Related papers: A computer based classification of caps in PG(4,3)
In this paper we present the complete classification of caps in PG(4,2). These results have been obtained using a computer based exhaustive search that exploits projective equivalence.
In this paper we present the complete classification of caps in PG(5,2). These results have been obtained using a computer based exhaustive search that exploits projective equivalence.
We describe an algorithm for testing the completeness of caps in PG(r; q), q even. It allowed us to check that the 95256-cap in PG(12; 4) recently found by Fu el al. (see [14]) is complete.
Some new families of small complete caps in $PG(N,q)$, $q$ even, are described. By using inductive arguments, the problem of the construction of small complete caps in projective spaces of arbitrary dimensions is reduced to the same problem…
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…
In a geometry, a maximal cap is a collection of points of largest size containing no lines. In AG(4,3), maximal caps contain 20 points. The 81 points of AG(4,3) can be partitioned into 4 mutually disjoint maximal caps together with a single…
Image classification has become one of the main tasks in the field of computer vision technologies. In this context, a recent algorithm called CapsNet that implements an approach based on activity vectors and dynamic routing between…
We define a cap in the affine geometry AG(n,2) to be a subset in which every collection of four points is in general position. In this paper, we classify, up to affine equivalence, all caps in AG(7,2) of size k greater than or equal to 10.…
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…
A cap set in projective or affine geometry over a finite field is a set of points no three of which are collinear. In this paper, we propose a new construction for complete cap sets that yields a cap set of size 124928 in the affine…
Capsule networks are designed to present the objects by a set of parts and their relationships, which provide an insight into the procedure of visual perception. Although recent works have shown the success of capsule networks on simple…
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…
Point clouds, being the simple and compact representation of surface geometry of 3D objects, have gained increasing popularity with the evolution of deep learning networks for classification and segmentation tasks. Unlike human, teaching…
In this paper, we deal with plane curves with cusps. It is well known that there are various types of cusps. Among them, we investigate criteria for $(n, n+1)$ cusps with respect to several differential conditions and relations between…
This paper introduces the 3DCapsule, which is a 3D extension of the recently introduced Capsule concept that makes it applicable to unordered point sets. The original Capsule relies on the existence of a spatial relationship between the…
In this paper we prove the existence of a complete cap of ${\rm PG}(4n+1, q)$ of size $2(q^{2n+1}-1)/(q-1)$, for each prime power $q>2$. It is obtained by projecting two disjoint Veronese varieties of ${\rm PG}(2n^2+3n, q)$ from a suitable…
In this paper, we introduce a fundamental substructure of maximal caps in the affine geometry $AG(4,3)$ that we call \emph{demicaps}. Demicaps provide a direct link to particular partitions of $AG(4,3)$ into 4 maximal caps plus a single…
We propose an interpretable Capsule Network, iCaps, for image classification. A capsule is a group of neurons nested inside each layer, and the one in the last layer is called a class capsule, which is a vector whose norm indicates a…
In this paper, we propose a capsule-based neural network model to solve the semantic segmentation problem. By taking advantage of the extractable part-whole dependencies available in capsule layers, we derive the probabilities of the class…
In PG(2,32) the following two results are proven by a computer aided search. (i) Uniqueness of hyperfocused 12-arcs, up to projectivities; (ii) Non-existence of hyperfocused 14-arcs. The existence problem for hyperfocused 16-arcs remains…