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Tovohery Hajatiana Randrianarisoa

Publications and source records attributed to Tovohery Hajatiana Randrianarisoa.

7 recordsLinked to original sources
arXiv2022

The Euler characteristic, $q$-matroids, and a Möbius function

We first give two new proofs of an old result that the reduced Euler characteristic of a matroid complex is equal to the Möbius number of the lattice of cycles of the matroid up to the sign. The purpose has been to find a model to establish an analogous result for the case of $q$-matroids and we find a relation between the Euler characteristic of the simplicial chain complex associated to a $q$-matroid complex and the lattice of $q$-cycles of the $q$-matroid. We use this formula to find the complete homology over $\mathbb{Z}$ of this shellable simplicial complex. We give a characterization of nonzero Euler characteristic for such order complexes. Finally, based on these results we remark why singular homology of a $q$-matroid equipped with order topology may not be effective to describe the $q$-cycles unlike the classical case of matroids.

math.CO↗
arXiv2022

Antipodal two-weight rank metric codes

We consider the class of linear antipodal two-weight rank metric codes and discuss their properties and characterization in terms of $t$-spreads. It is shown that the dimension of such codes is $2$ and the minimum rank distance is at least half of the length. We construct antipodal two-weight rank metric codes from certain MRD codes. A complete classification of such codes is obtained, when the minimum rank distance is equal to half of the length. As a consequence of our construction of two-weight rank metric codes, we get some explicit two-weight Hamming metric codes.

cs.IT↗
arXiv2022

Some Matroids Related to Sum-Rank Metric Codes

We introduce the notion of sum-matroids and show its association with sum-rank metric codes. As a consequence, some results for sum-rank metric codes by Martínez-Peñas are generalized for sum-matroids. The sum-matroids generalize the notions of matroids and $q$-matroids. We define the generalized weights for sum-matroids and prove a Wei-type duality theorem which generalizes the analogous cases for matroids and q-matroids.

math.CO↗
arXiv2021

Constructions of optimal rank-metric codes from automorphisms of rational function fields

We define a class of automorphisms of rational function fields of finite characteristic and employ these to construct different types of optimal linear rank-metric codes. The first construction is of generalized Gabidulin codes over rational function fields. Reducing these codes over finite fields, we obtain maximum rank distance (MRD) codes which are not equivalent to generalized twisted Gabidulin codes. We also construct optimal Ferrers diagram rank-metric codes which settles further a conjecture by Etzion and Silberstein.

cs.IT↗
arXiv2018

A Decoding Algorithm for Rank Metric Codes

In this work we will present algorithms for decoding rank metric codes. First we will look at a new decoding algorithm for Gabidulin codes using the property of Dickson matrices corresponding to linearized polynomials. We will be using a Berlekamp-Massey-like algorithm in the process. We will show the difference between our and existing algorithms. Apart from being a new algorithm, it is also interesting that it can be modified to get a decoding algorithm for general twisted Gabidulin codes.

cs.IT↗
arXiv2014

The number of matrices over $\mathbb{F}_q$ with irreducible characteristic polynomial

Let $\mathbb{F}_q$ be a finite field with $q$ elements. M. Gerstenhaber and Irving Reiner has given two different methods to show the number of matrices with a given characteristic polynomial. In this talk, we will give another proof for the particular case where the characteristic polynomial is irreducible. The number of such matrices is important to know the efficiency of an algorithm to factor polynomials using Drinfeld modules.

math.AC↗