arXiv subjects

Stefano Marcugini

Publications and source records attributed to Stefano Marcugini.

At least 37 records · Page 2Linked to original sources

arXiv2020

On integral weight spectra of the MDS codes cosets of weight 1, 2, and 3

The weight of a coset of a code is the smallest Hamming weight of any vector in the coset. For a linear code of length $n$, we call integral weight spectrum the overall numbers of weight $w$ vectors, $0\le w\le n$, in all the cosets of a fixed weight. For maximum distance separable (MDS) codes, we obtained new convenient formulas of integral weight spectra of cosets of weight 1 and 2. Also, we give the spectra for the weight 3 cosets of MDS codes with minimum distance $5$ and covering radius $3$.

cs.IT↗

arXiv2020

Optimal additive quaternary codes of low dimension

An additive quaternary $[n,k,d]$-code (length $n,$ quaternary dimension $k,$ minimum distance $d$) is a $2k$-dimensional F_2-vector space of $n$-tuples with entries in $Z_2\times Z_2$ (the $2$-dimensional vector space over F_2) with minimum Hamming distance $d.$ We determine the optimal parameters of additive quaternary codes of dimension $k\leq 3.$ The most challenging case is dimension $k=2.5.$ We prove that an additive quaternary $[n,2.5,d]$-code where $d<n-1$ exists if and only if $3(n-d)\geq \lceil d/2\rceil +\lceil d/4\rceil +\lceil d/8\rceil$. In particular we construct new optimal $2.5$-dimensional additive quaternary codes. As a by-product we give a direct proof for the fact that a binary linear $[3m,5,2e]_2$-code for $e<m-1$ exists if and only if the Griesmer bound $3(m-e)\geq \lceil e/2\rceil +\lceil e/4\rceil+\lceil e/8\rceil$ is satisfied.

math.CO↗

arXiv2020

Tables, bounds and graphics of short linear codes with covering radius 3 and codimension 4 and 5

The length function $\ell_q(r,R)$ is the smallest length of a $q$-ary linear code of codimension (redundancy) $r$ and covering radius $R$. The $d$-length function $\ell_q(r,R,d)$ is the smallest length of a $q$-ary linear code with codimension $r$, covering radius $R$, and minimum distance $d$. By computer search in wide regions of $q$, we obtained following short codes of covering radius $R=3$: $[n,n-4,5]_q3$ quasi-perfect MDS codes, $[n,n-5,5]_q3$ quasi-perfect Almost MDS codes, and $[n,n-5,3]_q3$ codes. In computer search, we use the step-by-step leximatrix and inverse leximatrix algorithms to obtain parity check matrices of codes. The new codes imply the following new upper bounds (called lexi-bounds) on the length and $d$-length functions: $$\ell_q(4,3)\le\ell_q(4,3,5)<2.8\sqrt[3]{\ln q}\cdot q^{(4-3)/3}=2.8\sqrt[3]{\ln q}\cdot\sqrt[3]{q}=2.8\sqrt[3]{q\ln q}~\text{for}~11\le q\le7057;$$ $$\ell_q(5,3)\le\ell_q(5,3,5)<3\sqrt[3]{\ln q}\cdot q^{(5-3)/3}=3\sqrt[3]{\ln q}\cdot\sqrt[3]{q^2}=3\sqrt[3]{q^2\ln q}~~\text{ for }~37\le q\le839.$$ Moreover, we improve the lexi-bounds, applying randomized greedy algorithms, and show that $$\ell_q(4,3)\le \ell_q(4,3,5)< 2.61\sqrt[3]{q\ln q}~\text{ if }~13\le q\le4373;$$ $$\ell_q(4,3)\le \ell_q(4,3,5)< 2.65\sqrt[3]{q\ln q}~\text{ if }~4373<q\le7057;$$ $$\ell_q(5,3)<2.785\sqrt[3]{q^2\ln q}~\text{ if }~11\le q\le401;$$ $$\ell_q(5,3)\le\ell_q(5,3,5)<2.884\sqrt[3]{q^2\ln q}~\text{ if }~401<q\le839.$$ The codes, obtained in this paper by leximatrix and inverse leximatrix algorithms, provide new upper bounds (called density lexi-bounds) on the smallest covering density $μ_q(r,R)$ of a $q$-ary linear code of codimension $r$ and covering radius $R$: $$μ_q(4,3)<3.3\cdot\ln q~~\text{ for }~11\le q\le7057;$$ $$μ_q(5,3)<4.2\cdot\ln q~~\text{ for }~37\le q\le839.$$

cs.IT↗

arXiv2020

On planes through points off the twisted cubic in $\mathrm{PG}(3,q)$ and multiple covering codes

Let $\mathrm{PG}(3,q)$ be the projective space of dimension three over the finite field with $q$ elements. Consider a twisted cubic in $\mathrm{PG}(3,q)$. The structure of the point-plane incidence matrix in $\mathrm{PG}(3,q)$ with respect to the orbits of points and planes under the action of the stabilizer group of the twisted cubic is described. This information is used to view generalized doubly-extended Reed-Solomon codes of codimension four as asymptotically optimal multiple covering codes.

math.CO↗

arXiv2019

Projective Automorphism Groups of Nonsingular Quartic Surfaces

For every p >= 5, we determine all Z_p-invariant nonsingular quartic surfaces in the three dimensional projective space over an algebraically closed field of characteristic zero. In some cases, we also determine their full projective automorphism groups.

math.AG↗

arXiv2019

New covering codes of radius $R$, codimension $tR$ and $tR+\frac{R}{2}$, and saturating sets in projective spaces

The length function $\ell_q(r,R)$ is the smallest length of a $ q $-ary linear code of codimension $r$ and covering radius $R$. In this work we obtain new constructive upper bounds on $\ell_q(r,R)$ for all $R\ge4$, $r=tR$, $t\ge2$, and also for all even $R\ge2$, $r=tR+\frac{R}{2}$, $t\ge1$. The new bounds are provided by infinite families of new covering codes with fixed $R$ and increasing codimension. The new bounds improve upon the known ones. We propose a general regular construction (called ``Line+Ovals'') of a minimal $ρ$-saturating $((ρ+1)q+1)$-set in the projective space $\mathrm{PG}(2ρ+1,q)$ for all $ρ\ge0$. Such a set corresponds to an $[Rq+1,Rq+1-2R,3]_qR$ locally optimal$^1$ code of covering radius $R=ρ+1$. Basing on combinatorial properties of these codes regarding to spherical capsules$^1$, we give constructions for code codimension lifting and obtain infinite families of new surface-covering$^1$ codes with codimension $r=tR$, $t\ge2$. In addition, we obtain new 1-saturating sets in the projective plane $\mathrm{PG}(2,q^2)$ and, basing on them, construct infinite code families with fixed even radius $R\ge2$ and codimension $r=tR+\frac{R}{2}$, $t\ge1$. ($^1$ see the definitions in Section 1)

math.CO↗

arXiv2018

Additive quaternary codes related to exceptional linear quaternary codes

We study additive quaternary codes whose parameters are close to those of the extended cyclic [12; 6; 6]4-code or to the quaternary linear codes generated by the elliptic quadric in PG(3; 4) or its dual. In particular we characterize those codes in the category of additive codes and construct some additive codes whose parameters are better than those of any linear quaternary code.

math.CO↗

arXiv2018

Classification of minimal 1-saturating sets in PG(v, 2), 2 <= v <= 6

The classification of all the minimal 1-saturating sets in PG(v, 2) for 2 <= v <= 5, and the classification of the smallest and of the second smallest minimal 1-saturating sets in PG(6, 2) are presented. These results have been found using a computer-based exhaustive search.

math.CO↗

arXiv2018

Tables, bounds and graphics of sizes of complete arcs in the plane $\mathrm{PG}(2,q)$ for all $q\le321007$ and sporadic $q$ in $[323761\ldots430007]$ obtained by an algorithm with fixed order of points (FOP)

In the previous works of the authors, a step-by-step algorithm FOP which uses any fixed order of points in the projective plane $\mathrm{PG}(2,q)$ is proposed to construct small complete arcs. In each step, the algorithm adds to a current 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 $\mathrm{PG}(2,q)$ 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 $\mathrm{PG}(2,q)$ is not relevant. A complete lexiarc in $\mathrm{PG}(2,q)$ is a complete arc obtained by the algorithm FOP using the lexicographical order of points. In this work, we collect and analyze the sizes of complete lexiarcs in the following regions: \begin{align*}& \textbf{all } q\le321007,~ q \mbox{ prime power}; & 15 \mbox{ sporadic $q$'s in the interval }[323761\ldots430007], \mbox{ see (1.10)}. \end{align*} In the work [9], the smallest known sizes of complete arcs in $\mathrm{PG}(2,q)$ are collected for all $q\leq160001$, $q$ prime power. The sizes of complete arcs, collected in this work and in [9], provide the following upper bounds on the smallest size $t_{2}(2,q)$ of a complete arc in the projective plane $\mathrm{PG}(2,q)$: \begin{align*} t_{2}(2,q)&<0.998\sqrt{3q\ln q}<1.729\sqrt{q\ln q}&\mbox{ for }&&7&\le q\le160001;\\ t_{2}(2,q)&<1.05\sqrt{3q\ln q}<1.819\sqrt{q\ln q}&\mbox{ for }&&7&\le q\le321007. \end{align*} Our investigations and results allow to conjecture that the bound $t_{2}(2,q)<1.05\sqrt{3q\ln q}<1.819\sqrt{q\ln q}$ holds for all $q\ge7$. It is noted that sizes of the random complete arcs and complete lexiarcs behave similarly. This work can be considered as a continuation and development of the paper [11].

math.CO↗

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

Upper bounds on the smallest size of a complete cap in $\mathrm{PG}(N,q)$, $N\ge3$, under a certain probabilistic conjecture

In the projective space $\mathrm{PG}(N,q)$ over the Galois field of order $q$, $N\ge3$, an iterative step-by-step construction of complete caps by adding a new point on every step is considered. It is proved that uncovered points are evenly placed on the space. A natural conjecture on an estimate of the number of new covered points on every step is done. For a part of the iterative process, this estimate is proved rigorously. Under the conjecture mentioned, new upper bounds on the smallest size $t_{2}(N,q)$ of a complete cap in $\mathrm{PG}(N,q)$ are obtained, in particular, \begin{align*} t_{2}(N,q)<\frac{\sqrt{q^{N+1}}}{q-1}\left(\sqrt{(N+1)\ln q}+1\right)+2\thicksim q^\frac{N-1}{2}\sqrt{(N+1)\ln q},\quad N\ge3. \end{align*} A connection with the Birthday problem is noted. The effectiveness of the new bounds is illustrated by comparison with sizes of complete caps obtained by computer in wide regions of $q$.

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↗

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

A combinatorial construction of an M_{12}-invariant code

In this work we summarized some recent results to be included in a forthcoming paper. A ternary [66,10,36]_3-code admitting the Mathieu group M_{12} as a group of automorphisms has recently been constructed by N. Pace. We give a construction of the Pace code in terms of $M_{12}$ as well as a combinatorial description in terms of the small Witt design, the Steiner system S(5,6,12). We also present a proof that the Pace code does indeed have minimum distance 36.

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

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

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↗