Arcs In PG Research Articles

Classical arcs in PG( r, q) for 11 ⩽ q ⩽ 19

The largest arcs in PG( r, q) have length q + 1 and they are classical, if 11 ⩽ q ⩽ 19, q ≠ 16 and 1 ⩽ r ⩽ q − 2 or if q = 16 and 4 ⩽ r ⩽ q − 5.

Cyclic arcs and pseudo-cyclic MDS codes

Cyclic arcs (defined by Storme and Van Maldghem, [1994]) and pseudo- cyclic MDS codes are equivalent objects. We survey known results on the existence of cyclic arcs. Some new results on cyclic arcs in PG(2, q) are also given.

Multiple Blocking Sets and Arcs in Finite Planes

This paper contains two main results relating to the size of a multiple blocking set in PG(2, q). The first gives a very general lower bound, the second a much better lower bound for prime planes. The latter is used to consider maximum sizes of (k, n)-arcs in PG(2, 11) and PG(2, 13), some of which are determined. In addition, a summary is given of the value of mn(2, q) for q ≤ 13.

K-Arcs and dual k-arcs

We study the relation between k-arcs and dual k-arcs. In particular, we look at the ( q+2)-arcs in PG(2, q) and in PG( q−2, q), q even. It will be shown that the problem of finding all the o-polynomials of the ( q+2)-arcs in PG(2, q), q even, is equivalent with the search for all the points of PG( q−2, q), q even, which extend the normal rational curve {(1, t,… t q−2 )‖∈GF( q) +} to a ( q+2)-arc.

Cyclic Arcs in PG (2, q )

B.C. Kestenband [9], J.C. Fisher, J.W.P. Hirschfeld, and J.A. Thas [3], E. Boros, and T. Szönyi [1] constructed complete (q 2 − q + l)-arcs in PG(2, q 2), q ≥ 3. One of the interesting properties of these arcs is the fact that they are fixed by a cyclic protective group of order q 2 − q + 1. We investigate the following problem: What are the complete k-arcs in PG(2, q) which are fixed by a cyclic projective group of order k? This article shows that there are essentially three types of those arcs, one of which is the conic in PG(2, q), q odd. For the other two types, concrete examples are given which shows that these types also occur.

M.D.S. codes and arcs in PG( n, q) with q even: An improvement of the bounds of Bruen, Thas, and Blokhuis

All bounds of Bruen, Thas, and Blokhuis in [ Invent. Math. 92 (1988), 441–459] are considerably improved: we show that 3 q> may be replaced by √ q 2 + 3 4 .

Complete k-arcs in PG( n, q), q even

This paper investigates the completeness of k-arcs in PG( n, q) q even. We determine all values of k for which there exists a complete k-arc in PG( n,q), q − 2 ⩾ n > q − q 2 − 11 4 . This is proven by using the duality principle between k-arcs in PG( n, q) and dual k-arcs in PG( k − n − 2, q) ( k⩾ n+4). The theorems show that the classification of all complete k-arcs in PG( n, q), q even and q − 2 ⩾ n > q − q 2 − 11 4 , is closely related to the classification of all ( q+2)-arcs in PG(2, q).

Orbits of arcs in PG(N, K) under projectivities

The symmetric group on k symbols is made to operate on a certain set of matrices in such a way that its orbits are in one-to-one correspondence with the orbits of the k-arcs of an N-dimensional projective space under the group of projectivities. This leads to a formula for the number of such orbits.

Minimal blocking sets in finite planes

A result on enclosed (k,n)-arcs in PG(2,q) is used to characterise all minimal blocking sets of sizes =q +n in PG(2,q) with at least twon-secants. This also permits a slight extension of the bounds on blocking sets established by Blokhuis and Brouwer.

Completeness of Normal Rational Curves

The completeness of normal rational curves, considered as (q + 1)-arcs in PG(n, q), is investigated. Previous results of Storme and Thas are improved by using a result by Kovacs. This solves the problem completely for large prime numbers q and odd nonsquare prime powers q e p2h+1 with p prime, p\geq p_0(h), h\geq 1, where p0(h) is an odd prime number which depends on h.

Arcs in PG(n, q), MDS-codes and three fundamental problems of B. Segre ? Some extensions

To each arc of PG(n, q) an algebraic hypersurface is associated. Using this tool new results on complete arcs are obtained. Since arcs and linear MDS-codes are equivalent objects, these results can be translated in terms of codes.

On M.D.S. codes, arcs inPG(n, q) withq even, and a solution of three fundamental problems of B. Segre

On M.D.S. codes, arcs inPG(n, q) withq even, and a solution of three fundamental problems of B. Segre

Synthesis and characterization of stoichiometric CdPS 3

New upper bounds on the smallest size t2(2,q) of a complete arc in the projective plane PG(2,q) are obtained for 853≤q≤5107 and q∈T1∪T2, where T1={173,181,193,229,243,257,271,277,293,343,373,409,443,449,457,461,463,467,479,487,491,499,529,563,569,571,577,587,593,599,601,607,613,617,619,631,641,661,673,677,683,691,709}, and T2={5119,5147,5153,5209,5231,5237,5261,5279,5281,5303,5347,5641,5843,6011,8192}. From these new bounds it follows that for q≤2593 and q=2693,2753, the relation t2(2,q)<4.5q holds. Also, for q≤5107 we have t2(2,q)<4.79q. It is shown that for 23≤q≤5107 and q∈T2∪{214,215,218}, the inequality t2(2,q)<qln0.75q is true. Moreover, the results obtained allow us to conjecture that this estimate holds for all q≥23. The new upper bounds are obtained by finding new small complete arcs with the help of a computer search using randomized greedy algorithms. Also new constructions of complete arcs are proposed. These constructions form families of k-arcs in PG(2,q) containing arcs of all sizes k in a region kmin≤k≤kmax, where kmin is of order 13q or 14q while kmax has order 12q. The completeness of the arcs obtained by the new constructions is proved for q≤2063. There is reason to suppose that the arcs are complete for all q>2063. New sizes of complete arcs in PG(2,q) are presented for 169≤q≤349 and q=1013,2003.

A class of ((p n?p m)(p n?1), p n?p m)-arcs in PG(2, p n)

A (k, d)-arc in PG(2, q) is a set of k points such that some d, but no d+1, of them are collinear. An outstanding problem is to find the maximum value of k for which a (k, d)-arc exists. A construction is given for a class of (k, pn−pm)-arcs in PG(2, pn). These arcs constitute a lower bound on the maximum possible value of k, and a subset of them is shown to be optimal.

Elementary proofs of two fundamental theorems of B. Segre without using the Hasse-Weil theorem

An interesting bound on the number of points of a plane algebraic curve C of PG(2, q) is obtained, without using the deep theorem of Hasse-Weil. This bound is used to prove a well-known theorem by Segre on complete k arcs in PG(2, q), q even.

On small {;k; q};-arcs in planes of order q2

In this paper we examine some properties of complete {; k; q};-arcs in projective planes of order q 2 . In particular, we derive a lower bound for k , and we exhibit a family of arcs having low values of k which exist in every such plane having a Baer subplane. In addition we resolve the existence problem for complete {; k; 3 };-arcs in PG (2, 9).