Complete Caps Research Articles

Binary codes and caps

The connection between maximal caps (sometimes called complete caps) and certain binary codes called quasi-perfect codes is described. We provide a geometric approach to the foundational work of Davydov and Tombak who have obtained the exact possible sizes of large maximal caps. A new self-contained proof of the existence and the structure of the largest maximal nonaffine cap in ℙG(n, 2) is given. Combinatorial and geometric consequences are briefly sketched. Some of these, such as the connection with families of symmetric-difference free subsets of a finite set will be developed elsewhere. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 275–284, 1998

Binary codes and caps

The connection between maximal caps (sometimes called complete caps) and certain binary codes called quasi-perfect codes is described. We provide a geometric approach to the foundational work of Davydov and Tombak who have obtained the exact possible sizes of large maximal caps. A new self-contained proof of the existence and the structure of the largest maximal nonaffine cap in ℙG(n, 2) is given. Combinatorial and geometric consequences are briefly sketched. Some of these, such as the connection with families of symmetric-difference free subsets of a finite set will be developed elsewhere. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 275–284, 1998

Complete arcs and caps in some papers of Giuseppe Tallini and Walking with Giuseppe Tallini along the streets of “Augusta Perusia”

Complete arcs and caps in some papers of Giuseppe Tallini and Walking with Giuseppe Tallini along the streets of “Augusta Perusia”

Small complete caps in PG( r, q), r ⩾ 3

A very difficult problem for complete caps in PG( r,q) is to determine their minimum size. The results on this topic are still scarce and in this paper we survey the best results now known. Furthermore, we construct new interesting sporadic examples of complete caps in PG(3, q) and in PG(4, q) such that their size are smaller than the currently known. As a consequence, we get that the Pellegrino's conjecture on the minimal size of a complete k-cap in PG(3, q), q odd, is in general false.

Caps in finite projective spaces of odd order

In this paper upper bounds on the number of points of large complete caps in PG(n, q), (q odd,n≥3) are derived. The previously known bounds of Hirschfeld and Segre are improved using recent bounds on the cardinality of the second largest complete cap in the plane PG(2,q).

A class of completek-caps inPG(3,q) forq an odd prime

The main problem on caps, posed originally by Segre in the fifties, is to determine the values of k for which there exists a complete k-cap. Very few results on this problem are known. The cardinality of the largest cap(s) and the smallest complete cap(s) are crucial. In this paper it is shown that there exist complete k-caps in PG(3, q), q an odd prime ≥ 5 or q = 9, such that k = (q2 + q + 6)/3 or k = (q2 + 2q + 6)/3. These complete caps are smaller than those currently known for q odd.

Small Complete Caps in Spaces of Even Characteristic

In 1959, Segre constructed a complete (3q+2)-cap inPG(3, q),qeven. This showed that the size of the smallest completek-cap inPG(3, q),qeven, is almost equal to the trivial lower bound which is of order[formula]. Generalizing the construction of Segre, complete (qn+3(qn−1+…+q)+2)-caps inPG(2n, q),qeven,q⩾4, and complete (3(qn+…+q)+2)-caps inPG(2n+1, q),qeven,q⩾4, are constructed. This shows that in all spacesPG(2n+1, q),qeven, the size of the smallest completek-cap is almost equal to the trivial lower bound which is of order[formula].

Some families of complete caps of quadrics and symplectic polarities over GF(8)

A cap on a quadric is a set of its points whose pairwise joins are all chords. A cap is complete if it is not part of a larger one. The only field for which all complete quadric caps are known is GF(2). Those caps are small; the biggest for each quadric is of order the dimension of the ambient space. Apart from information about ovoids in dimensions at most 7, little else is known. Here, the evidence is increased by providing caps over GF(2α), α odd, which, if α>1, have size of order the dimension cubed. In particular, complete caps are obtained for the quadrics Q 2m (8), Q + 8+7 (8), Q - 8+3 (8), Q + 8+1 (8) and Q - 8+5 (8). These caps on Q + 8+7 (8) and Q - 8+3 (8) are complete on any Q n(8) of which their quadrics are sections; so is that that of Q 4l+2(8) for any Q 2n (8) of which Q 4l+2(8) is a section with the same kernel. From the correspondence with Q 2n (8) complete caps are obtained for symplectic polarities over GF(8).

Buckling behaviour of ring-loaded shallow spherical shells with a centre hole

Results are presented of theoretical investigations into the elastic symmetrical buckling of clamped shallow spherical shells with a centre hole under a ring load, which are obtained by using both successive iterative techniques and cubic B-spline approximations. To find out the effect of the centre hole on buckling behaviour, clamped shallow shells with different boundary conditions at the hole edge (the centre hole edge is free or may be reinforced with a rigid ring) have been studied. It is shown that the presence of the centre hole causes opposed effects on the buckling behaviour for two kinds of clamped shells studied. This paper also investigates the limiting case in which the radius of the centre hole approaches zero. The results obtained are qualitatively similar to those observed by Evan-Iwanowski and Loo [Syracuse University Report, pp. 834–835, 1962] and Mescall [ Proc. 4th South-eastern Conf. on Appl. Mech., pp. 183–197, 1968] for complete shallow spherical caps subjected to ring load. However, detailed investigations indicate that the buckling load for a ring-loaded shell is in accord with that for a concentrated loaded shell when the ratio of loaded radius to shell base radius is less than a critical value. Upon an increase of the radius ratio beyond this critical value, the buckling load drops discontinuously due to the appearance of a new bump on the initial part of the load-deflection curve. The new maximum (i.e. buckling load) rises rapidly with the increase of radius ratio while the original maximum seems almost unaffected. For still larger loaded radius, the centre deflection becomes negative at low load and there is a point at which the slope of the load-deflection curve becomes infinite. Thus the corresponding buckling load rises drastically.

Linear codes with covering radius 2 and other new covering codes

Infinite families of linear binary codes with covering radius R=2 and minimum distance d=3 and d=4 are given. Using the constructed codes with d=3, R=2, families of covering codes with R>2 are obtained. The parameters of many constructed codes with R<or=2 are better than the parameters of known codes. The parity check matrices of constructed codes with d=4, R=2 are equivalent to complete caps in projective geometry.<<ETX>>

Codes projectifs à deux poids, “caps” complets et ensembles de différences

Using certain sets of points of a finite projective geometry some results are obtained from properties of two-weight projective codes. A problem concerning complete caps is solved. A deep connection between binary uniformily packed codes and difference sets over elementary Abelian 2-groups is established and a characterization of these difference sets is given.

On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two

On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two