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.

An upper bound of Singleton type for componentwise products of linear codes

We give an upper bound that relates the minimum weight of a nonzero componentwise product of codewords from some given number of linear codes, with the dimensions of these codes. Its shape is a direct generalization of the classical Singleton bound.Comment: 9 pages; major improvements in v3: now works for an arbitrary number of codes, and the low-weight codeword can be taken in product form; submitted to IEEE Trans. Inform. Theor

Asymptotically good binary linear codes with asymptotically good self-intersection spans

If C is a binary linear code, let C^2 be the linear code spanned by intersections of pairs of codewords of C. We construct an asymptotically good family of binary linear codes such that, for C ranging in this family, the C^2 also form an asymptotically good family. For this we use algebraic-geometry codes, concatenation, and a fair amount of bilinear algebra. More precisely, the two main ingredients used in our construction are, first, a description of the symmetric square of an odd degree extension field in terms only of field operations of small degree, and second, a recent result of Garcia-Stichtenoth-Bassa-Beelen on the number of points of curves on such an odd degree extension field.Comment: 18 pages; v2->v3: expanded introduction and bibliography + various minor change

On the construction of elliptic Chudnovsky-type algorithms for multiplication in large extensions of finite fields

International audienceWe indicate a strategy in order to construct bilinear multiplication algorithms of type Chudnovsky in large extensions of any finite field. In particular, using the symmetric version of the generalization of Randriambololona specialized on the elliptic curves, we show that it is possible to construct such algorithms with low bilinear complexity. More precisely, if we only consider the Chudnovsky-type algorithms of type symmetric elliptic, we show that the symmetric bilinear complexity of these algorithms is in O(n(2q)^log * q (n)) where n corresponds to the extension degree, and log * q (n) is the iterated logarithm. Moreover, we show that the construction of such algorithms can be done in time polynomial in n. Finally, applying this method we present the effective construction, step by step, of such an algorithm of multiplication in the finite field F 3^57. Index Terms Multiplication algorithm, bilinear complexity, elliptic function field, interpolation on algebraic curve, finite field