On Almost Complete Caps in PG(N, q)

Transcription

1 BLGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 8 No 5 Special Thematic Issue on Optimal Codes and Related Topics Sofia 08 Print ISSN: 3-970; Online ISSN: DOI: 0478/cait On Almost Complete Caps in PG(N ) Alexander A Davydov Stefano Marcugini Fernanda Pambianco Kharkevich Institute for Information Transmission Problems Russian Academy of Sciences Bol shoi Karetnyi pereulok 9 Mosco 705 Russian Federation Dipartimento di Matematica e Informatica niversità degli Studi di Perugia Via Vanvitelli Perugia 063 Italy s: adav@iitpru stefanomarcugini@unipgit fernandapambianco@unipgit Abstract: We propose the concepts of almost complete subset of an elliptic uadric in the proective space PG(3 ) and of almost complete cap in the space PG(N ) N 3 as generalizations of the concepts of almost complete subset of a conic and of almost complete arc in PG( ) pper bounds of the smallest size of the introduced geometrical obects are obtained by probabilistic and algorithmic methods Keyords: Proective space complete cap complete arc almost complete cap almost complete arc Introduction Let PG(N ) be the N-dimensional proective space over the Galois field F of order A cap in PG(N ) is a set of points no three of hich are collinear An n-cap of PG(N ) is complete if it is not contained in an (n + )-cap of PG(N ) Caps in PG( ) are called also arcs A point P of PG(N ) is covered by a cap N PG(N ) if P lies on a bisecant of The space PG(N ) contains N points An n-arc in PG(N ) ith n > N + is a set of n points such that no N + points belong to the same hyperplane of PG(N ) An n-arc of PG(N ) is complete if it is not contained in an (n + )-arc of PG(N ) In PG(N ) ith N a normal rational curve is any ( + )-arc proectively euivalent to the arc {( t t t N ): t F } {(0 0 )} For an introduction to the spaces on finite fields see [8-] and the references therein The concept of almost complete subset of a fixed irreducible conic in the plane PG( ) is considered in [] see also [3 4] and the references therein 54

2 Definition In PG( ) an almost complete subset of a fixed irreducible conic is a proper subset of the conic covering all the points of PG( ) except for the remaining points of the conic and its nucleus if is even Almost complete subsets of conics are useful for the classical problems of completeness of normal rational curves and extendability of generalized doublyextended Reed-Solomon codes Let t() be the smallest size of an almost complete subset of a conic in PG( ) Let an [n k d] code be a -ary linear code of length n dimension k and minimum distance d In [4] it is proved that under the condition 3 N + t() every normal rational curve in PG(N ) is a complete ( + )-arc (This assertion is euivalent to the folloing one: no [ + N + N + ] generalized doublyextended Reed-Solomon code can be extended to a [ + N + N + ] Maximal Distance Separable (MDS) code [3]) In [3] the folloing upper bound is obtained: t( ) (3ln ln ln ln 3) 4 3 ln 3ln The concept of almost complete arc in PG( ) is considered in [5] here arcs of an infinite family ( ) are called almost complete if #points not covered by ( ) () lim 0 #points of the plane PG( ) An almost complete subset of a conic is an almost complete arc as the number of points not covered by it is smaller than Almost complete arcs are useful for investigations of upper bounds on the smallest size of saturating sets and complete arcs in PG( ) In this ork e generalize both the aforementioned concepts Definition (i) In PG(3 ) an Almost Complete subset of the elliptic Quadric (ACQ-subset for short) is a proper subset of covering all the points of PG(3 ) except for the remaining points of (ii) In PG(N ) N a cap is almost complete if the number of points not covered by is not greater than θ N Note that if caps of Definition (ii) form an infinite family of caps ( ) in the spaces PG(N ) ith groing then it holds that (cf ()) #points not covered by ( ) N lim 0 #points of the space PG( ) An ACQ-subset is an almost complete cap as the number of points not covered by it is smaller than + Let d() be the smallest size of an ACQ-subset in PG(3 ) Let v(n ) be the smallest size of an almost complete cap in PG(N ) This ork is devoted to upper bounds on d() and v(n ) The main results of this ork are presented in Theorem (i) In PG(3 ) for the smallest size of an ACQ-subset e have N 55

3 () d( ) ( ) 6ln( ) 6ln (3) 56 (ii) In PG(N ) for the smallest size of an almost complete cap it holds that N N v( N ) N ln N ln N Moreover an almost complete cap of size at most NN ln can be constructed by a step-by-step greedy algorithm that in every step adds to the running cap a point providing the maximal possible (for the given step) number of ne covered points One see that the bounds () and (3) asymptotically coincide ith each other As far as it is knon to the authors ACQ-subsets and almost complete caps in PG(N ) N 3 are not considered in the literature Therefore it remains for us only to compare the bounds () and (3) ith the knon bounds on the smallest size t (N ) of a complete cap in PG(N ) Of course one should remember that these bounds are obtained for obects hich are similar to the almost complete caps but not the same In [4] it is proved that (4) N t (N ) < c log 300 ith a constant c independent of In [] see also [6] under some probabilistic conecture it is shon that N N N t( N ) ( N )ln ( N )ln 3 N We see that N ln is essentially smaller than c side the bound N N ln N log 300 On the other (that is proved rigorously) is greater than the conectural bound (4) So the bounds of Theorem obtained in this ork seem to be reasonable These ne concepts and the methods of their investigation can be useful for bounds and constructions of small saturating sets and small complete caps including a rigorous proof of the conectural bound (4) This paper is organized as follos In Section the bound () is proved by probabilistic methods In Section 3 the bound (3) is obtained by an algorithmic approach Some results of this ork ere briefly presented in [7] An upper bound on the smallest size of an almost complete subset of an elliptic uadric in PG(3 ) Let > 0 be a fixed integer Let be an elliptic uadric in PG(3 ) Consider a random ( + )-point subset + The total number of such subsets is

4 A fixed point A of PG(3 ) \ is covered by + if it belongs to a bisecant of + We denote by Prob( ) the probability of some event We estimate := Prob(A not covered by +) as the ratio of the number of ( + )-point subsets of not covering A over the total number of subsets of ith size ( + ) A set + does not cover A if and only if every line through A contains at most one point of + ( ) Through any point A PG(3 ) \ there are bisecants and + tangents of [9] Every bisecant has to places to put a point of + hile a tangent has the only one For simplicity of presentation e assume that a tangent also has to places to put a point of + (This ill slightly orsen our estimates) Therefore ( ) / ( ) / here the numerator estimates from above the number of (+)-point subsets of not covering A By straightforard calculations (5) i i i i i i0 i i0 i i0 From (5) using the ineuality x e x for x 0 e obtain that ( i )/( ) i0 ( () )/( ) e e nder the condition 4 4 (6) it holds that ( ) ( ) ( ) ( ) hence ( ( ) )/( ) ( ) /( ) e e The set + is not ACQ-subset if at least one point of PG(3 ) \ is not covered by it As PG(3 ) \ = 3 + e have 57

5 Prob( + is not ASQ-subset) Prob A not covered APG(3 )\ Q ( 3 + )π < ( + ) 3 e ( ) /(+) The probability that all the points of PG(3 ) \ are covered by + is Prob( + is ACQ-subset) > ( + ) 3 e ( ) /(+) This probability is larger than 0 if one takes = ( ) 6ln( ) here the condition (6) holds This shos that there exists an ACQ-subset + ith size + ( ) 6ln( ) + + Theorem (i) is proved 3 An upper bound on the smallest size of an almost complete cap in PG(N ) Assume that in PG(N ) N a cap is constructed by a step-by-step greedy algorithm (Algorithm for short) hich in every step adds to the cap a point providing the maximal possible (for the given step) number of ne covered points Such approach is considered in [ 6] After the -th step of Algorithm a -cap is obtained that does not cover exactly points Denote by ( ) the set of points of PG(N ) that are not covered by a cap By above # ( ) Let the cap consist of points A A A Let A ( ) be the point that ill be included into the cap in the ( + )-st step A point A + defines a bundle ( A ) of unisecants to hich are denoted as AA AA AA here AA i is the unisecant connecting A + ith the cap point A i Every unisecant contains + points Except for A A all the points on the unisecants in the bundle are candidates to be ne covered points in the ( + )-st step We call {A +} and ( A ) \ ( {A +}) respectively the head and the basic part of the bundle ( A ) For a given cap in total there are # ( ) distinct bundles Let (A +) be the number of ne covered points in the ( + )-st step ie A # ( ) # ( { A } ) In future e consider continuous approximations of the discrete functions A # ( ) # ( { A }) keeping the same notations We take into account that all points that are not covered by a cap lie on unisecants to the cap 58

6 In total there are θ N lines through every point of PG(N ) Therefore through every point A i of there is a pencil ( A i ) of θ N ( ) unisecants to Σ here i = The total number T of the unisecants to is Σ (8) T = (θ N + ) Let γ be the number of uncovered points on the -th unisecant = T Σ Every uncovered point lies on exactly unisecants; due to this multiplicity on Σ all unisecants there are in total uncovered points here T (9) By (8) (9) the age number γ of uncovered points on a unisecant is (0) T N A unisecant belongs to γ distinct bundles as every uncovered point on may be the head of a bundle Moreover provides γ (γ ) uncovered points to the basic parts of all these bundles The noted points are counted ith multiplicity Taking into account the multiplicity in all the bundles there are () ( A ) ( ) A T uncovered points here is the total numbers of all the heads By (9) () T T T A A ( ) ( ) For a cap e denote by ( ) the age value of (A +) by all # ( ) uncovered points A + ie () T ( A ) ( A ) A A ( ) # ( ) here the ineuality is obvious by sense; also note that (3) (4) T T We denote a loer estimate of ( ) see Lemma belo as follos: rigor ( ) : max N if N N if N 59

7 Lemma For a -cap the folloing holds: This ineuality alays holds rigor (5) ( ) ( ) In (5) e have the euality rigor (6) ( ) ( ) 60 N if and only if every unisecant contains the same number of uncovered points here is integer N N In (5) the euality rigor (7) ( ) ( ) holds if and only if each unisecant contains at most one uncovered point P r o o f By Cauchy-Scharz-Bunyakovsky ineuality it holds that T T T here euality holds if and only if all γ coincide In this case γ = N N for all and moreover the ratio is integer No by (8) (9) e have T N that together ith (9) () (3) (4) gives (5)-(7) Remark One can treat the estimates (5) (6) as follos A bundle contains unisecants having a common point its head Therefore the age number of uncovered points in a bundle is γ ( ) here γ is defined in (0) and the term takes into account the common point By (7) and Lemma N N hence (8) N N N N ( N ) ( N ) N N By (8)

8 ; N (9) N N 3 N N N N ; N N N N N N N N f ( ) here i f ( ) i N Remark The function f () and its approximations including () appear in distinct tasks of Probability Theory eg in the Birthday problem (or the Birthday paradox) [5 3] Actually let the year contain θ N days and let all birthdays occur ith the same probability Then ( + ) = f () here ( + ) is the P N probability that no to persons from + random persons have the same birthday Moreover if birthdays occur ith different probabilities e have ( + ) < f () [5] By (9) taking into account that = θ N < θ N = θ N + N e have (0) + < N f () + θ N sing the ineuality x e x for x 0 e obtain i/ () f () N < e = e Let i ( )/ N / N e N n () N ln NN ln N ln Then by (0)-() = θ N ln N ; / N N ; e + < θ N + ; + θ N So the number of points of PG(N ) not covered by the cap + is at most θ N P N P N 6

9 We have proved Theorem (ii) Acknoledgements: The research of A A Davydov as carried out at the IITP RAS at the expense of the Russian Foundation for Sciences (proect ) The research of S Marcugini and F Pambianco as supported in part by Ministry for Education niversity and Research of Italy (MIR) (Proect Geometrie di Galois e strutture di incidenza ) and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INDAM) R e f e r e n c e s B a r t o l i D A A D a v y d o v G F a i n a A A K r e s h c h u k S M a r c u g i n i F P a m b i a n c o pper Bounds on the Smallest Size of a Complete Arc in PG( ) under a Certain Probabilistic Conecture Problems Inform Transm Vol No 4 pp B a r t o l i D A A D a v y d o v G F a i n a S M a r c u g i n i F P a m b i a n c o Conectural pper Bounds on the Smallest Size of a Complete ap in PG(N ) N 3 Electron Notes Discrete Math Vol pp B a r t o l i D A A D a v y d o v S M a r c u g i n i F P a m b i a n c o On the Smallest Size of an Almost Complete Subset of a Conic in PG( ) and Extendability of Reed-Solomon Codes arxiv: [mathco] 06 4 B a r t o l i D G F a i n a S M a r c u g i n i F P a m b i a n c o A Construction of Small Complete Caps in Proective Spaces J Geom Vol No pp C l e v e n s o n M L W W a t k i n s Maorization and the Birthday Ineuality Math Magazine Vol No 3 pp D a v y d o v A A G F a i n a S M a r c u g i n i F P a m b i a n c o pper Bounds on the Smallest Size of a Complete Cap in PG(N ) N 3 under a Certain Probabilistic Conecture arxiv: [mathco] 07 7 D a v y d o v A A S M a r c u g i n i F P a m b i a n c o pper Bounds on the Smallest Size of an Almost Complete Cap in PG(N ) In: Proc of 8th International Workshop on Optimal Codes and Related Topics OC 7 (in Second International Conference Mathematics Days in Sofia ) Sofia Bulgaria 07 pp H i r s c h f e l d J W P Proective Geometries over Finite Fields Second Edition Oxford Oxford niversity Press H i r s c h f e l d J W P Finite Proective Spaces of Three Dimensions Oxford Oxford niversity Press H i r s c h f e l d J W P L S t o r m e The Packing Problem in Statistics Coding Theory and Finite Geometry: pdate 00 In: A Blokhuis J W P Hirschfeld et al Eds Proc of 4th Isle of Thorns Conf Chelood Gate 000 Kluer Academic Publisher Boston Finite Geometries Developments of Mathematics Vol 3 00 pp 0-46 H i r s c h f e l d J W P J A T h a s Open Problems in Finite Proective Spaces Finite Fields and their Appl Vol 3 05 No pp 44-8 K o v à c s S J Small Saturated Sets in Finite Proective Planes Rend Mat (Roma) Vol 99 No pp S a y r a f i e z a d e h M The Birthday Problem Revisited Math Magazine Vol No 3 pp S t o r m e L Completeness of Normal Rational Curves J Algebraic Combin Vol 99 No pp g h i E Small Almost Complete Arcs Discrete Math Vol No 3 pp Received 50907; Accepted 007 6