scholarly journals Closed sets of finitary functions between finite fields of coprime order

2020 ◽  
Vol 81 (4) ◽  
Author(s):  
Stefano Fioravanti
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Linear Transformation ◽  
Minimal Polynomial ◽  
Invariant Subspaces ◽  
Closed Sets ◽  
The Right ◽  
Coprime Order

AbstractWe investigate the finitary functions from a finite field $$\mathbb {F}_q$$ F q to the finite field $$\mathbb {F}_p$$ F p , where p and q are powers of different primes. An $$(\mathbb {F}_p,\mathbb {F}_q)$$ ( F p , F q ) -linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these subsets of functions through the invariant subspaces of the vector space $$\mathbb {F}_p^{\mathbb {F}_q\backslash \{0\}}$$ F p F q \ { 0 } with respect to a certain linear transformation with minimal polynomial $$x^{q-1} - 1$$ x q - 1 - 1 . Furthermore we prove that each of these subsets of functions is generated by one unary function.

scholarly journals Closed sets of finitary functions between products of finite fields of coprime order

2021 ◽  
Vol 82 (4) ◽  
Author(s):  
Stefano Fioravanti
Keyword(s):  
Finite Fields ◽  
Upper Bound ◽  
Finite Product ◽  
Closed Sets ◽  
The Right ◽  
Multiplicative Monoid ◽  
Coprime Order

AbstractWe investigate the finitary functions from a finite product of finite fields $$\prod _{j =1}^m\mathbb {F}_{q_j} = {\mathbb K}$$ ∏ j = 1 m F q j = K to a finite product of finite fields $$\prod _{i =1}^n\mathbb {F}_{p_i} = {\mathbb {F}}$$ ∏ i = 1 n F p i = F , where $$|{\mathbb K}|$$ | K | and $$|{\mathbb {F}}|$$ | F | are coprime. An $$({\mathbb {F}},{\mathbb K})$$ ( F , K ) -linearly closed clonoid is a subset of these functions which is closed under composition from the right and from the left with linear mappings. We give a characterization of these subsets of functions through the $${\mathbb {F}}_p[{\mathbb K}^{\times }]$$ F p [ K × ] -submodules of $$\mathbb {F}_p^{{\mathbb K}}$$ F p K , where $${\mathbb K}^{\times }$$ K × is the multiplicative monoid of $${\mathbb K}= \prod _{i=1}^m {\mathbb {F}}_{q_i}$$ K = ∏ i = 1 m F q i . Furthermore we prove that each of these subsets of functions is generated by a set of unary functions and we provide an upper bound for the number of distinct $$({\mathbb {F}},{\mathbb K})$$ ( F , K ) -linearly closed clonoids.

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 On $k$-simplexes in $(2k-1)$-dimensional vector spaces over finite fields

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Nous Montrons ◽  
International Audience

International audience We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{ 2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence. Nous montrons que si la cardinalité d'un sous-ensemble de l'espace vectoriel à $(2k-1)$ dimensions sur un corps fini à $q$ éléments est $\gg q^{2k-1-\frac{1}{ 2k}}$, alors il contient une proportion non-nulle de tous les $k$-simplexes de congruence.

ON THE POSSIBILITY OF GROUP-THEORETIC DESCRIPTION OF AN EQUIVALENCE RELATION CONNECTED TO THE PROBLEM OF COVERING SUBSETS IN FINITE FIELDS WITH COSETS OF LINEAR SUBSPACES

2019 ◽  
Vol 53 (1 (248)) ◽  
pp. 23-27
Author(s):  
D.S. Sargsyan
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Equivalence Relation ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Linear Subspaces ◽  
Theoretic Description

Let $ F^{n}_{q} $ be an $ n $-dimensional vector space over a finite field $ F_q $ . Let $ C(F^{n}_{q} ) $ denote the set of all cosets of linear subspaces in $ F^{n}_{q} $. Cosets $ H_1, H_2, \ldots H_s $ are called exclusive if $ H_i \nsubseteq H_j $, $ 1 \mathclose{\leq} i \mathclose{<} j \mathclose{\leq} s $. A permutation $ f $ of $ C(F^{n}_{q} ) $ is called a $ C $-permutation, if for any exclusive cosets $ H, H_1, H_2, \ldots H_s $ such that $ H \subseteq H_1 \cup H_2 \cup \cdots \cup H_s $ we have:i) cosets $ f(H), f(H_1), f(H_2), \ldots, f(H_s) $ are exclusive;ii) cosets $ f^{-1}(H), f^{-1}(H_1), f^{-1}(H_2), \ldots, f^{-1}(H_s) $ are exclusive;iii) $ f(H) \subseteq f(H_1) \cup f(H_2) \cup \cdots \cup f(H_s) $;vi) $ f^{-1}(H) \subseteq f^{-1}(H_1) \cup f^{-1}(H_2) \cup \cdots \cup f^{-1}(H_s) $.In this paper we show that the set of all $ C $-permutations of $ C(F^{n}_{q} ) $ is the General Semiaffine Group of degree $ n $ over $ F_q $.

scholarly journals On the Number of Orthogonal Systems in Vector Spaces over Finite Fields

10.37236/907 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Graph Theoretic ◽  
Theoretic Proof ◽  
Orthogonal Systems

Iosevich and Senger (2008) showed that if a subset of the $d$-dimensional vector space over a finite field is large enough, then it contains many $k$-tuples of mutually orthogonal vectors. In this note, we provide a graph theoretic proof of this result.

Rearrangements of affine subspaces in vector spaces over finite fields

2011 ◽  
Vol 141 (4) ◽  
pp. 709-715
Author(s):  
Anthony Carbery ◽  
Daniel Wilheim
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces ◽  
Characteristic Functions ◽  
Affine Subspaces ◽  
Open Questions

We consider the Lp norms of sums of characteristic functions of affine subspaces of a vector space V over a finite field under certain restrictions on p, dim V and the dimensions of the subspaces involved. We investigate the conditions under which these norms are increased when the affine subspaces are replaced by their parallel translates passing through 0. Applications to extremal configurations for Kakeya maximal-type inequalities are given and open questions are raised.

scholarly journals Averages in vector spaces over finite fields

2008 ◽  
Vol 144 (1) ◽  
pp. 13-27 ◽  
Author(s):  
ANTHONY CARBERY ◽  
BRENDAN STONES ◽  
JAMES WRIGHT
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces

AbstractWe study the analogues of the problems of averages and maximal averages over a surface in $\mathbb{R}^{n}$ when the euclidean structure is replaced by that of a vector space over a finite field, and obtain optimal results in a number of model cases.

The Invariant Subspace Lattice of a Linear Transformation

1967 ◽  
Vol 19 ◽  
pp. 810-822 ◽  
Author(s):  
L. Brickman ◽  
P. A. Fillmore
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Invariant Subspaces ◽  
Arbitrary Field ◽  
Dimensional Vector ◽  
Subspace Lattice ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Algebraic Properties

The purpose of this paper is to study the lattice of invariant subspaces of a linear transformation on a finite-dimensional vector space over an arbitrary field. Among the topics discussed are structure theorems for such lattices, implications between linear-algebraic properties and lattice-theoretic properties, nilpotent transformations, and the conditions for the isomorphism of two such lattices. These topics correspond roughly to §§2, 3, 4, and 5 respectively.

scholarly journals DECOMPOSING ELEMENTS OF A RIGHT SELF-INJECTIVE RING

2013 ◽  
Vol 12 (06) ◽  
pp. 1350014 ◽  
Author(s):  
FEROZ SIDDIQUE ◽  
ASHISH K. SRIVASTAVA
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Division Ring ◽  
Linear Transformations ◽  
One Dimensional ◽  
Factor Ring ◽  
Theoretic Characterization

It was proved independently by both Wolfson [An ideal theoretic characterization of the ring of all linear transformations, Amer. J. Math.75 (1953) 358–386] and Zelinsky [Every linear transformation is sum of nonsingular ones, Proc. Amer. Math. Soc.5 (1954) 627–630] that every linear transformation of a vector space V over a division ring D is the sum of two invertible linear transformations except when V is one-dimensional over ℤ2. This was extended by Khurana and Srivastava [Right self-injective rings in which each element is sum of two units, J. Algebra Appl.6(2) (2007) 281–286] who proved that every element of a right self-injective ring R is the sum of two units if and only if R has no factor ring isomorphic to ℤ2. In this paper we prove that if R is a right self-injective ring, then for each element a ∈ R there exists a unit u ∈ R such that both a + u and a - u are units if and only if R has no factor ring isomorphic to ℤ2 or ℤ3.

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