scholarly journals Corrigendum to “On the numerical range of matrices over a finite field” [Linear Algebra Appl. 512 (2017) 162–171]

2018 ◽  
Vol 556 ◽  
pp. 421-427 ◽  
Author(s):  
E. Ballico
Keyword(s):  
Finite Field ◽  
Linear Algebra ◽  
Numerical Range

scholarly journals Monomial codes seen as invariant subspaces

2017 ◽  
Vol 15 (1) ◽  
pp. 1099-1107 ◽  
Author(s):  
María Isabel García-Planas ◽  
Maria Dolors Magret ◽  
Laurence Emilie Um
Keyword(s):  
Finite Field ◽  
Linear Algebra ◽  
Linear Transformation ◽  
Cyclic Codes ◽  
Invariant Subspaces ◽  
Hyperinvariant Subspaces ◽  
Computationally Expensive ◽  
The Relationship

Abstract It is well known that cyclic codes are very useful because of their applications, since they are not computationally expensive and encoding can be easily implemented. The relationship between cyclic codes and invariant subspaces is also well known. In this paper a generalization of this relationship is presented between monomial codes over a finite field 𝔽 and hyperinvariant subspaces of 𝔽n under an appropriate linear transformation. Using techniques of Linear Algebra it is possible to deduce certain properties for this particular type of codes, generalizing known results on cyclic codes.

scholarly journals Ordinary K3 surfaces over a finite field

2020 ◽  
Vol 2020 (761) ◽  
pp. 141-161
Author(s):  
Lenny Taelman
Keyword(s):  
Finite Field ◽  
Linear Algebra ◽  
Abelian Varieties ◽  
K3 Surfaces ◽  
Small Degree ◽  
Stable Reduction

AbstractWe give a description of the category of ordinary K3 surfaces over a finite field in terms of linear algebra data over {{\mathbf{Z}}}. This gives an analogue for K3 surfaces of Deligne’s description of the category of ordinary abelian varieties over a finite field, and refines earlier work by N.O. Nygaard and J.-D. Yu. Our main result is conditional on a conjecture on potential semi-stable reduction of K3 surfaces over p-adic fields. We give unconditional versions for K3 surfaces of large Picard rank and for K3 surfaces of small degree.

scholarly journals Operator norms determined by their numerical ranges

1971 ◽  
Vol 17 (3) ◽  
pp. 249-255
Author(s):  
Colin M. McGregor
Keyword(s):  
Linear Algebra ◽  
Operator Algebras ◽  
Numerical Range ◽  
Sufficient Conditions ◽  
Unital Algebra ◽  
Complete Answer ◽  
Counter Example ◽  
Operator Norms

This paper owes its origin to the following question posed by A. M. Sinclair, “If a linear algebra with identity has two equivalent unital algebra norms, |.|1 and |.|2, whose corresponding numerical radii, v1 and v2, are equal on the whole algebra, are |.|1 and |.|2 related? Are they, for example, necessarily equal?” We do not give a complete answer to this question but are able to give sufficient conditions on algebras of operators for v1 = v2 to imply |.|1 = |.|2 That this implication does not hold for an arbitrary algebra with identity is demonstrated by means of a counter-example. The result for operator algebras is used to deduce some essentially non numerical range results for equivalent operator norms.

scholarly journals Fast Jacobian Group Operations for C3,4 Curves over a Large Finite Field

2007 ◽  
Vol 10 ◽  
pp. 307-328 ◽  
Author(s):  
Fatima K. Abu Salem ◽  
Kamal khuri-makdisi
Keyword(s):  
Finite Field ◽  
Linear Algebra ◽  
Algebraic Curve ◽  
Vector Spaces ◽  
Explicit Formulae ◽  
Group Law

Let C be an arbitrary smooth algebraic curve of genus g over a large finite field F. The authors of this paper revisit fast addition algorithms in the Jacobian of C due to Khuri-Makdisi [math.NT/0409209, to appear in Mathematics of Computation]. The algorithms, which reduce to linear algebra in vector spaces of dimension O(g) once |K| ≫ g and which asymptotically require O(g2.376) field operations using fast linear algebra, are shown to perform efficiently even for certain low genus curves. Specifically, the authors provide explicit formulae for performing the group law on Jacobians of C3,4 curves of genus 3. They show show that, typically, the addition of two distinct elements in the Jacobian of a C3,4 curve requires 117 multiplications and 2 inversions in K, and an element can be doubled using 129 multiplications and 2 inversions in K. This represents an improvement of approximately 20% over previous methods.

ON THE VANISHING IDEAL OF AN ALGEBRAIC TORIC SET AND ITS PARAMETRIZED LINEAR CODES

2012 ◽  
Vol 11 (04) ◽  
pp. 1250072 ◽  
Author(s):  
ELISEO SARMIENTO ◽  
MARIA VAZ PINTO ◽  
RAFAEL H. VILLARREAL
Keyword(s):  
Finite Field ◽  
Projective Space ◽  
Complete Intersection ◽  
Linear Algebra ◽  
Upper Bound ◽  
Minimum Distance ◽  
Linear Codes ◽  
Intersection Property ◽  
Vanishing Ideal ◽  
Algebraic Invariants

Let K be a finite field and let X be a subset of a projective space, over the field K, which is parametrized by monomials arising from the edges of a clutter. We show some estimates for the degree-complexity, with respect to the revlex order, of the vanishing ideal I(X) of X. If the clutter is uniform, we classify the complete intersection property of I(X) using linear algebra. We show an upper bound for the minimum distance of certain parametrized linear codes along with certain estimates for the algebraic invariants of I(X).

scholarly journals The Hermitian Null-range of a Matrix over a Finite Field

2018 ◽  
Vol 34 ◽  
pp. 205-216 ◽  
Author(s):  
Edoardo Ballico
Keyword(s):  
Finite Field ◽  
Prime Power ◽  
Hermitian Form ◽  
Numerical Range

Let $q$ be a prime power. For $u=(u_1,\dots ,u_n), v=(v_1,\dots ,v_n)\in \mathbb {F} _{q^2}^n$, let $\langle u,v\rangle := \sum _{i=1}^{n} u_i^qv_i$ be the Hermitian form of $\mathbb {F} _{q^2}^n$. Fix an $n\times n$ matrix $M$ over $\mathbb {F} _{q^2}$. In this paper, it is considered the case $k=0$ of the set $\mathrm{Num} _k(M):= \{\langle u,Mu\rangle \mid u\in \mathbb {F} _{q^2}^n, \langle u,u\rangle =k\}$. When $M$ has coefficients in $\mathbb {F} _q$ the paper studies the set $\mathrm{Num} _k(M)_q:= \{\langle u,Mu\rangle \mid u\in \mathbb {F} _q^n,\langle u,u\rangle =k\}\subseteq \mathbb {F} _q$. The set $\mathrm{Num} _1(M)$ is the numerical range of $M$, previously introduced in a paper by Coons, Jenkins, Knowles, Luke, and Rault (case $q$ a prime $p\equiv 3\pmod{4}$), and by the author (arbitrary $q$). In this paper, it is studied in details $\mathrm{Num} _0(M)$ and $\mathrm{Num} _k(M)_q$ when $n=2$. If $q$ is even, $\mathrm{Num} _0(M)_q$ is easily described for arbitrary $n$. If $q$ is odd, then either $\mathrm{Num} _0(M)_q =\{0\}$, or $\mathrm{Num} _0(M)_q=\mathbb {F} _q$, or $\sharp (\mathrm{Num} _0(M)_q)=(q+1)/2$.

Wave Packet Transform over Finite Fields

2015 ◽  
Vol 30 ◽  
pp. 507-529 ◽  
Author(s):  
Arash Ghaani Farashahi
Keyword(s):  
Finite Group ◽  
Wave Packet ◽  
Finite Field ◽  
Linear Algebra ◽  
Finite Fields ◽  
Algebra Approach

In this article we introduce the notion of finite wave packet groups over finite fields as the finite group of dilations, translations, and modulations. Then we will present a unified theoretical linear algebra approach to the theory of wave packet transform (WPT) over finite fields. It is shown that each vector defined over a finite field can be represented as a coherent sum of finite wave packet group elements as well.

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