Small complete caps in Galois affine spaces

Some new families of caps in Galois affine spaces AG(N, q) of dimension N≡ 0(mod 4) and odd order q are constructed. Such caps are proven to be complete by using some new ideas depending on the concept of a regular point with respect to a complete plane arc. As a corollary, an improvement on the currently known upper bounds on the size of the smallest complete caps in AG(N, q) is obtained. This research was performed within the activity of GNSAGA of the Italian INDAM, with the financial support of the Italian Ministry MIUR project “Strutture geometriche, combinatorica e loro applicazioni”, PRIN 2004–2005.     

On multiple caps in finite projective spaces

In this paper, we consider new results on (k, n)-caps with n > 2. We provide a lower bound on the size of such caps. Furthermore, we generalize two product constructions for (k, 2)-caps to caps with larger n. We give explicit constructions for good caps with small n. In particular, we determine the largest size of a (k, 3)-cap in PG(3, 5), which turns out to be 44. The results on caps in PG(3, 5) provide a solution to four of the eight open instances of the main coding theory problem for q = 5 and k = 4.     

Embedding the planar structure of 4-dimensional linear spaces into projective spaces

It is known that a linear spaces of dimensiond has at least as many hyperplanes as points with equality if it is a (possibly degenerate) projective space. If there are only a few more hyperplanes than points, then the linear space can still be embedded in a projective space of the same dimension. But even if the difference between the number of hyperplanes and points is too big to ensure an embedding, it seems likely that the linear space is closely related to a projective space. We shall demonstrate this in the cased=4.     

On cyclic Meir-Keeler contractions in metric spaces

Cyclic Meir-Keeler contractions are considered under the recently introduced WUC and HW properties on pairs of subsets of metric spaces. We show that, in contrast with previous results in the theory, best proximity point theorems under these properties do not directly extend from cyclic contractions to cyclic Meir-Keeler contractions. We obtain, however, a positive result for cyclic Meir-Keeler contractions under additional properties which is shown to be an extension of already existing results for cyclic contractions. Moreover, we give examples supporting the necessity of our additional conditions.     

Complete caps in projective spaces PG (n, q)

A computer search in the finite projective spaces PG(n, q) for the spectrum of possible sizes k of complete k-caps is done. Randomized greedy algorithms are applied. New upper bounds on the smallest size of a complete cap are given for many values of n and q. Many new sizes of complete caps are obtained.     

Minimal 1-saturating sets and complete caps in binary projective spaces

In binary projective spaces PG(v,2), minimal 1-saturating sets, including sets with inner lines and complete caps, are considered. A number of constructions of the minimal 1-saturating sets are described. They give infinite families of sets with inner lines and complete caps in spaces with increasing dimension. Some constructions produce sets with an interesting symmetrical structure connected with inner lines, polygons, and orbits of stabilizer groups. As an example we note an 11-set in PG(4,2) called “Pentagon with center”. The complete classification of minimal 1-saturating sets in small geometries is obtained by computer and is connected with the constructions described.     

On ternary Kloosterman sums modulo 12

Let K(a) denote the Kloosterman sum on . It is easy to see that for all . We completely characterize those for which , and . The simplicity of the characterization allows us to count the number of the belonging to each of these three classes. As a byproduct we offer an alternative proof for a new class of quasi-perfect ternary linear codes recently presented by Danev and Dodunekov.     

Several kinds of large cyclic subspace codes via Sidon spaces

Subspace codes have attracted much attention in recent years due to their applications to error correction in random network coding. In this paper, we construct several kinds of large cyclic subspace codes via Sidon spaces and large subspace codes via unions of some Sidon spaces. Therefore, some known results are extended.     

Uniqueness of some cyclic projective planes

For n < 41 and for {121, 125, 128, 169, 256, 1024}, every cyclic projective plane of order n is desarguesian.      

Maximum distance separable codes and arcs in projective spaces

Given any linear code C over a finite field GF(q) we show how C can be described in a transparent and geometrical way by using the associated Bruen-Silverman code.Then, specializing to the case of MDS codes we use our new approach to offer improvements to the main results currently available concerning MDS extensions of linear MDS codes. We also sharply limit the possibilities for constructing long non-linear MDS codes. Our proofs make use of the connection between the work of Rédei [L. Rédei, Lacunary Polynomials over Finite Fields, North-Holland, Amsterdam, 1973. Translated from the German by I. Földes. [18]] and the Rédei blocking sets that was first pointed out over thirty years ago in [A.A. Bruen, B. Levinger, A theorem on permutations of a finite field, Canad. J. Math. 25 (1973) 1060-1065]. The main results of this paper significantly strengthen those in [A. Blokhuis, A.A. Bruen, J.A. Thas, Arcs in PG(n,q), MDS-codes and three fundamental problems of B. Segre—Some extensions, Geom. Dedicata 35 (1-3) (1990) 1-11; A.A. Bruen, J.A. Thas, A.Blokhuis, On M.D.S. codes, arcs in PG(n,q) with q even, and a solution of three fundamental problems of B. Segre, Invent. Math. 92 (3) (1988) 441-459].     

A symmetric Roos bound for linear codes

The van Lint-Wilson AB-method yields a short proof of the Roos bound for the minimum distance of a cyclic code. We use the AB-method to obtain a different bound for the weights of a linear code. In contrast to the Roos bound, the role of the codes A and B in our bound is symmetric. We use the bound to prove the actual minimum distance for a class of dual BCH codes of length q²−1 over Fq. We give cyclic codes [63,38,16] and [65,40,16] over F₈ that are better than the known [63,38,15] and [65,40,15] codes.     

On the number of lines in 4-dimensional linear spaces

Assuming a weak non-degeneracy condition, we show that a linear spaceL of dimension at least 4 withv=q ⁴+q ³+q ²+q+1 points,q > 1 any positive real number, has at least (q²+1)v lines with equality if and only ifq is a prime power andL = PG(4,q).Dedicated to H. Mäurer on the occasion of his 60th birthday