arXiv subjects

Ramamonjy Andriamifidisoa

Publications and source records attributed to Ramamonjy Andriamifidisoa.

4 recordsLinked to original sources

arXiv2026

Construction of Multicyclic Codes of Arbitrary Dimension $r$ via Idempotents: A Unified Combinatorial-Algebraic Approach

We propose a unified method to construct multicyclic codes of arbitrary dimension $r$ over $\mathbb{F}_q$. The approach relies on $r$-dimensional primitive idempotents defined as tensor products of univariate ones, combined with multidimensional cyclotomic orbits. This establishes a direct equivalence between combinatorial and algebraic descriptions, yields a natural polynomial basis, and provides an optimal product bound generalizing BCH and Reed-Solomon bounds. An efficient constructive algorithm is presented and illustrated by optimal 3-dimensional codes.

math.AC↗

arXiv2021

Discrete linear Algebraic Dynamical Systems

The vector space of the multi-indexed sequences over a field and the vector space of the sequences with finite support are dual to each other, with respect to a \textit{scalar product}, which we used to define \textit{orthogonals} in these spaces. The closed subspaces in the first vector space are then the orthogonals of subsets in the second space. Using power series and polynomials, we prove that the \textit{polynomial operator in the shift} which U. Oberst and J. C. Willems have introduced to define time invariant discrete linear dynamical systems is the functorial adjoint of the polynomial multiplication. These results are generalized to the case of vectors of sequences and vectors of power series and polynomials. We end this paper by describing discrete linear algebraic dynamical systems.

math.DS↗

arXiv2019

A Dynamical System-based Key Equation for Decoding One-Point Algebraic-Geometry Codes

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.

cs.IT↗

arXiv2014

Duality of Discrete Topological Vector Spaces

For a field $\ef$, the discrete topological vector spaces over $\ \ef$ are essentially of the form $\ef^α$ where $α$ is an ordinal. With additional appropriate properties, they are isomorphic to $\ef^{(β)}$ where $β$ is again an ordinal. Finally, the categories of the vector spaces of the the first and the second type are equivalent.

math.AC↗