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Daniele Bartoli

Publications and source records attributed to Daniele Bartoli.

At least 91 records · Page 5Linked to original sources
arXiv2017

On Almost Complete Subsets of a Conic in $\mathrm{PG}(2,q)$, Completeness of Normal Rational Curves and Extendability of Reed-Solomon Codes

A subset $\mathcal{S}$ of a conic $\mathcal{C}$ in the projective plane $\mathrm{PG}(2,q)$ is called almost complete (AC-subset for short) if it can be extended to a larger arc in $\mathrm{PG}(2,q)$ only by the points of $\mathcal{C}\setminus \mathcal{S}$ and by the nucleus of $\mathcal{C}$ when $q$ is even. New upper bounds on the smallest size $t(q)$ of an AC-subset are obtained, in particular, \begin{align*} &t(q)<\sqrt{q(3\ln q+\ln\ln q +\ln3)}+\sqrt{\frac{q}{3\ln q}}+4\thicksim\sqrt{3q\ln q};&t(q)<1.835\sqrt{q\ln q}.\end{align*} The new bounds are used to increase regions of pairs $(N,q)$ for which it is proved that every normal rational curve in $\mathrm{PG}(N,q)$ is a complete $(q+1)$-arc or, equivalently, that no $[q+1,N+1,q-N+1]_q$ generalized doubly-extended Reed-Solomon code can be extended to a $[q+2,N+1,q-N+2]_q$ MDS code.

math.CO↗
arXiv2017

Linear codes from Denniston maximal arcs

In this paper we construct functional codes from Denniston maximal arcs. For $q=2^{4n+2}$ we obtain linear codes with parameters $[(\sqrt{q}-1)(q+1),5,d]_q$ where $\lim_{q \to +\infty} d=(\sqrt{q}-1)q-3\sqrt{q}$. We also find for $q=16,32$ a number of linear codes which appear to have larger minimum distance with respect to the known codes with same length and dimension.

math.CO↗
arXiv2017

Permutation polynomials, fractional polynomials, and algebraic curves

In this note we prove a conjecture by Li, Qu, Li, and Fu on permutation trinomials over $\mathbb{F}_3^{2k}$. In addition, new examples and generalizations of some families of permutation polynomials of $\mathbb{F}_{3^k}$ and $\mathbb{F}_{5^k}$ are given. We also study permutation quadrinomials of type $Ax^{q(q-1)+1} + Bx^{2(q-1)+1} + Cx^{q} + x$. Our method is based on the investigation of an algebraic curve associated with a {fractional polynomial} over a finite field.

math.CO↗
arXiv2017

Exceptional Scattered Polynomials

Let $f$ be an $\mathbb{F}_q$-linear function over $\mathbb{F}_{q^n}$. If the $\mathbb{F}_q$-subspace $U= \{ (x^{q^t}, f(x)) : x\in \mathbb{F}_{q^n} \}$ defines a maximum scattered linear set, then we call $f$ a scattered polynomial of index $t$. As these polynomials appear to be very rare, it is natural to look for some classification of them. We say a function $f$ is an exceptional scattered polynomial of index $t$ if the subspace $U$ associated with $f$ defines a maximum scattered linear set in $\mathrm{PG}(1, q^{mn})$ for infinitely many $m$. Our main results are the complete classifications of exceptional scattered monic polynomials of index $0$ (for $q>5$) and of index $1$. The strategy applied here is to convert the original question into a special type of algebraic curves and then to use the intersection theory and the Hasse-Weil theorem to derive contradictions.

math.CO↗
arXiv2017

AG codes and AG quantum codes from the GGS curve

In this paper, algebraic-geometric (AG) codes associated with the GGS maximal curve are investigated. The Weierstrass semigroup at all $\mathbb F_{q^2}$-rational points of the curve is determined; the Feng-Rao designed minimum distance is computed for infinite families of such codes, as well as the automorphism group. As a result, some linear codes with better relative parameters with respect to one-point Hermitian codes are discovered. Classes of quantum and convolutional codes are provided relying on the constructed AG codes.

math.CO↗
arXiv2017

On the metric dimension of affine planes, biaffine planes and generalized quadrangles

In this paper the metric dimension of (the incidence graphs of) particular partial linear spaces is considered. We prove that the metric dimension of an affine plane of order $q\geq13$ is $3q-4$ and describe all resolving sets of that size if $q\geq 23$. The metric dimension of a biaffine plane (also called a flag-type elliptic semiplane) of order $q\geq 4$ is shown to fall between $2q-2$ and $3q-6$, while for Desarguesian biaffine planes the lower bound is improved to $8q/3-7$ under $q\geq 7$, and to $3q-9\sqrt{q}$ under certain stronger restrictions on $q$. We determine the metric dimension of generalized quadrangles of order $(s,1)$, $s$ arbitrary. We derive that the metric dimension of generalized quadrangles of order $(q,q)$, $q\geq2$, is at least $\max\{6q-27,4q-7\}$, while for the classical generalized quadrangles $W(q)$ and $Q(4,q)$ it is at most $8q$.

math.CO↗
arXiv2017

Upper bounds on the smallest size of a saturating set in projective planes and spaces of even dimension

In a projective plane $Π_{q}$ (not necessarily Desarguesian) of order $q$, a point subset $\mathcal{S}$ is saturating (or dense) if any point of $Π_{q}\setminus \mathcal{S}$ is collinear with two points in $\mathcal{S}$. Modifying an approach of [31], we proved the following upper bound on the smallest size $s(2,q)$ of a saturating set in $Π_{q}$: \begin{equation*} s(2,q)\leq \sqrt{(q+1)\left(3\ln q+\ln\ln q +\ln\frac{3}{4}\right)}+\sqrt{\frac{q}{3\ln q}}+3. \end{equation*} The bound holds for all q, not necessarily large. By using inductive constructions, upper bounds on the smallest size of a saturating set in the projective space $\mathrm{PG}(N,q)$ with even dimension $N$ are obtained. All the results are also stated in terms of linear covering codes.

math.CO↗
arXiv2017

Complete permutation polynomials from exceptional polynomials

We classify complete permutation polynomials of type $aX^{\frac{q^n-1}{q-1}+1}$ over the finite field with $q^n$ elements, for $n+1$ a prime and $n^4 < q$. For the case $n+1$ a power of the characteristic we study some known families. We also classify indecomposable exceptional polynomials of degree $8$ and $9$.

math.CO↗
arXiv2016

Algebraic Geometric codes from Kummer Extensions

For Kummer extensions defined by $y^m = f (x)$, where $f (x)$ is a separable polynomial over the finite field $\mathbb{F}_q$, we compute the number of Weierstrass gaps at two totally ramified places. For many totally ramified places we give a criterion to find pure gaps at these points and present families of pure gaps. We then apply our results to construct many points algebraic geometric codes with good parameters.

math.AG↗
arXiv2016

Tables, bounds and graphics of the smallest known sizes of complete caps in the spaces $\mathrm{PG}(3,q)$ and $\mathrm{PG}(4,q)$

In this paper we present and analyze computational results concerning small complete caps in the projective spaces $\mathrm{PG}(N,q)$ of dimension $N=3$ and $N=4$ over the finite field of order $q$. The results have been obtained using randomized greedy algorithms and the algorithm with fixed order of points (FOP). The computations have been done in relatively wide regions of $q$ values; such wide regions are not considered in literature for $N=3,4$. The new complete caps are the smallest known. Basing on them, we obtained new upper bounds on $t_2(N,q)$, the minimum size of a complete cap in $\mathrm{PG}(N,q)$, in particular, \begin{align*} &t_{2}(N,q)<\sqrt{N+2}\cdot q^{\frac{N-1}{2^{\vphantom{H}}}}\sqrt{\ln q},\quad q\in L_{N},\quad N=3,4,\\ &t_{2}(N,q)<\left(\sqrt{N+1}+\frac{1.3}{\ln (2q)}\right)q^{\frac{N-1}{2^{\vphantom{H}}}}\sqrt{\ln q},\quad q\in L_{N},\quad N=3,4, \end{align*} where \begin{align*} &L_{3}:=\{q\le 4673, ~q\ \textrm{prime}\} \cup \{5003,6007,7001,8009\},\\ &L_{4}:=\{q\le 1361, ~q\ \textrm{prime}\} \cup \{1409\}. \end{align*} Our investigations and results allow to conjecture that these bounds hold for all $q$.

math.CO↗
arXiv2016

Multi-Point AG Codes on the GK Maximal Curves

In this paper we investigate multi-point Algebraic-Geometric codes associated to the GK maximal curve, starting from a divisor which is invariant under a large automorphism group of the curve. We construct families of codes with large automorphism groups.

math.CO↗
arXiv2016

On upper bounds on the smallest size of a saturating set in a projective plane

In a projective plane $Π_{q}$ (not necessarily Desarguesian) of order $q,$ a point subset $S$ is saturating (or dense) if any point of $Π_{q}\setminus S$ is collinear with two points in$~S$. Using probabilistic methods, the following upper bound on the smallest size $ s(2,q)$ of a saturating set in $Π_{q}$ is proved: \begin{equation*} s(2,q)\leq 2\sqrt{(q+1)\ln (q+1)}+2\thicksim 2\sqrt{q\ln q}. \end{equation*} We also show that for any constant $c\ge 1$ a random point set of size $k$ in $Π_{q}$ with $ 2c\sqrt{(q+1)\ln(q+1)}+2\le k<\frac{q^{2}-1}{q+2}\thicksim q$ is a saturating set with probability greater than $1-1/(q+1)^{2c^{2}-2}.$ Our probabilistic approach is also applied to multiple saturating sets. A point set $S\subset Π_{q}$ is $(1,μ)$-saturating if for every point $Q$ of $Π_{q}\setminus S$ the number of secants of $S$ through $Q$ is at least $μ$, counted with multiplicity. The multiplicity of a secant $ \ell $ is computed as ${\binom{\#(\ell \,\cap S)}{2}}.$ The following upper bound on the smallest size $s_{μ}(2,q)$ of a $(1,μ)$-saturating set in $Π_{q}$ is proved: \begin{equation*} s_{μ}(2,q)\leq 2(μ+1)\sqrt{(q+1)\ln (q+1)}+2\thicksim 2(μ+1)\sqrt{ q\ln q}\,\text{ for }\,2\leq μ\leq \sqrt{q}. \end{equation*} By using inductive constructions, upper bounds on the smallest size of a saturating set (as well as on a $(1,μ)$-saturating set) in the projective space $PG(N,q)$ are obtained. All the results are also stated in terms of linear covering codes.

math.CO↗
arXiv2016

A family of semifields in characteristic 2

We construct and describe the basic properties of a family of semifields in characteristic $2.$ The construction relies on the properties of projective polynomials over finite fields. We start by associating non-associative products to each such polynomial. The resulting presemifields form the degenerate case of our family. They are isotopic to the Knuth semifields which are quadratic over left and right nucleus. The non-degenerate members of our family display a very different behaviour. Their left and right nucleus agrees with the center, the middle nucleus is quadratic over the center. None of those semifields is isotopic or Knuth equivalent to a commutative semifield. As a by-product we obtain the complete taxonomy of the characteristic $2$ semifields which are quadratic over the middle nucleus, bi-quadratic over the left and right nucleus and not isotopic to twisted fields. This includes {determining} when two such semifields are isotopic and the order of the autotopism group.

math.CO↗
arXiv2015

Maximum scattered linear sets and complete caps in Galois spaces

Explicit constructions of infinite families of scattered ${\mathbb F}_q$--linear sets in $PG(r-1,q^t)$ of maximal rank $\frac{rt}2$, for $t$ even, are provided. When $q=2$ and $r$ is odd, these linear sets correspond to complete caps in $AG(r,2^t)$ fixed by a translation group of size $2^{\frac{rt}2}$. The doubling construction applied to such caps gives complete caps in $AG(r+1,2^t)$ of size $2^{\frac{rt}2+1}$. For Galois spaces of even dimension greater than $2$ and even square order, this solves the long-standing problem of establishing whether the theoretical lower bound for the size of a complete cap is substantially sharp.

math.CO↗
arXiv2015

Completeness of cubic curves in PG(2, q), q <= 81

Theoretical results are known about the completeness of a planar algebraic cubic curve as a (n,3)-arc in PG(2,q). They hold for q big enough and sometimes have restriction on the characteristic and on the value of the j-invariant. We determine the completeness of all cubic curves for q <= 81.

math.CO↗
arXiv2015

Constant dimension codes from Riemann-Roch spaces

Some families of constant dimension codes arising from Riemann-Roch spaces associated to particular divisors of a curve $\X$ are constructed. These families are generalizations of the one constructed by Hansen

cs.IT↗
arXiv2015

Tables, bounds and graphics of the smallest known sizes of complete arcs in the plane $\mathrm{PG}(2,q)$ for all $q\le160001$ and sporadic $q$ in the interval $[160801\ldots 430007]$

In the projective planes $\mathrm{PG}(2,q)$, we collect the smallest known sizes of complete arcs for the regions \begin{align*} &\mbox{all } q\le160001,~~ q \mbox{ prime power};\\ &Q_{4}=\{34 \mbox{ sporadic }q'\mbox{s in the interval }[160801\ldots430007], \mbox{ see Table 3}\}. \end{align*} For $q\le160001$, the collection of arc sizes is complete in the sense that arcs for all prime powers are considered. This proves new upper bounds on the smallest size $t_{2}(2,q)$ of a complete arc in $\mathrm{PG}(2,q)$, in particular \begin{align*} t_{2}(2,q)&<0.998\sqrt{3q\ln q}<1.729\sqrt{q\ln q}&\mbox{ for }&&7&\le q\le160001;~~(1) \\ t_{2}(2,q)&<\sqrt{q}\ln^{0.7295}q&\mbox{ for }&&109&\le q\le160001;~~(2)\\ t_{2}(2,q)&<\sqrt{q}\ln^{c_{up}(q)}q,~~c_{up}(q)=\frac{0.27}{\ln q}+0.7,&\mbox{ for }&&19&\le q\le160001;~~(3)\\ t_{2}(2,q)&<0.6\sqrt{q}\ln^{φ_{up}(q;0.6)} q,~~φ_{up}(q;0.6)=\frac{1.5}{\ln q}+0.802,&\mbox{ for }&&19&\le q\le160001.~~(4) \end{align*} Moreover, the bounds (2) -- (4) hold also for $q\in Q_{4}$. Also, \begin{align*} t_{2}(2,q)&<1.006\sqrt{3q\ln q}<1.743\sqrt{q\ln q}&\mbox{ for }&&q\in Q_{4}.~~(5) \end{align*} Our investigations and results allow to conjecture that the bounds (2) -- (5) hold for all $q\geq109$.

math.CO↗