2019/05/31 by Ramamonjy Andriamifidisoa, Andriamifidisoa, Ramamonjy, Rufine Marius Lalasoa +3
Computer Science · Mathematics · #37N35 #94B27 - Secondary: 39A06 #FOS: Computer and information sciences #Information Theory (cs.IT) #Primary: 11T66 #cs.IT #math.IT #msc:11T66 #msc:37N35 #msc:39A06 #msc:94B27
paper · pdf · doi:10.48550/arxiv.1906.01428
19 pages
arxiv created 2019/05/31 · arxiv updated 2019/06/05
A closer look at linear recurring sequences allowed us to define the multiplication of a univariate polynomial and a sequence, viewed as a power series with another variable, resulting in another sequence. Extending this operation, one gets the multiplication of matrices of multivariate polynomials and vectors of powers series. A dynamical system, according to U. Oberst is then the kernel of the linear mapping of modules defined by a polynomial matrix by this operation. Applying these tools in the decoding of the so-called one point algebraic-geometry codes, after showing that the syndrome array, which is the general transform of the error in a received word is a linear recurring sequence, we construct a dynamical system. We then prove that this array is the solution of Cauchy's homogeneous equations with respect to the dynamical system. The aim of the Berlekamp-Massey-Sakata Algorithm in the decoding process being the determination of the syndrome array, we have proved that in fact, this algorithm solves the Cauchy's homogeneous equations with respect to a dynamical system.