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Daniele Bartoli

Publications and source records attributed to Daniele Bartoli.

108 records · Page 6Linked to original sources
arXiv2015

Further results on multiple coverings of the farthest-off points

Multiple coverings of the farthest-off points ($(R,μ)$-MCF codes) and the corresponding $(ρ,μ)$-saturating sets in projective spaces $PG(N,q)$ are considered. We propose and develop some methods which allow us to obtain new small $(1,μ)$-saturating sets and short $(2,μ)$-MCF codes with $μ$-density either equal to 1 (optimal saturating sets and almost perfect MCF-codes) or close to 1 (roughly $1+1/cq$, $c\ge1$). In particular, we provide new algebraic constructions and some bounds. Also, we classify minimal and optimal $(1,μ)$-saturating sets in $PG(2,q)$, $q$ small.

math.CO↗
arXiv2014

2-semiarcs in $\mathrm{PG}(2,q)$, $q\leq 13$

A $2$-semiarc is a pointset ${\mathcal S}_k$ with the property that the number of tangent lines to ${\mathcal S}_k$ at each of its points is two. Using some theoretical results and computer aided search, the complete classification of $2$-semiarcs in PG$(2,q)$ is given for $q\leq 7,$ the spectrum of their sizes is determined for $q\leq 9$, and some results about the existence are proven for $q=11$ and $q=13.$ For several sizes of $2$-semiarcs in $\mathrm{PG}(2,q)$, $q\leq 7$, classification results have been obtained by theoretical proofs.

math.CO↗
arXiv2014

A probabilistic construction of small complete caps in projective spaces

In this work complete caps in $PG(N,q)$ of size $O(q^{\frac{N-1}{2}}\log^{300} q)$ are obtained by probabilistic methods. This gives an upper bound asymptotically very close to the trivial lower bound $\sqrt{2}q^{\frac{N-1}{2}}$ and it improves the best known bound in the literature for small complete caps in projective spaces of any dimension. The result obtained in the paper also gives a new upper bound for $l(m,2,q)_4$, that is the minimal length $n$ for which there exists an $[n,n-m, 4]_q2$ covering code with given $m$ and $q$.

math.CO↗
arXiv2014

Tables of sizes of random complete arcs in the plane $PG(2,q)$

Tables of sizes of random complete arcs in the plane $PG(2,q)$ are given. The sizes are close to the smallest known sizes of complete arcs in $PG(2,q)$, in particular, to ones constructed by Algorithm FOP (fixed order of points). The random arcs are obtained in the region $\{3\leq q\leq 46337,~q \mbox{prime}\}$.

math.CO↗
arXiv2013

Semiovals in PG(2,8) and PG(2,9)

The classification of all semiovals and blocking semiovals in $\mathrm{PG}(2,8)$ and in $\mathrm{PG}(2,9)$ of size less than $17$ is determined. Also, some information on the stabilizer groups and the intersection sizes with lines is given.

math.CO↗
arXiv2013

Complete arcs and complete caps from cubics with an isolated double point

Small complete arcs and caps in Galois spaces over finite fields $\fq$ with characteristic greater than 3 are constructed from cubic curves with an isolated double point. For $m$ a divisor of $q+1$, complete plane arcs of size approximately $q/m$ are obtained, provided that $(m,6)=1$ and $m<\{1}{4}q^{1/4}$. If in addition $m=m_1m_2$ with $(m_1,m_2)=1$, then complete caps of size approximately $\{m_1+m_2}{m}q^{N/2}$ in affine spaces of dimension $N\equiv 0 \pmod 4$ are constructed.

math.CO↗
arXiv2013

Small complete caps from nodal cubics

Bicovering arcs in Galois affine planes of odd order are a powerful tool for constructing complete caps in spaces of higher dimensions. In this paper we investigate whether some arcs contained in nodal cubic curves are bicovering. For $m_1$, $m_2$ coprime divisors of $q-1$, bicovering arcs in $AG(2,q)$ of size $k\le (q-1)\frac{m_1+m_2}{m_1m_2}$ are obtained, provided that $(m_1m_2,6)=1$ and $m_1m_2<\sqrt[4]{q}/3.5$. Such arcs produce complete caps of size $kq^{(N-2)/2}$ in affine spaces of dimension $N\equiv 0 \pmod 4$. For infinitely many $q$'s these caps are the smallest known complete caps in $AG(N,q)$, $N \equiv 0 \pmod 4$.

math.CO↗
arXiv2012

Classification of minimal 1-saturating sets in $PG(2,q)$, $q\leq 23$

Minimal 1-saturating sets in the projective plane $PG(2,q)$ are considered. They correspond to covering codes which can be applied to many branches of combinatorics and information theory, as data compression, compression with distortion, broadcasting in interconnection network, write-once memory or steganography (see \cite{Coh} and \cite{BF2008}). The full classification of all the minimal 1-saturating sets in PG(2,9) and PG(2,11) and the classification of minimal 1-saturating sets of smallest size in PG(2,q), $16\leq q\leq 23$ are given. These results have been found using a computer-based exhaustive search that exploits projective equivalence properties.

math.CO↗
arXiv2011

Upper bounds on the smallest size of a complete arc in the plane PG(2,q)

New upper bounds on the smallest size t_{2}(2,q) of a complete arc in the projective plane PG(2,q) are obtained for q <= 9109. From these new bounds it follows that for q <= 2621 and q = 2659,2663,2683,2693,2753,2801, the relation t_{2}(2,q) < 4.5\sqrt{q} holds. Also, for q <= 5399 and q = 5413,5417,5419,5441,5443,5471,5483,5501,5521, we have t_{2}(2,q) < 4.8\sqrt{q}. Finally, for q <= 9067 it holds that t_{2}(2,q) < 5\sqrt{q}. The new upper bounds are obtained by finding new small complete arcs with the help of a computer search using randomized greedy algorithms.

math.CO↗
arXiv2011

On sizes of complete arcs in PG(2,q)

New upper bounds on the smallest size t_{2}(2,q) of a complete arc in the projective plane PG(2,q) are obtained for 853 <= q <= 4561 and q\in T1\cup T2 where T1={173,181,193,229,243,257,271,277,293,343,373,409,443,449,457, 461,463,467,479,487,491,499,529,563,569,571,577,587,593,599,601,607,613,617,619,631, 641,661,673,677,683,691, 709}, T2={4597,4703,4723,4733,4789,4799,4813,4831,5003,5347,5641,5843,6011,8192}. From these new bounds it follows that for q <= 2593 and q=2693,2753, the relation t_{2}(2,q) < 4.5\sqrt{q} holds. Also, for q <= 4561 we have t_{2}(2,q) < 4.75\sqrt{q}. It is showed that for 23 <= q <= 4561 and q\in T2\cup {2^{14},2^{15},2^{18}}, the inequality t_{2}(2,q) < \sqrt{q}ln^{0.75}q is true. Moreover, the results obtained allow us to conjecture that this estimate holds for all q >= 23. The new upper bounds are obtained by finding new small complete arcs with the help of a computer search using randomized greedy algorithms. Also new constructions of complete arcs are proposed. These constructions form families of k-arcs in PG(2,q) containing arcs of all sizes k in a region k_{min} <= k <= k_{max} where k_{min} is of order q/3 or q/4 while k_{max} has order q/2. The completeness of the arcs obtained by the new constructions is proved for q <= 1367 and 2003 <= q <= 2063. There is reason to suppose that the arcs are complete for all q > 1367. New sizes of complete arcs in PG(2,q) are presented for 169 <= q <= 349 and q=1013,2003.

math.CO↗
arXiv2010

New quantum caps in PG(4,4)

Calderbank, Rains, Shor and Sloane (see \cite{Sloane}) showed that error-correction is possible in the context of quantum computations. Quantum stabilizer codes are a class of additive quaternary codes in binary projective spaces, which are self-orthogonal with respect to the symplectic form. A geometric description is given in \cite{Bierbra}, where also the notion of quantum cap is introduced. Quantum caps correspond to the special case of quantum stabilizer codes of distance $d=4$ when the code is linear over GF(4). In the present paper we review the translation from quantum error-correction to symplectic geometry and study quantum codes in PG(4,4) where we construct complete quantum caps with 20, 29, 30, 32, 33, 34, 36 and 38 points and incomplete quantum caps with 10, 12, 13, 20, 23, 24, 25 and 26 points and we prove the non existence of 11-quantum caps. In particular the quantum caps of sizes 36 and 38 yield positive answers to the existence questions of quantum codes $[[36,26,4]]$ and $[[38,28,4]]$ that remained open in the data base \cite{codetable}.

quant-ph↗
arXiv2009

The Spectrum of Quantum Caps in PG(4,4)

We prove the non existence of quantum caps of sizes 37 and 39. This completes the spectrum of quantum caps in PG(4, 4). This also implies the non existence of linear [[37,27,4]] and [[39,29,4]]-codes. The problem of the existence of non linear quantum codes with such parameters remains still open.

math.CO↗