Fernando Piñero

University of Puerto Rico at Ponce | UPRP · Department of Mathematics

Employing underlying geometric and algebraic structures allows for constructing bespoke codes for local recovery of erasures. We survey techniques for enriching classical codes with additional machinery, such as using lines or curves in projective space for local recovery sets or products of curves to enhance the availability of data.

In this paper we prove that the class of Goppa codes whose Goppa polynomial is of the form \(g(x) = \textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}}\) where \(\textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}}\) is a trace polynomial from a field extension of degree \(m \ge 3\) has a better minimum distance than what the...

Photosynthetic picoeukaryotes (PPEs) are characterized by a reduction in cell and genome size but are free living, in contrast to many other organisms that have undergone such reductions. The relative abundance of PPEs in the oceans remains to be determined, as do the evolutionary imperatives behind their cell and genome reduction. Their enigmatic...

In this article we improve the dimension and minimum distance bound of the the Hermitian Lifted Codes LRCs construction from L\'opez, Malmskog, Matthews, Pi\~nero and Wooters via elementary univariarte polynomial division. They gave an asymptotic rate estimate of $0.007$. We improve the rate estimate to $0.1$ using univariate polynomial division

Let q be a prime power, let F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> q </sub> denote the finite field of q elements, and let F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> q ²</sub> denote the field of q <sup xmlns:mml="http:...

The polar orthogonal Grassmann code $C(\mathbb{O}_{3,6})$ is the linear code associated to the Grassmann embedding of the Dual Polar space of $Q^+(5,q)$. In this manuscript we study the minimum distance of this embedding. We prove that the minimum distance of the polar orthogonal Grassmann code $C(\mathbb{O}_{3,6})$ is $q^3-q^3$ for $q$ odd and $q^...

In this article, we consider decoding Grassmann codes, linear codes associated to the Grassmannian and its embedding in a projective space. We look at the orbit structure of Grass-mannian arising from the multiplicative group F* <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> q ^m</sub> in...

In this article we introduce a new class of linear codes, called affine symplectic Grassmann codes, and determine their parameters, automorphism group, minimum distance codewords, dual code and other key features. These linear codes are defined from affine part of polar symplectic Grassmann codes. They combine polar symplectic Grassmann codes and a...

In this article we prove that a class of Goppa codes whose Goppa polynomial is of the form $g(x) = \Tr(x)$ (i.e. $g(x)$ is a trace polynomial from a field extension of degree $m \geq 3$) has a better minimum distance then than what the Goppa bound $d \geq 2deg(g(x))+1$ implies. Our improvement is based on finding another Goppa polynomial $h$ such t...

The Grassmannian is an important object in Algebraic Geometry. One of the many techniques used to study the Grassmannian is to build a vector space from its points in the projective embedding and study the properties of the resulting linear code. We introduce a new class of linear codes, called Affine Hermitian Grassman Codes. These codes are the l...

In this paper, we construct codes for local recovery of erasures with high availability and constant-bounded rate from the Hermitian curve. These new codes, called Hermitian-lifted codes, are evaluation codes with evaluation set being the set of Fq2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepac...

The current best known $[239, 21], \, [240, 21], \, \text{and} \, [241, 21]$ binary linear codes have minimum distance 98, 98, and 99 respectively. In this article, we introduce three binary Goppa codes with Goppa polynomials $(x^{17} + 1)^6, (x^{16} + x)^6,\text{ and } (x^{15} + 1)^6$. The Goppa codes are $[239, 21, 103], \, [240, 21, 104], \, \te...

Recently, Skabelund defined new maximal curves which are cyclic extensions of the Suzuki and Ree curves. Previously, the now well-known GK curves were found as cyclic extensions of the Hermitian curve. In this paper, we consider locally recoverable codes constructed from these new curves, complementing that done for the GK curve. Locally recoverabl...

In this manuscript, we consider decoding Grassmann codes, linear codes associated to Grassmannian of planes in an affine space. We look at the orbit structure of Grassmannian arising from the natural action of multiplicative group of certain finite field extension. We project the corresponding Grassmann code onto these orbits to obtain a few subcod...

Recently, Skabelund defined new maximal curves which are cyclic extensions of the Suzuki and Ree curves. Previously, the now well-known GK curves were found as cyclic extensions of the Hermitian curve. In this paper, we consider locally recoverable codes constructed from these new curves, complementing that done for the GK curve. Locally recoverabl...

In this paper, we construct codes for local recovery of erasures with high availability and constant-bounded rate from the Hermitian curve. These new codes, called Hermitian-lifted codes, are evaluation codes with evaluation set being the set of $\mathbb{F}_{q^2}$-rational points on the affine curve. The novelty is in terms of the functions to be e...

The photosynthetic picoeukaryotes (PPEs) comprise a rare example of free-living eukaryotes that have undergone genome reduction. Here, we examine a duality in the process; the proposed driver of genome reduction (the Black Queen hypothesis, BQH), and the resultant impact of genome information loss (the Proteomic Constraint hypothesis, PCH). The BQH...

We consider the linear code corresponding to a special affine part of the Grassmannian \({G_{2,m}}\), which we denote by \({C^{\mathcal {A}}(2, m)}\). This affine part is the complement of the Schubert divisor of \({G_{2,m}}\). In view of this, we show that there is a projection of Grassmann code onto the affine Grassmann code which is also a linea...

We consider the linear code corresponding to a special affine part of the Grassmannian G 2,m , which we denote by C A (2, m). This affine part is the complement of the Schubert divisor of G 2,m. In view of this, we show that there is a projection of Grassmann code onto the affine Grassmann code which is also a linear isomorphism. This implies that...

Locally repairable codes (LRCs) have received significant recent attention as a method of designing data storage systems robust to server failure. Optimal LRCs offer the ideal trade-off between minimum distance and locality, a measure of the cost of repairing a single codeword symbol. For optimal LRCs with minimum distance greater than or equal to...

We consider the Schubert code \(C_{\alpha }(2, m)\) associated to the \(\mathbb {F}_q\)-rational points of the Schubert variety \(\Omega _{\alpha }(2,m)\) in the Grassmannian \(G_{2,m}\). A correspondence between codewords of \(C_{\alpha }(2, m)\) and skew-symmetric matrices of certain special form is given. Using this correspondence, we give a for...

In this work we extend results on the automorphism group of Schubert varieties. We consider the Schubert conditions which define a Schubert variety. It turns out that an automorphism of the Grassmannian fixes a Schubert variety contained in it if and only if it fixes certain elements of the flag used to define the Schubert conditions.

In this article we prove that Schubert union codes are Tanner codes constructed with the point–line incidence geometry that Schubert varieties inherit from the Grassmannian. We do this by first finding an lengthening algorithm for Tanner codes. This algorithm finds the entries of a codeword of a Tanner code from the entries in a given subset of its...

In this article we study the duals of Grassmann codes, certain codes coming from the Grassmannian variety. Exploiting their structure, we are able to count and classify all their minimum weight codewords. In this classification the lines lying on the Grassmannian variety play a central role. Related codes, namely the affine Grassmann codes, were in...

In this article we prove that Schubert union codes are Tanner codes constructed with the point--line incidence geometry that Schubert varieties inherit from the Grassmannian. We do this by first finding an lengthening algorithm for Tanner codes. This algorithm finds the entries of a codeword of a Tanner code from the entries in a given subset of it...

In this article we study a class of graph codes with cyclic code component codes as affine variety codes. Within this class of Tanner codes we find some optimal binary codes. We use a particular subgraph of the point-line incidence plane of \(\mathbf {A}(2,q)\) as the Tanner graph, and we are able to describe the codes succinctly using Gröbner base...

We present a fast algorithm using Gröbner basis to compute the dimensions of subfield subcodes of Hermitian codes. With these algorithms we are able to compute the exact values of the dimension of all subfield subcodes up to q ≤ 32 and length up to 215. We show that some of the subfield subcodes of Hermitian codes are at least as good as the previo...

We study a class of graph based codes with Reed-Solomon component codes as affine variety codes. We give a formulation of the exact dimension of graph codes in general. We give an algebraic description of these codes which makes the exact computation of the dimension of the graph codes easier.

In this article we describe how to find the parameters of subfield subcodes of extended Norm–Trace codes (ENT codes). With a Gröbner basis of the ideal of the Fqr rational points of the extended Norm–Trace curve one can determine the dimension of the subfield subcodes or the dimension of the trace code. We also find a BCH–like bound from the minimu...