The Fermat curve x^n+y^n+z^n: the most symmetric non-singular algebraic plane curve

A projective non-singular plane algebraic curve of degree d<=4 is called maximally symmetric if it attains the maximum order of the automorphism groups for complex non-singular plane algebraic curves of degree d. For d<=7, all such curves are known. Up to projectivities, they are the Fermat curve for d=5,7, see \cite{kmp1,kmp2}, the Klein quartic for d=4, see \cite{har}, and the Wiman sextic for d=6, see \cite{doi}. In this paper we work on projective plane curves defined over an algebraically closed field of characteristic zero, and we extend this result to every d>=8 showing that the Fermat curve is the unique maximally symmetric non-singular curve of degree d with d>=8, up to projectivity. For d=11,13,17,19, this characterization of the Fermat curve has already been obtained, see \cite{kmp2}.Comment: 21 page

Classification of minimal 1-saturating sets in PG(2,q)PG(2,q), q≤23q\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.Comment: 4 page

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

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 $q$.Comment: 32 Page

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.Comment: 10 pages, 7 table

A computer based classification of caps in PG(4,3)

In this paper we present the complete classification of caps in PG(4,3). These results have been obtained using a computer based exhaustive search that exploits projective equivalence.Comment: 9 page

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.Comment: 5 page

The non-existence of a [[13,5,4]] quantum stabilizer code

We solve one of the oldest problems in the theory of quantum stabilizer codes by proving the non-existence of quantum [[13,5,4]]-codes

The maximum and the minimum size of complete (n,3)-arcs in PG(2,16)

In this work we solve the packing problem for complete (n,3)-arcs in PG(2,16), determining that the maximum size is 28 and the minimum size is 15. We also performed a partial classification of the extremal size of complete (n,3)-arcs in PG(2,16).Comment: 3 page

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.Comment: 13 page