- Perugia, Umbria, Italy
ABSTRACT Some constructions and bounds on the sizes of semiovals contained in the Hermitian curve are given. A construction of an infinite family of 2-blocking sets of the Hermitian curve is also presented.
In the projective plane PG(2, q), an iterative construction of complete arcs by adding a new point on every step is considered. The working mechanism of a step-by-step greedy algorithm constructing small complete arcs in PG(2, q) is... more
In the projective plane PG(2, q), an iterative construction of complete arcs by adding a new point on every step is considered. The working mechanism of a step-by-step greedy algorithm constructing small complete arcs in PG(2, q) is explained. It is proven that uncovered points are evenly placed on the plane. For more than half of the steps of the iterative process, an estimation of the number of new covered points is proven. A natural conjecture that the estimation holds for the rest of steps is done. From that, upper bounds on the smallest size t2(2, q) of a complete arc in PG(2, q) are obtained. In particular, $t_{2}(2,q)<\sqrt{q}\sqrt{3\ln q+\ln \ln q+\ln 3}+\sqrt{\frac{q}{3\ln q}}+3$, $t_{2}(2,q)<1.885\sqrt{q\ln q}.$ A connection with the Birthday problem is noted. The effectiveness of the new bounds is illustrated by comparison with the smallest known sizes of complete arcs obtained in the recent works of the authors via computer search for a huge region of q.
Multiple coverings of the farthest-off points ($(R,\mu)$-MCF codes) and the corresponding $(\rho,\mu)$-saturating sets in projective spaces $PG(N,q)$ are considered. We propose and develop some methods which allow us to obtain new small... more
Multiple coverings of the farthest-off points ($(R,\mu)$-MCF codes) and the corresponding $(\rho,\mu)$-saturating sets in projective spaces $PG(N,q)$ are considered. We propose and develop some methods which allow us to obtain new small $(1,\mu)$-saturating sets and short $(2,\mu)$-MCF codes with $\mu$-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 $(1,\mu)$-saturating sets in $PG(2,q)$, $q$ small.
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... more
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 $AG(N,q)$, $N \equiv 0 \pmod 4$.
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ABSTRACT Let KK be an algebraically closed field of characteristic p&gt;0p&gt;0, and let XX be a curve over KK of genus g≥2g≥2. Assume that p&gt;2p&gt;2 and that XX admits a non-singular plane model. The following result... more
ABSTRACT Let KK be an algebraically closed field of characteristic p&gt;0p&gt;0, and let XX be a curve over KK of genus g≥2g≥2. Assume that p&gt;2p&gt;2 and that XX admits a non-singular plane model. The following result is proven: if XX has more than 3(2g2+g)(8g+1+3) automorphisms, then XX is birationally equivalent to a Hermitian curve.
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ABSTRACT Small complete arcs and caps in Galois spaces over finite fields F-q with characteristic greater than three are constructed from singular cubic curves. For m a divisor of q + 1 or q - 1, complete plane arcs of size approximately... more
ABSTRACT Small complete arcs and caps in Galois spaces over finite fields F-q with characteristic greater than three are constructed from singular cubic curves. For m a divisor of q + 1 or q - 1, complete plane arcs of size approximately q/m are obtained, provided that (m, 6) = 1 and m &lt; 1/4q(1/4). If in addition m = m(1)m(2) with (m(1), m(2)) = 1, then complete caps in affine spaces of dimension N equivalent to 0 (mod 4) with roughly m(1)+m(2)/m q(N/2) points are described. These results substantially widen the spectrum of qs for which complete arcs in AG(2, q) of size approximately q(3/4) can be constructed. Complete caps in AG(N, q) with roughly q((4N-1)/8) points are also provided. For infinitely many qs, these caps are the smallest known complete caps in AG(N, q), N equivalent to 0 (mod 4).
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ABSTRACT The classification of all semiovals and blocking semiovals in PG(2, 8) is determined. Also, new theoretical results on the structure of semiovals containing a -secant and some nonexistence results are presented.
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ABSTRACT In a three-dimensional Galois space of odd order q, the smallest complete caps appeared in the literature have size approximately qq/2 and were presented by Pellegrino in 1998. In this paper, a major gap in the proof of their... more
ABSTRACT In a three-dimensional Galois space of odd order q, the smallest complete caps appeared in the literature have size approximately qq/2 and were presented by Pellegrino in 1998. In this paper, a major gap in the proof of their completeness is pointed out. On the other hand, we show that a variant of Pellegrinoʼs method provides the smallest known complete caps for each odd q between 100 and 30 000.
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ABSTRACT In recent years, functional codes have received much attention. In his PhD thesis, F.A.B. Edoukou investigated various functional codes linked to quadrics and Hermitian varieties defined in finite projective spaces (Edoukou, PhD... more
ABSTRACT In recent years, functional codes have received much attention. In his PhD thesis, F.A.B. Edoukou investigated various functional codes linked to quadrics and Hermitian varieties defined in finite projective spaces (Edoukou, PhD Thesis, 2007). This work was continued in (Edoukou et al., Des Codes Cryptogr 56:219–233, 2010; Edoukou et al., J Pure Appl Algebr 214:1729–1739, 2010; Hallez and Storme, Finite Fields Appl 16:27–35, 2010), where the results of the thesis were improved and extended. In particular, Hallez and Storme investigated the functional codes ${C_2(\mathcal{H})}$ , with ${\mathcal{H}}$ a non-singular Hermitian variety in PG(N, q 2). The codewords of this code are defined by evaluating the points of ${\mathcal{H}}$ in the quadratic polynomials defined over ${\mathbb{F}_{q^2}}$ . We now present the similar results for the functional code ${C_{Herm}(\mathcal{Q})}$ . The codewords of this code are defined by evaluating the points of a non-singular quadric ${\mathcal{Q}}$ in PG(N, q 2) in the polynomials defining the Hermitian varieties of PG(N, q 2).
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In the previous works of the authors, a step-by-step algorithm FOP which uses any fixed order of points in the projective plane PG(2, q) is proposed for constructing small complete arcs. In every step, the algorithm adds to a running arc... more
In the previous works of the authors, a step-by-step algorithm FOP which uses any fixed order of points in the projective plane PG(2, q) is proposed for constructing small complete arcs. In every step, the algorithm adds to a running arc the first point in the fixed order not lying on the bisecants of the arc. The algorithm is based on the intuitive postulate that PG(2, q) contains a sufficient number of relatively small complete arcs. Also, in the previous papers, it is shown that the type of order on the points of PG(2, q) is not relevant. In this work we collect the sizes of complete lexiarcs (i.e. complete arcs in PG(2, q) obtained by the algorithm FOP with the lexicographical orders of points) in the following regions: all q ≤ 301813, q prime power; 23 sporadic q’s in the interval [301897 . . . 430007], see Table 2. In the work [9], the smallest known sizes of complete arcs in PG(2, q) are collected for all q ≤ 160001, q prime power. The sizes of complete arcs collected in this...
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In the projective planes PG(2,q), we collect the smallest known sizes of complete arcs for the regions all q ≤ 160001,, q prime power; Q_{4}={34 sporadic q's in the interval [160801...430007]}. For q ≤ 160001, the collection of arc... more
In the projective planes PG(2,q), we collect the smallest known sizes of complete arcs for the regions all q ≤ 160001,, q prime power; Q_{4}={34 sporadic q's in the interval [160801...430007]}. For q ≤ 160001, 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 PG(2,q), in particular t_{2}(2,q)<0.998\sqrt{3q\ln q}<1.729\sqrt{q\ln q} for 7\le q\le160001; (1) t_{2}(2,q)<\sqrt{q}\ln^{0.7295}q 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, for 19\le q\le160001; (3) t_{2}(2,q)<0.6\sqrt{q}\ln^{\varphi_{up}(q;0.6)} q, \varphi_{up}(q;0.6)=\frac{1.5}{\ln q}+0.802, for 19&\le q\le160001. (4) Moreover, the last 3 bounds hold also for q\in Q_{4}. Also, t_{2}(2,q)<1.006\sqrt{3q\ln q}<1.743\sqrt{q\ln q} for q\in Q_{4}. (5) Our investigations and results allow to conjecture that the bounds (2)--...
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... more
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<= q<= 46337, q prime}.
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The following upper bound on the smallest size $s(2,q)$ of a saturating set in a projective plane $\Pi _{q}$ (not necessary Desarguesian) of order $q$ is proven. \begin{equation*} s(2,q)\leq 2\sqrt{(q+1)\ln q}+2\thicksim 2\sqrt{q\ln q}.... more
The following upper bound on the smallest size $s(2,q)$ of a saturating set in a projective plane $\Pi _{q}$ (not necessary Desarguesian) of order $q$ is proven. \begin{equation*} s(2,q)\leq 2\sqrt{(q+1)\ln q}+2\thicksim 2\sqrt{q\ln q}. \end{equation*} Our approach is probabilistic rather than geometric. The proof uses some known results on the classical Birthday problem, and also provides the following result. In $\Pi _{q}$ any point $k$-set, with $ k\leq 2c\sqrt{(q+1)\ln q}+2$ and an universal constant $c\geq 1,$ is a saturating set with probability \begin{equation*} p>1-\frac{1}{q^{2c^{2}-2}}. \end{equation*}
In the recent works of the authors, an algorithm FOP using any fixed order of points in $PG(2,q)$ is proposed for constructing small complete arcs. The algorithm is based on an intuitive postulate that $PG(2,q)$ contains a sufficient... more
In the recent works of the authors, an algorithm FOP using any fixed order of points in $PG(2,q)$ is proposed for constructing small complete arcs. The algorithm is based on an intuitive postulate that $PG(2,q)$ contains a sufficient number of relatively small complete arcs. Also, in these works, it is shown that the type of order on the points of $PG(2,q)$ is not relevant. In this work we collect the sizes of complete arcs obtained by the algorithm FOP with the lexicographical and the Singer orders of points in the following regions: Lexicographical order: $3\le q\le67993$, $q$ prime; Lexicographical order: 43 sporadic prime $q$'s in the interval $[69997\ldots190027]$. Singer order: $5\le q\le40009$, $q$ prime.