On Generalized Galois Cyclic Orbit Flag Codes

Flag codes that are orbits of a cyclic subgroup of the general linear group acting on flags of a vector space over a finite field, are called cyclic orbit flag codes. In this paper, we present a new contribution to the study of such codes, by focusing this time on the generating flag. More precisely, we examine those ones whose generating flag has at least one subfield among its subspaces. In this situation, two important families arise: the already known Galois flag codes, in case we have just fields, or the generalized Galois flag codes in other case. We investigate the parameters and properties of the latter ones and explore the relationship with their underlying Galois flag code.

Irrationality of the Angles of Kloosterman Sums over Finite Field

We prove that the angles of Kloosterman sums over arbitrary finite field are incommensurable with the constant π. In particular, this implies that the Weil bound for Kloosterman sums over finite fields cannot be reached.

On cutting blocking sets and their codes

Let PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} be the r-dimensional projective space over the finite field GF ⁡ ( q ) {\operatorname{GF}(q)} . A set 𝒳 {\mathcal{X}} of points of PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} is a cutting blocking set if for each hyperplane Π of PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} the set Π ∩ 𝒳 {\Pi\cap\mathcal{X}} spans Π. Cutting blocking sets give rise to saturating sets and minimal linear codes, and those having size as small as possible are of particular interest. We observe that from a cutting blocking set obtained in [20], by using a set of pairwise disjoint lines, there arises a minimal linear code whose length grows linearly with respect to its dimension. We also provide two distinct constructions: a cutting blocking set of PG ⁡ ( 3 , q 3 ) {\operatorname{PG}(3,q^{3})} of size 3 ⁢ ( q + 1 ) ⁢ ( q 2 + 1 ) {3(q+1)(q^{2}+1)} as a union of three pairwise disjoint q-order subgeometries, and a cutting blocking set of PG ⁡ ( 5 , q ) {\operatorname{PG}(5,q)} of size 7 ⁢ ( q + 1 ) {7(q+1)} from seven lines of a Desarguesian line spread of PG ⁡ ( 5 , q ) {\operatorname{PG}(5,q)} . In both cases, the cutting blocking sets obtained are smaller than the known ones. As a byproduct, we further improve on the upper bound of the smallest size of certain saturating sets and on the minimum length of a minimal q-ary linear code having dimension 4 and 6.

Galois representations on the cohomology of hyper-Kähler varieties

We show that the André motive of a hyper-Kähler variety X over a field $$K \subset {\mathbb {C}}$$ K ⊂ C with $$b_2(X)>6$$ b 2 ( X ) > 6 is governed by its component in degree 2. More precisely, we prove that if $$X_1$$ X 1 and $$X_2$$ X 2 are deformation equivalent hyper-Kähler varieties with $$b_2(X_i)>6$$ b 2 ( X i ) > 6 and if there exists a Hodge isometry $$f:H^2(X_1,{\mathbb {Q}})\rightarrow H^2(X_2,{\mathbb {Q}})$$ f : H 2 ( X 1 , Q ) → H 2 ( X 2 , Q ) , then the André motives of $$X_1$$ X 1 and $$X_2$$ X 2 are isomorphic after a finite extension of K, up to an additional technical assumption in presence of non-trivial odd cohomology. As a consequence, the Galois representations on the étale cohomology of $$X_1$$ X 1 and $$X_2$$ X 2 are isomorphic as well. We prove a similar result for varieties over a finite field which can be lifted to hyper-Kähler varieties for which the Mumford–Tate conjecture is true.

p-Adic estimates of abelian Artin L-functions on curves

The purpose of this article is to prove a ‘Newton over Hodge’ result for finite characters on curves. Let X be a smooth proper curve over a finite field $\mathbb {F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve. Consider a nontrivial finite character $\rho :\pi _1^{et}(V) \to \mathbb {C}^{\times }$ . In this article, we prove a lower bound on the Newton polygon of the L-function $L(\rho ,s)$ . The estimate depends on monodromy invariants of $\rho $ : the Swan conductor and the local exponents. Under certain nondegeneracy assumptions, this lower bound agrees with the irregular Hodge filtration introduced by Deligne. In particular, our result further demonstrates Deligne’s prediction that the irregular Hodge filtration would force p-adic bounds on L-functions. As a corollary, we obtain estimates on the Newton polygon of a curve with a cyclic action in terms of monodromy invariants.

Constructions and Necessities of Some Permutation Polynomials over Finite Fields

Let F q denote the finite field with q elements. Permutation polynomials over finite fields have important applications in many areas of science and engineering such as coding theory, cryptography, and combinatorial design. The study of permutation polynomials has a long history, and many results are obtained in recent years. In this paper, we obtain some further results about the permutation properties of permutation polynomials. Some new classes of permutation polynomials are constructed, and the necessities of some permutation polynomials are studied.

A Pseudorandom Sequence Generated over a Finite Field Using The Möbius Function

The aim of this paper is to generate and examine a pseudorandom sequence over a finite field using the Möbius function. In the main part of the paper, after generating a number of sequences using the Möbius function, we examine the sequences’ pseudorandomness using autocorrelation and prove that the second half of any sequence in $\mathbb{F}_{3^n}$ is the same as the first, but for the sign of the terms. I reach the conclusion, that it is preferable to generate sequences in fields of the form $\mathbb{F}_{3^n}$, thereby obtaining a sequence of the numbers $-1$,$0$,$1$, each of which appear in the same amounts. There is a variety of applications of the discussed pseudorandom generator and other generators such as cryptography or randomized algorithms.

Elliptic curve over a finite ring generated by 1 and an idempotent element $e$ with coefficients in the finite field $\mathbb{F}_{3^d}$

An elliptic curve over a ring $\mathcal{R}$ is a curve in the projective plane $\mathbb{P}^{2}(\mathcal{R})$ given by a specific equation of the form $f(X, Y, Z)=0$ named the Weierstrass equation, where $f(X, Y, Z)=Y^2Z+a_1XYZ+a_3YZ^2-X^3-a_2X^2Z-a_4XZ^2-a_6Z^3$ with coefficients $a_1, a_2, a_3, a_4, a_6$ in $\mathcal{R}$ and with an invertible discriminant in the ring $\mathcal{R}.$ %(see \cite[Chapter III, Section 1]{sil1}). In this paper, we consider an elliptic curve over a finite ring of characteristic 3 given by the Weierstrass equation: $Y^2Z=X^3+aX^2Z+bZ^3$ where $a$ and $b$ are in the quotient ring $\mathcal{R}:=\mathbb{F}_{3^d}[X]/(X^2-X),$ where $d$ is a positive integer and $\mathbb{F}_{3^d}[X]$ is the polynomial ring with coefficients in the finite field $\mathbb{F}_{3^d}$ and such that $-a^3b$ is invertible in $\mathcal{R}$.

On matroids that are transversal and cotransversal

<p>There are distinct differences between classes of matroids that are closed under principal extensions and those that are not Finite-field-representable matroids are not closed under principal extensions and they exhibit attractive properties like well-quasi-ordering and decidable theories (at least for subclasses with bounded branch-width). Infinite-field-representable matroids, on the other hand, are closed under principal extensions and exhibit none of these behaviours. For example, the class of rank-3 real representable matroids is not well-quasi-ordered and has an undecidable theory. The class of matroids that are transversal and cotransversal is not closed under principal extensions or coprincipal coextentions, so we expect it to behave more like the class of finite-field-representable matroids. This thesis is invested in exploring properties in the aforementioned class. A major idea that has inspired the thesis is the investigation of well-quasi-ordered classes in the world of matroids that are transversal and cotransversal. We conjecture that any minor-closed class with bounded branch-width containing matroids that are transversal and cotransversal is well-quasi-ordered. In Chapter 8 of the thesis, we prove this is true for lattice-path matroids, a well-behaved class that falls in this intersection. The general class of lattice-path matroids is not well-quasi-ordered as it contains an infinite antichain of so-called ‘notch matroids’. The interesting phenomenon that we observe is that this is essentially the only antichain in this class, that is, any minor-closed family of lattice-path matroids that contains only finitely many notch matroids is well-quasi-ordered. This answers a question posed by Jim Geelen. Another question that drove the research was recognising fundamental transversal matroids, since these matroids are also cotransversal. We prove that this problem in general is in NP and conjecture that it is NP-complete. We later explore this question for the classes of lattice-path and bicircular matroids. We are successful in finding polynomial-time algorithms in both classes that identify fundamental transversal matroids. We end this part by investigating the intersection of bicircular and cobicircular matroids. We define a specific class - whirly-swirls - and conjecture that eventually any matroid in the above mentioned intersection belongs to this class.</p>

On matroids that are transversal and cotransversal

<p>There are distinct differences between classes of matroids that are closed under principal extensions and those that are not Finite-field-representable matroids are not closed under principal extensions and they exhibit attractive properties like well-quasi-ordering and decidable theories (at least for subclasses with bounded branch-width). Infinite-field-representable matroids, on the other hand, are closed under principal extensions and exhibit none of these behaviours. For example, the class of rank-3 real representable matroids is not well-quasi-ordered and has an undecidable theory. The class of matroids that are transversal and cotransversal is not closed under principal extensions or coprincipal coextentions, so we expect it to behave more like the class of finite-field-representable matroids. This thesis is invested in exploring properties in the aforementioned class. A major idea that has inspired the thesis is the investigation of well-quasi-ordered classes in the world of matroids that are transversal and cotransversal. We conjecture that any minor-closed class with bounded branch-width containing matroids that are transversal and cotransversal is well-quasi-ordered. In Chapter 8 of the thesis, we prove this is true for lattice-path matroids, a well-behaved class that falls in this intersection. The general class of lattice-path matroids is not well-quasi-ordered as it contains an infinite antichain of so-called ‘notch matroids’. The interesting phenomenon that we observe is that this is essentially the only antichain in this class, that is, any minor-closed family of lattice-path matroids that contains only finitely many notch matroids is well-quasi-ordered. This answers a question posed by Jim Geelen. Another question that drove the research was recognising fundamental transversal matroids, since these matroids are also cotransversal. We prove that this problem in general is in NP and conjecture that it is NP-complete. We later explore this question for the classes of lattice-path and bicircular matroids. We are successful in finding polynomial-time algorithms in both classes that identify fundamental transversal matroids. We end this part by investigating the intersection of bicircular and cobicircular matroids. We define a specific class - whirly-swirls - and conjecture that eventually any matroid in the above mentioned intersection belongs to this class.</p>