Complexity of multivariate polynomial evaluation

We describe a method to evaluate multivariate polynomials over a finite field and discuss its multiplicative complexity

Decoding of Projective Reed-Muller Codes by Dividing a Projective Space into Affine Spaces

A projective Reed-Muller (PRM) code, obtained by modifying a (classical) Reed-Muller code with respect to a projective space, is a doubly extended Reed-Solomon code when the dimension of the related projective space is equal to 1. The minimum distance and dual code of a PRM code are known, and some decoding examples have been represented for low-dimensional projective space. In this study, we construct a decoding algorithm for all PRM codes by dividing a projective space into a union of affine spaces. In addition, we determine the computational complexity and the number of errors correctable of our algorithm. Finally, we compare the codeword error rate of our algorithm with that of minimum distance decoding.Comment: 17 pages, 4 figure

Fast algorithm for border bases of Artinian Gorenstein algebras

Given a multi-index sequence $$\sigma$$, we present a new efficient algorithm to compute generators of the linear recurrence relations between the terms of $$\sigma$$. We transform this problem into an algebraic one, by identifying multi-index sequences, multivariate formal power series and linear functionals on the ring of multivariate polynomials. In this setting, the recurrence relations are the elements of the kerne l$I$\sigma of the Hankel operator $H$\sigma associated to $$\sigma$$. We describe the correspondence between multi-index sequences with a Hankel operator of finite rank and Artinian Gorenstein Algebras. We show how the algebraic structure of the Artinian Gorenstein algebra $A$\sigma$$associated to the sequence$$\sigma yields the structure of the terms $\sigma\alpha$for all$$\alpha$ $\in$ N n$. This structure is explicitly given by a border basis of$A$\sigma$$, which is presented as a quotient of the polynomial ring$K[x 1 ,. .. , xn$] by the kernel$I$\sigma$$of the Hankel operator$H$\sigma$$. The algorithm provides generators of$I$\sigma$$constituting a border basis, pairwise orthogonal bases of$A$\sigma$$ and the tables of multiplication by the variables in these bases. It is an extension of Berlekamp-Massey-Sakata (BMS) algorithm, with improved complexity bounds. We present applications of the method to different problems such as the decomposition of functions into weighted sums of exponential functions, sparse interpolation, fast decoding of algebraic codes, computing the vanishing ideal of points, and tensor decomposition. Some benchmarks illustrate the practical behavior of the algorithm

A new approach to the Berlekamp-Massey-Sakata Algorithm. Improving Locator Decoding

We study the problem of the computation of Groebner basis for the ideal of linear recurring relations of a doubly periodic array. We find a set of indexes such that, along with some conditions, guarantees that the set of polynomials obtained at the last iteration in the Berlekamp-Massey-Sakata algorithm is exactly a Groebner basis for the mentioned ideal. Then, we apply these results to improve locator decoding in abelian codes.Comment: 21 page

Guessing Linear Recurrence Relations of Sequence Tuples and P-recursive Sequences with Linear Algebra

International audienceGiven several $n$-dimensional sequences, we first present an algorithmfor computing the Gröbner basis of their module of linear recurrencerelations.A P-recursive sequence $(u_{\mathbf{i}})_{\mathbf{i}\in\mathbb{N}^n}$satisfies linear recurrence relations with polynomial coefficients in$\mathbf{i}$, as defined by Stanley in 1980. Calling directlythe aforementioned algorithm on the tuple ofsequences $\left((\mathbf{i}^{\mathbf{j}}\,u_{\mathbf{i}})_{\mathbf{i}\in\mathbb{N}^n}\right)_{\mathbf{j}}$for retrieving the relations yields redundant relations.Since the module of relations of aP-recursive sequence also has an extra structure of a $0$-dimensional rightideal of an Ore algebra, we design a more efficient algorithm that takesadvantage of this extra structure forcomputing the relations.Finally, we show how to incorporate Gröbner bases computations in anOre algebra $\mathbb{K}\langle t_1,\ldots,t_n,x_1,\ldots,x_n\rangle$, withcommutators $x_k\,x_{\ell}-x_{\ell}\,x_k=t_k\,t_{\ell}-t_{\ell}\,t_k=t_k\,x_{\ell}-x_{\ell}\,t_k=0$ for $k\neq\ell$ and$t_k\,x_k-x_k\,t_k=x_k$, into the algorithm designed for P-recursivesequences. This allows us to compute faster the Gr\"obner basis of the ideal spanned by the first relations,such as in \textsc{2D}/\textsc{3D}-space walks examples