Upper bounds on the smallest size of a complete cap in PG(N, q), N 3, under a certain probabilistic conjecture

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1 AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 72(3) (2018), Pages Upper bounds on the smallest size of a complete cap in PG(N, q), N 3, under a certain probabilistic conjecture Alexander A. Davydov Institute for Information Transmission Problems (Kharkevich institute) Russian Academy of Sciences, Mosco Russian Federation adav@iitp.ru Giorgio Faina Stefano Marcugini Fernanda Pambianco Dipartimento di Matematica e Informatica Università degli Studi di Perugia Perugia 06123, Italy giorgio.faina@unipg.it stefano.marcugini@unipg.it fernanda.pambianco@unipg.it Abstract In the N-dimensional projective space PG(N, q) over the Galois field of order q, N 3, an iterative step-by-step construction of complete caps by adding a ne point in every step is considered. It is proved that uncovered points are uniformly distributed in the space. A natural conjecture on an estimate of the number of ne covered points in every step is done. For a part of the iterative process, this estimate is proved rigorously. Under the conjecture mentioned, ne upper bounds on the smallest size t 2 (N, q) of a complete cap in PG(N, q), N 3, are obtained, in particular, t 2 (N, q) < q N+1 ( ) (N + 1) ln q + 1 q q N 1 2 (N + 1) ln q. A connection ith the Birthday problem is noted. The effectiveness of the ne bounds is illustrated by comparison ith sizes of complete caps obtained by computer in ide regions of q. ISSN: c The author(s). Released under the CC BY 4.0 International License

2 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), Introduction Let PG(N, q) be the N-dimensional projective space over the Galois field F q of order q. A k-cap in PG(N, q) is a set of k points no three of hich are collinear. A k-cap is complete if it is not contained in a (k + 1)-cap. Caps in PG(2, q) are also called arcs and they have been idely studied; see [4,5,7,8,20,28,30 33,39]. Let AG(N, q) be the N-dimensional affine space over F q. If N > 2 only a fe constructions and bounds are knon for small complete caps in PG(N, q) and AG(N, q); see [1 3,6,10 14, 20 32, 35, 36, 38, 39] for surveys and results. Caps have been intensively studied for their connection ith coding theory [30, 31, 34]. A linear q-ary code of length n, dimension k, and minimum distance d is denoted by [n, k, d] q. If a parity-check matrix of a linear q-ary code is obtained by taking as columns the homogeneous coordinates of the points of a cap in PG(N, q), then the code has minimum distance d = 4 (ith the exceptions of the complete 5-cap in PG(3, 2) and 11-cap in PG(4, 3) giving rise to the [5, 1, 5] 2 and [11, 6, 5] 3 codes). Complete n-caps in PG(N, q) correspond to non-extendable [n, n N 1, 4] q quasi-perfect codes of covering radius 2 [17,19]. If N = 2 these codes are Minimum Distance Separable (MDS); for N = 3 they are Almost MDS since their Singleton defect is equal to 1. For fixed N, the covering density of the mentioned codes decreases ith decreasing n. Thus small complete caps have a better covering quality than the big ones. Note also that caps are connected ith quantum codes; see, for instance, [15, 40]. In general, a central problem concerning caps is to determine the spectrum of the possible sizes of complete caps in a given space; see [30, 31] and the references therein. Of particular interest for applications to coding theory is the loer part of the spectrum, as small complete caps correspond to short quasi-perfect linear codes ith small covering density. Let t 2 (N, q) be the smallest size of a complete cap in PG(N, q). A hard open problem in the study of projective spaces is the determination of t 2 (N, q). The exact values of t 2 (N, q), N 3, are knon only for very small q. For instance, t 2 (3, q) is knon only for q 7; see [20, Table 3]. This ork is devoted to upper bounds on t 2 (N, q), N 3. The trivial loer bound for t 2 (N, q) is 2q N 1 2. Constructions of complete caps hose sizes are close to this loer bound are knon for the folloing cases: q = 2 and N arbitrary; q = 2 m > 2 and N odd; q is even square [14, 20, 21, 25, 27, 35, 38]. Using a modification of the probabilistic approach of [33] for the projective plane, the upper bound t 2 (N, q) < cq N 1 2 log 300 q, here c is a constant independent of q, has been obtained in [13]. Computer assisted results on small complete caps in PG(N, q) and AG(N, q) are given in [6, 10 12, 20, 22, 24, 36]. The main result of this paper is given by Theorem 1.1 based on Theorem 4.5.

3 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), Theorem 1.1 (The main result.) Let t 2 (N, q) be the smallest size of a complete cap in the projective space PG(N, q). Let D 1 be a constant independent of q. (i) Under Conjecture 3.3(i), in PG(N, q) ith N 3, e have q N+1 ( ) t 2 (N, q) < D (N + 1) ln q Dq N 1 2 (N + 1) ln q. (1.1) q 1 (ii) Under Conjecture 3.3(ii), the bound (1.1) ith D = 1 holds, i.e. t 2 (N, q) < q N+1 ( ) (N + 1) ln q + 1 q q N 1 2 (N + 1) ln q, N 3. (1.2) Conjecture 1.2 In PG(N, q), N 3, the upper bound (1.2) holds for all q ithout any extra conditions and conjectures. This ork can be treated as a development of the paper [4]. Some results of this ork ere briefly presented in [9]. The paper is organized as follos. In Section 2, e describe the iterative step-bystep process for constructing caps. In Section 3, probabilities of events, that points of PG(N, q) are not covered by a current cap, are considered. It is proved that uncovered points are uniformly distributed in the space. A natural Conjecture 3.3 on the estimate of the number of ne covered points in every step of the iterative process is done. In Section 4, under the conjecture of Section 3 e give ne upper bounds on t 2 (N, q). In Section 5, e illustrate the effectiveness of the ne bounds comparing them ith the results of computer search from [10, 11]. For a part of the iterative process, a rigorous proof of Conjecture 3.3 is given in Section 6. In Section 7, the reasonableness of Conjecture 3.3 is discussed. It is shon that in the steps of the iterative process here the rigorous estimates give bad results, these estimates do not reflect the real situation effectively. The reason is that the rigorous estimates assume that the number of uncovered points on unisecants is the same for all unisecants. Hoever, in fact, there is a dispersion of the number of uncovered points on unisecants. Moreover, this dispersion gros in the iterative process. In Section 8, the results are discussed. 2 An iterative step-by-step process Assume that in PG(N, q), N 3, a complete cap is constructed by a step-by-step algorithm (Algorithm for short) hich adds one ne point to the cap in each step. As an example, e can mention the greedy algorithm that in every step adds to the cap a point providing the maximal possible (for the given step) number of ne covered points; see [7, 8, 20, 22]. Recall that a point of PG(N, q) is covered by a cap if the point lies on a bisecant of the cap. Clearly, all points of the cap are covered.

4 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), The space PG(N, q) contains θ N,q = qn+1 1 points. q 1 Assume that after the -th step of Algorithm, a -cap is obtained that does not cover exactly U points. Let S(U ) be the set of all -caps in PG(N, q) each of hich does not cover exactly U points. Evidently, the group of collineations P ΓL(N +1, q) preserves S(U ). Consider the ( +1)-st step of Algorithm. This step starts from a -cap K ith K S(U ). The choice K from S(U ) can be done in distinct ays. One ay is to choose randomly a -cap of S(U ) so that for every cap of S(U ) the 1 probability to be chosen is equal to. In this case, the set S(U #S(U ) ) is considered as an ensemble of random objects ith the uniform probability distribution. Anyhere here e depend on probabilities and mathematical expectations, such a random choice is supposed. On the other hand, sometimes e study values that are average or maximum for all caps of S(U ) ithout a random choice. Also, e can consider some properties that hold for all caps of S(U ). Finally, for practice calculations (e.g. for the illustration of investigations) e use the same cap adding to it one ne point in each step of the iterative process. Denote by U(K) the set of points of PG(N, q) that are not covered by a cap K. By definition, #U(K ) = U. Let the cap K consist of points A 1, A 2,..., A. Let A +1 U(K ) be the point that ill be included into the cap in the ( + 1)-st step. Remark 2.1 Belo e introduce a fe point subsets, depending on A +1, for hich e use the notation of the form M (A +1 ). Any uncovered point may be added to K. So, there exist U distinct subsets M (A +1 ). When a particular point A +1 is not relevant, one may use the short notation M. The same concerns the quantities (A +1 ) and introduced belo. A point A +1 defines a bundle B(A +1 ) of unisecants of K hich are denoted as A 1 A +1, A 2 A +1,..., A A +1, here A i A +1 is the unisecant connecting A +1 ith the cap point A i. Every unisecant contains q + 1 points. Except for A 1,..., A, all the points on the unisecants in the bundle are candidates to be ne covered points in the ( + 1)-st step. Denote by C (A +1 ) the point set of the candidates. By definition, C (A +1 ) = B(A +1 ) \ K, #C = (q 1) + 1. We call {A +1 } and B(A +1 ) \ (K {A +1 }), respectively, the head and the basic part of the bundle B(A +1 ). For a given cap K, in total, there are #U(K ) = U distinct bundles and, respectively, U distinct sets of candidates. Let (A +1 ) be the number of ne covered points in the ( + 1)-st step, i.e. (A +1 ) = #U(K ) #U(K {A +1 }) = #{C (A +1 ) U(K )}. (2.1)

5 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), In the future, e consider continuous approximations of the discrete functions #U(K ), (A +1 ), #U(K {A +1 }), and other ones keeping the same notations. 3 Probabilities of uncovering and key conjectures Let n (H) be the number of caps of S(U ) that do not cover a point H of PG(N, q). Each point H P G(N, q) ill be considered as a random object that is not covered by a randomly chosen -cap K ith some probability p (H) defined as p (H) = n (H) #S(U ). Lemma 3.1 The value n (H) is the same for all points H PG(N, q). Proof: Let K (H) S(U ) be the subset of -caps in S(U ) that do not cover H. By the definition, n (H) = #K (H). Let H i and H j be to distinct points of PG(N, q). In the group P ΓL(N + 1, q), denote by Ψ(H i, H j ) the subset of collineations sending H i to H j. Clearly, Ψ(H i, H j ) embeds the subset K (H i ) in K (H j ). Therefore, #K (H i ) #K (H j ). Vice versa, Ψ(H j, H i ) embeds K (H j ) in K (H i ), and e have #K (H j ) #K (H i ). Thus, #K (H i ) = #K (H j ), i.e. n (H i ) = n (H j ). So, n (H) can be considered as n. This means that the probability p (H) is the same for all points H; it may be considered as p = n #S(U ). In turn, since the probability to be uncovered is independent of a point, e conclude that, for a -cap K randomly chosen from S(U ), the fraction #U (K )/θ N,q of uncovered points of PG(N, q) is equal to the probability p that a point of PG(N, q) is not covered. In other ords, p = #U (K ) θ N,q = U θ N,q. (3.1) Equality (3.1) can also be explained as follos. By Lemma 3.1, the multiset consisting of all points that are not covered by all caps of S(U ) has cardinality n #P G(N, q), here #P G(N, q) = θ N,q. This cardinality can also be ritten as U #S(U ). Thus, n θ N,q = U #S(U ), hence n #S(U ) = U θ N,q. Let s (h) be the number of ones in a sequence of h random and independent 1/0 trials each of hich yields 1 ith the probability p. For the random variable s (h) e have the binomial probability distribution; the expected value of s (h) is E[s (h)] = hp = h U θ N,q. (3.2)

6 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), Remark 3.2 We can also consider the hypergeometric probability distribution, hich describes the probability of s (h) successes in h random and independent dras ithout replacement from a finite population of size θ N,q containing exactly U successes. The expected value of s (h) again is E[s (h)] = h U θ N,q = E[s (h)]. Note also that the average number of uncovered points among h points of PG(N, q) calculated over all ( θ N,q ) h combinations of h points is 1 h ( )( ) θn,q U U ) i = U h ( )( ) θn,q U U 1 ) = U ( θn,q ) 1 h 1 ) h i i h i i 1 ( θn,q h i=1 = h U θ N,q = E[s (h)]. ( θn,q h i=1 We denote by E,q the expected value of the number of uncovered points among (q 1) + 1 randomly taken points in PG(N, q), if the events to be uncovered are independent. By Lemma 3.1, taking into account (3.1), (3.2), e have ( θn,q h E,q = E[s ((q 1) + 1)] = ((q 1) + 1)p = ((q 1) + 1)U θ N,q. (3.3) In (2.1), e defined (A +1 ) as the number of ne covered points in the (+1)- st step. Since all candidates to be ne covered points lie on some bundle, they cannot be considered as randomly taken points for hich the events to be uncovered are independent. Thus, in the general case, the expected value E[ ] is not equal to E,q. On the other hand, there is a large number of random factors affecting the process, for instance, the relative positions and intersections of bisecants and unisecants. These factors especially act for groing q, hen the volume of the ensemble S(U ) and the number of distinct bundles B(A +1 ) are relatively large. Therefore, the variance of the random variable, in principle, implies the existence of bundles B(A +1 ) providing the inequality (A +1 ) > E[ ]. By these arguments (see also Section 7) Conjecture 3.3 seems to be reasonable and founded. Conjecture 3.3 (i) (The generalized conjecture.) For q large enough, in every (+1)-st step of the iterative process in PG(N, q) considered in Section 2, there exists a -cap K S(U ) such that one can find an uncovered point A +1 providing the inequality (A +1 ) E,q D here D 1 is a constant independent of q. (ii) (the basic conjecture) In (3.4), e have D = 1. = 1 D ((q 1) + 1)U θ N,q, (3.4)

7 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), Upper bounds on t 2 (N, q) We denote Q = θ N,q q 1 = qn+1 1 (q 1) 2. (4.1) By Conjecture 3.3, taking into account (2.1), (3.3), (3.4), e obtain #U(K {A +1 }) = #U(K ) (A +1 ) (4.2) ( ) ( ) ( (q 1) + 1 (q 1) U 1 < U 1 < U 1 ). Dθ N,q Dθ N,q DQ Clearly, #U(K 1 ) = U 1 = θ N,q 1. Using (4.2) iteratively, e have #U(K {A +1 }) (θ N,q 1)f q (; D) < θ N,q f q (; D) (4.3) here f q (; D) = i=1 ( 1 i ). (4.4) DQ Remark 4.1 The function f q (; D) and its approximations, including (4.8), appear in distinct tasks of probability theory, for example, in the Birthday problem [16, 18, 37]. Actually, let the year contain DQ days and let all birthdays occur ith the same probability. Then P DQ ( + 1) = f q(; D) here P DQ ( + 1) is the probability that no to persons from + 1 random persons have the same birthday. Moreover, if birthdays occur ith different probabilities e have P DQ ( + 1) < f q(; D) [18]. In the folloing discussion, e consider a truncated iterative process. The iterative process ends hen #U(K {A +1 }) ξ here ξ 1 is some value chosen to improve estimates. Then, to obtain a complete k-cap, it is sufficient to add to K at most ξ points. The size k of the complete cap obtained is as follos: + 1 k ξ under condition #U(K {A +1 }) ξ. (4.5) Theorem 4.2 Let f q (; D) be as in (4.4). Let ξ be a constant independent of ith ξ 1. Under Conjecture 3.3, in PG(N, q), here the value satisfies the inequality t 2 (N, q) ξ (4.6) f q (; D) ξ θ N,q. (4.7) Proof: By (4.3), to provide the inequality #U(K {A +1 }) ξ it is sufficient to find such that θ N,q f q (; D) ξ. No (4.6) follos from (4.5).

8 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), We find an upper bound on the smallest possible solution of inequality (4.7). The Taylor series of e α implies 1 α < e α for α 0, hence i=1 ( 1 i ) < DQ e i/dq = e (2 +)/2DQ < e 2 /2DQ. (4.8) i=1 Lemma 4.3 Let ξ be a constant independent of ith ξ 1. The value satisfies the inequality (4.7). 2DQ ln θ N,q ξ + 1 (4.9) Proof: By (4.4),(4.8), to provide (4.7) it is sufficient to find such that e 2 /2DQ ξ θ N,q. As should be an integer, in (4.9) one is added. Theorem 4.4 Let D 1 be a constant independent of q. Under Conjecture 3.3(i), t 2 (N, q) 2DQ ln θ N,q ξ here ξ is an arbitrarily chosen constant independent of. + ξ + 2, ξ 1, (4.10) Proof: The assertion follos from (4.6) and (4.9). We should choose ξ to obtain a relatively small value in the right part of (4.10). We consider the function of ξ of the form φ(ξ) = 2DQ ln θ N,q ξ + ξ + 2. Its derivative by ξ is Put φ (ξ) = 0. Then φ (ξ) = 1 1 ξ DQ 2 ln θ N,q ξ. ξ 2 = DQ 2 ln θ N,q 2 ln ξ = Dθ N,q 2(q 1)(ln θ N,q ln ξ). (4.11)

9 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), We find ξ in the form ξ = θn,q c ln θ N,q. By (4.11), c = q 1 (ln θ N,q + ln c + ln ln θ N,q ) = q 1 D ln θ N,q D So, for q large enough, one could take c = q 1 D, ξ = Dθ N,q = (q 1) ln θ N,q ( 1 + ln c + ln ln θ ) N,q. ln θ N,q D(q N+1 1) (q 1) 2 ln θ N,q. For simplicity, e put ξ = q N+1 q 1. (4.12) Theorem 4.5 Let D 1 be a constant independent of q. Under Conjecture 3.3(i), the folloing upper bound on the smallest size t 2 (N, q) of a complete cap in PG(N, q), N 3, holds: t 2 (N, q) < q N+1 ( ) D (N + 1) ln q + 1 q Dq N 1 2 (N + 1) ln q. (4.13) Proof: In (4.10), e take Q and ξ from (4.1) and (4.12) and obtain t 2 (N, q) < 2D qn+1 qn q 1 q N+1 ln + (q 1) 2 q q N+1 2 q 1 hence the relation (4.13) follos directly, as q N+1 1 < q N+1. From Theorem 4.5 e obtain Theorem Illustration of the effectiveness of the ne bounds In [10,11], for PG(N, q), N = 3, 4, q L N, complete caps are obtained by computer search. Here L 3 := {q 4673, q prime} {5003, 6007, 7001, 8009}, L 4 := {q 1361, q prime} {1409}. All the complete caps obtained satisfy the bound (4.13) ith D = 1 (equivalently, the bound (1.2)). Let t 2 (N, q) be the smallest knon size of a complete cap in PG(N, q); the sizes t 2 (N, q) can be found in [10]. In Figure 1, e compare the upper bound (1.2) ith the sizes t 2 (N, q). The top dashed-dotted curve, corresponding to (1.2), is strictly higher than the bottom curve t 2 (N, q).

10 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), x 10 4 a) PG(3,q) q 2 1 q 1 (2 lnq +1)+2 t 2 (3,q) x 10 5 b) PG(4,q) q q 2 q 1 q 1 ( 5lnq +1) t 2 (4,q) q ( q (N ) Figure 1: Bound t 2 (N, q) < N+1 q 1 + 1) ln q (top dashed-dotted curve) vs. the smallest knon sizes t 2 (N, q) of complete caps, q L N, N = 3, 4 (bottom curve). a) PG(3, q) b) PG(4, q) 6 A rigorous proof of Conjecture 3.3 for a part of the iterative process In the folloing discussion, e take into account the fact that all points not covered by a cap lie on unisecants of the cap. There are θ N 1,q lines through every point of PG(N, q). Therefore, through every point A i of K there is a pencil P(A i ) of θ N 1,q ( 1) unisecants of K, here i = 1, 2,...,. The total number T Σ of the unisecants of K is T Σ = (θ N 1,q + 1 ). (6.1) Let γ,j be the number of uncovered points on the j-th unisecant T j, j = 1, 2,..., T Σ. Observation 6.1 Every unisecant of K belongs to one and only one pencil P(A i ), i {1, 2,..., }. Every uncovered point belongs to one and only one unisecant from every pencil P(A i ), i = 1, 2,...,. Every uncovered point A lies on exactly unisecants hich form the bundle B(A) ith the head {A}. All unisecants from the same bundle belong to distinct pencils. A unisecant T j belongs to γ,j distinct bundles.

11 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), Every uncovered point lies on exactly unisecants; due to this multiplicity, in total, on all unisecants there are Γ Σ uncovered points, here Γ Σ = T Σ γ,j = U. (6.2) j=1 By (6.1) and (6.2), the average number γ aver of uncovered points on a unisecant is γ aver = ΓΣ T Σ = U θ N 1,q + 1. (6.3) A unisecant T j belongs to γ,j distinct bundles, because every uncovered point on T j may be the head of a bundle. Moreover, T j provides γ,j (γ,j 1) uncovered points to the basic parts of all these bundles. The noted points are counted ith multiplicity. Taking into account the multiplicity, in all U the bundles there are A +1 (A +1 ) = U + T Σ γ,j (γ,j 1) (6.4) uncovered points, here U is the total numbers of all the heads. By (6.2) and (6.4), T Σ T Σ T Σ (A +1 ) = U + γ,j 2 γ,j = U (1 ) + A +1 j=1 j=1 j=1 j=1 γ 2,j. For a cap K, e denote by aver (K ) the average value of (A +1 ) by all #U(K ) uncovered points A +1, i.e. aver (K ) = A +1 (A +1 ) #U(K ) = A +1 (A +1 ) U = T Σ γ,j 2 j=1 U (6.5) Also note that T Σ γ,j 2 j=1 T Σ γ,j = U. (6.6) j=1 We denote a loer estimate of aver rigor = { { (K ) := max 1, (K ), see Lemma 6.2 belo, as follos: } U θ N 1,q = (6.7) U θ N 1,q +1 if U θ N 1,q + 1, 1 if U < θ N 1,q + 1.

12 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), Lemma 6.2 For any -cap K S(U ), the folloing assertions hold: This inequality alays holds In (6.8), e have the equality aver (K ) rigor (K ). (6.8) aver (K ) = rigor (K ) = if and only if every unisecant contains the same number points here U θ N 1,q +1 In (6.8), the equality is an integer. U θ N 1,q (6.9) U θ N 1,q +1 of uncovered aver (K ) = rigor (K ) = 1 (6.10) holds if and only if each unisecant contains at most one uncovered point. Proof: By the Cauchy Scharz Bunyakovsky inequality, e have T Σ j=1 γ,j 2 T Σ T Σ γ,j 2 (6.11) here equality holds if and only if all γ,j coincide. In this case, γ,j = for all j and, moreover, the ratio U θ N 1,q +1 U θ N 1,q + 1 j=1 U θ N 1,q +1 is an integer. No, by (6.1) and (6.2), T Σ γ,j 2 j=1 U that together ith (6.2), (6.5), (6.6) and (6.7) gives (6.8) (6.10). Remark 6.3 One can treat the estimates (6.8) and (6.9) as follos. A bundle contains unisecants having a common point, its head. Therefore the average number of uncovered points in a bundle is γ aver ( 1) here γ aver is defined in (6.3) and the term 1 takes into account the common point. It is clear that for any -cap K S(U ), e have max A +1 (A +1 ) aver (K ). (6.12)

13 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), Corollary 6.4 The folloing inequality holds: { 1, max (A +1 ) max A +1 U θ N 1,q }. Let D 1 be a constant independent of q. Throughout the paper e denote D( 1)θ N,q (θ N 1,q + 1 ) Φ,q (D) = Dθ N,q (θ N 1,q + 1 )((q 1) + 1), Dθ N,q Υ,q (D) = (q 1) + 1. Lemma 6.5 Let D 1 be a constant independent of q. Let either of the folloing to conditions hold: Then, for any cap K of S(U ), e have Proof: By (6.7) and (6.8), aver U Φ,q (D), Υ,q (D) U. aver (K ) E,q D. (K ) rigor (K ) U θ N 1,q It is easy to see that under condition U Φ,q (D), e have U θ N 1,q ((q 1) + 1)U Dθ N,q 0. If U Υ,q (D) then E,q 1. On the other hand, the inequalities aver D (K ) rigor (K ) 1 alays hold. From Lemmas 6.2 and 6.5 e obtain the folloing corollary. Corollary 6.6 Let D 1 be a constant independent of q. Let either of the folloing to conditions hold: U Φ,q (D), Υ,q (D) U. Then, for any cap K of S(U ), there exists an uncovered point A +1 yielding the inequality (A +1 ) E,q D = ((q 1) + 1)U Dθ N,q. Proof: By definition of the average value (6.5), there is alays an uncovered point A +1 yielding the inequality (A +1 ) aver (K ); see also (6.12).

14 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), On reasonableness of Conjecture 3.3 In this section e sho (by reflections, calculations and figures) that in the steps of the iterative process here the rigorous estimates give bad results, these estimates do not reflect the real situation effectively. (i) First e ill illustrate the folloing: hen the rigorous bound (6.7) (6.8) is smaller than the expectation E,q, the average value aver (K ) of (6.5) is greater (and the maximum value max (A +1 ) is essentially greater) than E,q. A +1 We have calculated the values (A +1 ), defined in (2.1), for numerous iterative processes in PG(3, q) and PG(4, q). It is important that for all the calculations hich have been done, e have max (A +1 ) > E,q. A +1 Moreover, the ratio max (A +1 )/E,q has an increasing trend hen gros. A +1 Thus, the variance of the random value helps to get good results. The existence of points A +1 providing (A +1 ) > E,q is used by the greedy algorithms to obtain complete caps smaller than the bounds folloing from Conjecture 3.3. An illustration of the aforesaid is shon in Figure 2 here for a complete k-cap in PG(3, 101), k = 415, obtained by the greedy algorithm, the values δ min = δ aver min A +1 (A +1 ) E,q, δ max = = aver (K ), δ rigor E,q max A +1 (A +1 ) E,q, = rigor (K ) E,q, are presented. The horizontal axis shos the values of. The final region of the k iterative process hen U Υ,q (D) and E,q 1 is not completely shon. The D lines y = 1 and y = 1 correspond, respectively, to Conjecture 3.3(ii), here D = 1, 5 and Conjecture 3.3(i) ith D = 5. The signs correspond to the values Φ,q (D) and Υ,q (D) ith D = 1 and D = 5. It is important that, for all the steps of the iterative process, e have aver (K ) > E,q, i.e. δ aver > 1. In Figure 2, the region here e rigorously prove Conjecture 3.3 lies on the left of Φ,q (D) and on the right of Υ,q (D). This region takes 35% of the hole iterative process for D = 1 and 75% for D = 5. Note that the forms of curves δ max e calculated these values. and δ aver are similar for all q and N for hich (ii) No e consider the dispersion of the number of uncovered points on unisecants. The loer estimate in (6.8), based on (6.11), is attained in to cases: either every unisecant contains the same number of uncovered points or each unisecant contains at most one uncovered point.

15 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), P G(3, 101) 415-cap Φ,q (1) Φ,q (5) δ max δ aver δ min δ rigor 1 Υ,q (1) Υ,q (5) /k 1 5 Figure 2: Illustration of reasonableness of Conjecture 3.3. Values δ for a complete k-cap in PG(3, q), k = 415: δ max (top curve), δ aver (the second curve), δ min (the third curve), δ rigor (bottom dotted curve); lines y = 1 (for D = 1) and y = 1 (for D = 5) 5 The first situation holds in the starting steps of the iterative process only. Then the differences γ,j γ,i become nonzero. But hile the inequality U (D) Φ,q (D) holds, these differences are relatively small and the estimate (6.8) orks ell. When U decreases, the differences relatively increase, and the estimate becomes orse in the sense that actually aver (K ) is considerably greater than rigor (K ). The second situation is possible, in principle, hen U θ N 1,q + 1 and the average number γ aver of uncovered points on a unisecant is smaller than one; see (6.3). But in this stage of the iterative process variations in the values γ,j are relatively large; and again aver (K ) is considerably greater than rigor (K ). In the final region of the iterative process, here U Υ,q (D) and E,q 1, D the estimate (6.8) becomes reasonable once more. Thus, in the region Φ,q (D) > U > Υ,q (D) the loer estimate (6.8) does not reflect the real situation effectively. This leads the necessity to formulate Conjecture 3.3 as a (plausible) hypothesis. Let γ aver be defined in (6.3). Let γ max and γ min be, respectively, the maximum

16 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), and minimum of the number γ,j of uncovered points on a unisecant, i.e. γ max = max γ,j, j γ min = min j γ,j. An illustration of the fact that the numbers γ,j of uncovered points on unisecants lie in a relatively ide region is shon in Figure 3, here for a complete k-cap in PG(3, 101), k = 415, obtained by the greedy algorithm, the values γ max /γ aver and γ min /γ aver are presented. The horizontal axis shos the values of. Such k curves ere obtained for numerous iterative processes in PG(3, q) and PG(4, q). It is important that for all the calculations have been done, the forms of the curves are similar. Moreover, the value γ max /γ aver increases hen the ratio gros; in the k region 0.78 < < 0.95 (it is not shon in Figure 3), the value γmax k /γ aver increases from 20 to 590 for the 415-cap in PG(3, 101) PG(3, 101) 415-cap γ min /γ aver γ max /γ aver /k 1 0 Figure 3: Dispersion of the number γ,j of uncovered points on unisecants. Values γ max /γ aver (top solid curve), γ min /γ aver (bottom solid curve) and line y = 1 for a complete k-cap in PG(3, q), k = 415 Remark 7.1 It can be proved rigorously (using Observation 6.1) that if in some step of the iterative process every unisecant contains the same number of uncovered points then in the next step this situation does not hold. The calculations mentioned in this section and Figures 2, 3 illustrate the soundness of the key Conjecture Conclusion In the present paper, e have made an attempt to obtain a theoretical upper bound on t 2 (N, q) ith the main term of the form cq N 1 2 ln q, here c is a small constant

17 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), independent of q. The bound is based on explaining the mechanism of a step-by-step greedy algorithm for constructing complete caps in PG(N, q) and on quantitative estimations of the algorithm. For a part of the steps of the iterative process, these estimations have been proved rigorously. We made the natural (and ell-founded) conjecture that they hold for other steps too. Under this conjecture e have given ne upper bounds on t 2 (N, q) in the required form; see (1.1) and (1.2). We have illustrated the effectiveness of the ne bounds, comparing them ith the results of a computer search from [10, 11]; see Figure 1. We have not obtained a rigorous proof for precisely the part of the process here the variance of the random variable (A +1 ) determining the estimates implies the existence of points A +1 hich are considerably better than hat is necessary for fulfillment of the conjecture (see the curve δ max in Figure 2). In other ords, in the steps of the iterative process here the rigorous estimates give bad results, these estimates do not reflect the real situation effectively. The reason is that the rigorous estimates assume that the number of uncovered points on unisecants is the same for all unisecants. Hoever, there is a dispersion of the number of uncovered points on unisecants; see Section 7. Moreover, this dispersion gros in the iterative process. So Conjecture 3.3 seems to be reasonable. Acknoledgments The research of A. A. Davydov as carried out at the IITP RAS at the expense of the Russian Foundation for Sciences (project ). The research of G. Faina, S. Marcugini and F. Pambianco as supported in part by Ministry for Education, University and Research of Italy (MIUR) (Project Geometrie di Galois e strutture di incidenza ), by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA INDAM), and by the University of Perugia, (Projects Configurazioni Geometriche e Superfici Altamente Simmetriche and Codici lineari e strutture geometriche correlate, Base Research Fund 2015). The authors ould like to thank the anonymous referees and the editor for their helpful comments and suggestions. References [1] N. Anbar, D. Bartoli, M. Giulietti and I. Platoni, Small complete caps from singular cubics, J. Combin. Des. 22 (2014), [2] N. Anbar, D. Bartoli, M. Giulietti and I. Platoni, Small complete caps from singular cubics, II, J. Algebraic Combin. 41 (2015), [3] N. Anbar and M. Giulietti, Bicovering arcs and small complete caps from elliptic curves, J. Algebraic. Combin. 38 (2013),

18 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), [4] D. Bartoli, A. A. Davydov, G. Faina, A. A. Kreshchuk, S. Marcugini and F. Pambianco, Upper bounds on the smallest size of a complete arc in PG(2, q) under a certain probabilistic conjecture, Problems Inform. Transmission 50 (2014), [5] D. Bartoli, A. A. Davydov, G. Faina, A. A. Kreshchuk, S. Marcugini and F. Pambianco, Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search, J. Geom. 107 (2016), [6] D. Bartoli, A. A. Davydov, G. Faina, S. Marcugini and F. Pambianco, Ne upper bounds on the smallest size of a complete cap in the space PG(3, q), in Proc. VII Int. Workshop on Optimal Codes and Related Topics, OC2013, Albena, Bulgaria, 2013, [7] D. Bartoli, A. A. Davydov, G. Faina, S. Marcugini and F. Pambianco, On sizes of complete arcs in PG(2, q), Discrete Math. 312 (2012), [8] D. Bartoli, A. A. Davydov, G. Faina, S. Marcugini and F. Pambianco, Ne types of estimates for the smallest size of complete arcs in a finite Desarguesian projective plane, J. Geom. 106 (2015), [9] D. Bartoli, A. A. Davydov, G. Faina, S. Marcugini and F. Pambianco, Conjectural upper bounds on the smallest size of a complete cap in PG(N, q), N 3, Electron. Notes Discrete Math. 57 (2017), [10] D. Bartoli, A. A. Davydov, A. A. Kreshchuk, S. Marcugini and F. Pambianco, Tables, bounds and graphics of the smallest knon sizes of complete caps in the spaces PG(3,q) and PG(4,q), arxiv: [math.co] (2016). [11] D. Bartoli, A. A. Davydov, A. A. Kreshchuk, S. Marcugini and F. Pambianco, Upper bounds on the smallest size of a complete cap in PG(3, q) and PG(4, q), Electron. Notes Discrete Math. 57 (2017), [12] D. Bartoli, G. Faina and M. Giulietti, Small complete caps in three-dimensional Galois spaces, Finite Fields Appl. 24 (2013), [13] D. Bartoli, G. Faina, S. Marcugini and F. Pambianco, A construction of small complete caps in projective spaces, J. Geom. 108 (2017), [14] D. Bartoli, M. Giulietti, G. Marino and O. Polverino, Maximum scattered linear sets and complete caps in Galois spaces, Combinatorica 38 (2018), [15] D. Bartoli, S. Marcugini and F. Pambianco, Ne quantum caps in PG(4, 4), J. Combin. Des. 20 (2012), [16] D. Brink, A (probably) exact solution to the birthday problem, Ramanujan J. 28 (2012),

19 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), [17] R. A. Brualdi, S. Litsyn and V. S. Pless, Covering radius, in Handbook of Coding Theory (Eds.: V.S. Pless, W.C. Huffman and R.A. Brualdi), Vol. 1, Elsevier, Amsterdam, The Netherlands, 1998, [18] M. L. Clevenson and W. Watkins, Majorization and the Birthday inequality, Math. Mag. 64 (1991), [19] G. D. Cohen, I. S. Honkala, S. Litsyn and A. C. Lobstein, Covering Codes, Elsevier, Amsterdam, The Netherlands, [20] A. A. Davydov, G. Faina, S. Marcugini and F. Pambianco, On sizes of complete caps in projective spaces PG(n, q) and arcs in planes PG(2, q), J. Geom. 94 (2009), [21] A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, Ne inductive constructions of complete caps in PG(n, q), q even, J. Combin. Des. 18 (2010), [22] A. A. Davydov, S. Marcugini and F. Pambianco, Complete caps in projective spaces PG(n, q), J. Geom. 80 (2004), [23] A. A. Davydov and P. R. J. Östergård, Recursive constructions of complete caps, J. Statist. Plann. Inference 95 (2001), [24] G. Faina, F. Pasticci and L. Schmidt, Small complete caps in Galois spaces, Ars Combin. 105 (2012), [25] E. M. Gabidulin, A. A. Davydov and L. M. Tombak, Linear codes ith covering radius 2 and other ne covering codes, IEEE Trans. Inform. Theory 37 (1991), [26] M. Giulietti, Small complete caps in Galois affine spaces, J. Algebraic Combin. 25 (2007), [27] M. Giulietti, Small complete caps in PG(n, q), q even, J. Combin. Des. 15 (2007), [28] M. Giulietti, The geometry of covering codes: small complete caps and saturating sets in Galois spaces, in Surveys in Combinatorics 2013, London Math. Soc., Lec. Note Ser. 409, Cambridge University Press, Cambridge, 2013, [29] M. Giulietti and F. Pasticci, Quasi-perfect linear codes ith minimum distance 4, IEEE Trans. Inform. Theory 53 (2007), [30] J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces, J. Statist. Plann. Inference 72 (1998),

20 A.A. DAVYDOV ET AL. / AUSTRALAS. J. COMBIN. 72 (3) (2018), [31] J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces: update 2001, in Finite geometries, , Dev. Math., 3, Kluer Acad. Publ., Dordrecht, [32] J. W. P. Hirschfeld and J. A. Thas, Open problems in finite projective spaces, Finite Fields Appl. 32 (2015), [33] J. H. Kim and V. Vu, Small complete arcs in projective planes, Combinatorica 23 (2003), [34] I. Landjev and L. Storme, Galois geometry and coding theory, in Current Research Topics in Galois geometry (Eds.: J. De Beule and L. Storme), Chapter 8, Nova Science Publisher, 2011, [35] F. Pambianco and L. Storme, Small complete caps in spaces of even characteristic, J. Combin. Theory Ser. A 75 (1996), [36] I. Platoni, Complete caps in AG(3, q) from elliptic curves, J. Alg. Appl. 13 (2014), (8 pp.). [37] M. Sayrafiezadeh, The Birthday problem revisited, Math. Mag. 67 (1994), [38] B. Segre, On complete caps and ovaloids in three-dimensional Galois spaces of characteristic to, Acta Arith. 5 (1959), [39] T. Szőnyi, Arcs, caps, codes and 3-independent subsets, in Giornate di Geometrie Combinatorie (Eds.: G. Faina and G. Tallini), Università degli Studi di Perugia, Perugia, Italy, 1993, [40] V. D. Tonchev, Quantum codes from caps, Discrete Math. 308 (2008), (Received 10 Oct 2017; revised 20 Sep 2018)