extended version of our paper in Proceedings of 31st Annual IACR CRYPTO, Santa Barbara, Ca., USA, 2011. The results in Sections 5 and 6 did not appear in that paper. A first version of this paper has been widely circulated since November 200

Bilinear complexity of algebras and the Chudnovsky-Chudnovsky interpolation method

We give new improvements to the Chudnovsky-Chudnovsky method that provides upper bounds on the bilinear complexity of multiplication in extensions of finite fields through interpolation on algebraic curves. Our approach features three independent key ingredients: (1) We allow asymmetry in the interpolation procedure. This allows to prove, via the usual cardinality argument, the existence of auxiliary divisors needed for the bounds, up to optimal degree. (2) We give an alternative proof for the existence of these auxiliary divisors, which is constructive, and works also in the symmetric case, although it requires the curves to have sufficiently many points. (3) We allow the method to deal not only with extensions of finite fields, but more generally with monogenous algebras over finite fields. This leads to sharper bounds, and is designed also to combine well with base field descent arguments in case the curves do not have sufficiently many points. As a main application of these techniques, we fix errors in, improve, and generalize, previous works of Shparlinski-Tsfasman-Vladut, Ballet, and Cenk-Ozbudak. Besides, generalities on interpolation systems, as well as on symmetric and asymmetric bilinear complexity, are also discussed.Comment: 40 pages; difference with previous version: modified Lemma 5.