THE MAXIMUM SIZE OF A PARTIAL SPREAD II: UPPER BOUNDS

THE MAXIMUM SIZE OF A PARTIAL SPREAD II: UPPER BOUNDS ESMERALDA NĂSTASE MATHEMATICS DEPARTMENT XAVIER UNIVERSITY CINCINNATI, OHIO 4507, USA PAPA SISSOKHO MATHEMATICS DEPARTMENT ILLINOIS STATE UNIVERSITY NORMAL, ILLINOIS 61790, USA Abstact Let n and t be positive integes with t < n, and let q be a pime powe A patial (t )-spead of PG(n, q) is a set of (t )-dimensional subspaces of PG(n, q) that ae paiwise disjoint Let n (mod t) with 0 < t, and let Θ i = (q i )/(q ) We essentially pove that if < t Θ, then the maximum size of a patial (t )-spead of PG(n, q) is bounded fom above by (Θ n Θ t+)/θ t + q (q )(t 3) + 1 We actually give tighte bounds when cetain divisibility conditions ae satisfied These bounds impove on the peviously known uppe bound fo the maximum size patial (t )-speads of PG(n, q); fo instance, when Θ + 4 t Θ and q > The exact value of the maximum size patial (t )-spead has been ecently detemined fo t > Θ by the authos of this pape (see [1]) Keywods: Galois geomety; patial speads; subspace patitions; subspace codes Mathematics Subject Classification: 51E3; 05B5; 94B5 1 Intoduction Let n and t be positive integes with t < n, and let q be a pime powe Let PG(n, q) denote the (n )-dimensional pojective space ove the finite field F q A patial (t )-spead S of PG(n, q) is a collection of (t )-dimensional subspaces of PG(n, q) that ae paiwise disjoint If S contains all the points of PG(n, q), then it is called a (t )-spead It is well-known that a (t )-spead of PG(n, q) exists if and only if t divides n (eg, see [3, p 9]) Besides thei taditional elevance to Galois geomety [6, 11, 13, 17], patial (t )-speads ae used to build byte-coecting codes (eg, see [7, 16]), 1-pefect mixed eo-coecting codes (eg, see [15, 16]), othogonal aays (eg, see [4]), and subspace codes (eg, see [8, 10, 18]) Convention: Fo the est of the pape, we assume that q is a pime powe, and n, t, and ae integes such that n > t > 0 and n (mod t) We also use µ q (n, t) to denote the maximum size of any patial (t )-spead of PG(n, q) The poblem of detemining µ q (n, t) is a long standing open poblem Cuently, the best geneal uppe bound fo µ q (n, t) is given by the following theoem of Dake and Feeman [4] nastasee@xavieedu, psissok@ilstuedu 1

ESMERALDA NĂSTASE AND PAPA SISSOKHO Theoem 1 If > 0, then µ q (n, t) qn q t+ q t 1 + q ω, whee ω = 4q t (q t q ) + 1 (q t q + 1) The following esult is attibuted to Andé [1] and Sege [] fo = 0 Fo = 1, it is due to Hong and Patel [16] when q =, and Beutelspache [] when q > Theoem If 0 < t, then µ q (n, t) qn q t+ q t 1 + 1, and equality holds if {0, 1} In light of Theoem, it was late conjectued (eg, see [5, 16]) that the value of µ q (n, t) is given by the lowe bound in Theoem Howeve, this conjectue was dispoved by El-Zanati, Jodon, Seelinge, Sissokho, and Spence [9] who poved the following esult Theoem 3 If n 8 and n mod 3 =, then µ (n, 3) = n 5 7 + Recently, Kuz [19] poved the following theoem which upholds the lowe bound fo µ q (n, t) when q =, =, and t > 3 Theoem 4 If n > t > 3 and n mod t =, then µ (n, t) = n t+ t 1 + 1 Fo any intege i 1, let (1) Θ i = (q i )/(q ) Still ecently, the authos of this pape affimed the conjectue (eg, see [5, 16]) on the value of µ q (n, t) fo t > Θ and any pime powe q, by poving the following geneal esult (see [1]) Theoem 5 If t > Θ, then µ q (n, t) = qn q t+ q t 1 + 1 In light of Theoem 5, it emains to detemine the value of µ q (n, t) fo < t Θ In this pape, we apply the hypeplane aveaging method that we devised in [1] to pove the following esults 1 The est of the pape is devoted to thei poofs { q if q ((q )(t ) + c 1 ) Theoem 6 Let c 1 (t ) (mod q), 0 c 1 < q, and c = 0 if q ((q )(t ) + c 1 ) If < t Θ, then Consequently, µ q (n, t) qn q t+ µ q (n, t) qn q t+ + q (q )(t ) c 1 + c + q (q )(t 3) + 1 Remak 7 The best possible bound in Theoem 6 is obtained when t aq + 1 (mod q ), 1 a q (equivalently, when t 1 (mod q) but t 1 (mod q )) In this case, we can check that c 1 = q and c = 0, which implies that µ q (n, t) qn q t+ + q (q )(t ) This was aleady noted in [1, Lemma 10 and Remak 11] fo and t = Θ = (q 1)/(q 1) 1 Also see [0] fo a ecent pepint in this aea

THE MAXIMUM SIZE OF A PARTIAL SPREAD II: UPPER BOUNDS 3 Coollay 8 Let f q (n, t) denote the uppe bound fo µ q (n, t) in Theoem 1 and let g q (n, t)denote the uppe bound fo µ q (n, t) in Theoem 6 Let c 1 and c be as defined in Theoem 6 If and t Θ then q g q (n, t) f q (n, p) = (q )(t ) c 1 + c Consequently, fo Θ + 4 t Θ with q >, and fo Θ + 5 t Θ with q =, we have g q (n, t) f q (n, p) < 0, and thus the uppe bound fo µ q (n, t) given in Theoem 6 is tighte than the Dake Feeman bound in Theoem 1 In Section, we pesent some auxiliay esults fom the aea of subspace patitions, and in Section 3 we pove Theoem 6 and Coollay 8 Subspace patitions Let V = V (n, q) denote the vecto space of dimension n ove F q Fo any subspace U of V, let U denote the set of nonzeo vectos in U A d-subspace of V (n, q) is a d-dimensional subspace of V (n, q); this is equivalent to a (d )-subspace in PG(n, q) A subspace patition P of V, also known as a vecto space patition, is a collection of nontivial subspaces of V such that each vecto of V is in exactly one subspace of P (eg, see Heden [13] fo a suvey on subspace patitions) The size of a subspace patition P, denoted by P, is the numbe of subspaces in P Suppose that thee ae s distinct integes, d s > > d 1, that occu as dimensions of subspaces in a subspace patition P, and let n i denote the numbe of i-subspaces in P Then the expession [d n ds s,, d n d 1 1 ] is called the type of P Remak 9 A patial (t 1)-spead of PG(n 1, q) of size n t is a patial t-spead of V (n, q) of size n t This is equivalent to a subspace patition of V (n, q) of type [t nt, 1 n 1 ], whee n 1 = Θ n n t Θ t We will use this subspace patition fomulation in the poof of Lemma 14 Also, we will use the following theoem due to Heden [1] in the poof of Lemma 14 Theoem 10 [1, Theoem 1] Let P be a subspace patition of V (n, q) of type [d n ds s,, d n d 1 1 ], whee d s > > d 1 Then, (i) if q d d 1 does not divide n d1 and if d < d 1, then n d1 q d 1 + 1 (ii) if q d d 1 does not divide n d1 and d d 1, then eithe n d1 = (q d )/(q d 1 ) o n d1 > q d d 1 (iii) if q d d 1 divides n d1 and d < d 1, then n d1 q d q d 1 + q d d 1 (iv) if q d d 1 divides n d1 and d d 1, then n d1 q d To state the next lemmas, we need the following definitions Recall that fo any intege i 1, Θ i = (q i )/(q ) Then, fo i 1, Θ i is the numbe of 1-subspaces in an i-subspace of V (n, q) Let P be a subspace patition of V = V (n, q) of type [d n ds s,, d n d 1 1 ] Fo any hypeplane H of V, let b H,d be

4 ESMERALDA NĂSTASE AND PAPA SISSOKHO the numbe of d-subspaces in P that ae contained in H and set b H = [b H,ds,, b H,d1 ] Define the set B of hypeplane types as follows: B = {b H : H is a hypeplane of V } Fo any b B, let s b denote the numbe of hypeplanes of V of type b We will also use Lemma 11 and Lemma 1 by Heden and Lehmann [14] in the poof of Lemma 14 Lemma 11 [14, Equation (1)] Let P be a subspace patition of V (n, q) of type [d n ds s,, d n d 1 1 ] If H is a hypeplane of V (n, q) and b H,d is as defined above, then s P = 1 + b H,di q d i i=1 Lemma 1 [14, Equation () and Coollay 5] Let P be a subspace patition of V (n, q), and let B and s b be as defined above Then s b = Θ n, and fo 1 d n, we have b B b d s b = n d Θ n d b B 3 Poofs of the main esults Recall that q is a pime powe, and n, t, and ae integes such that n > t > 0, and n (mod t) To pove ou main esult, we fist need to pove the following two technical lemmas Lemma 13 Let x be an intege such that 0 < x < q Fo any positive intege i, let δ i = q i xq i Θ i xθ i Then the following popeties hold: (i) xq t Θ t = x q 1 (ii) fo 1 i t, we have 0 δ i < q i, q (x + δ i+1 ), and δ i = q 1 (x + δ i+1 ) mod q i (iii) δ i = 0 if and only if q i x Poof Let α and β be integes such that x = α(q ) + β, α 0, and 0 β < q Since 0 < x < q and < t hold by hypothesis, it follows that () 0 α < x < q < q t and α(q ) x < q < q t If β = 0, then by (), we obtain xq t α(q t ) Θ t = q t = α αq (3) t = α = x q Now suppose 1 β < q Fist, since β 1, it follows fom () that xq t [α(q ) + β](q t ) [α(q ) + 1](q t ) Θ t = q t (q ) q t (q ) = α + (qt ) α(q ) q t (q ) (4) = α + 1

THE MAXIMUM SIZE OF A PARTIAL SPREAD II: UPPER BOUNDS 5 Second, since β < q, it follows fom () and the popeties of the ceiling function that xq t [α(q ) + β](q t ) (α + 1)(q t ) Θ t = q t (q ) q t = α + 1 α + 1 (5) q t = α + 1 Then (4) and (5) imply that fo 1 β < q, xq t Θ t = α + 1 = which completes the poof of (i) x q We now pove (ii) Since 0 a a < 1 holds fo any eal numbe a, we have 0 q i xθ i q i xθ i < 1 = δ i = q i xq i Θ i xθ i < q i and δ i 0 By the definition of δ i, we have that and thus, x + δ i+1 = x + q i+1 xq i 1 Θ i+1 xθ i+1 = q(q i xq i 1 Θ i+1 xθ i ),, q 1 (x + δ i+1 ) q i xq i 1 Θ i+1 xθ i xθ i (6) q i xq i Θ i xθ i δ i (mod q i ) Finally, we pove (iii) Since gcd(q i, Θ i ) = 1 fo any positive intege i, we have δ i = q i xq i Θ i xθ i = 0 xq i Θ i = xq i Θ i q i x We now pove ou main lemma Lemma 14 Let x be a positive intege such that q x and q x Let l = (q n t q )/() If and t Θ x/(q ) +, then µ q (n, t) lq t + x Poof If x q, then Theoem 1 implies the nonexistence of a patial t-spead of size lq t + x Thus, we can assume that x < q Recall that Θ i = (q i )/(q ) fo any intege i 1 Fo an intege i, with i t, let (7) δ i = q i xq i Θ i xθ i Applying Lemma 13(i), we let (8) h := q t xθ t = x q

6 ESMERALDA NĂSTASE AND PAPA SISSOKHO The poof is by contadiction So assume that µ q (n, t) > lq t + x Then PG(n, q) has a (t )-patial spead of size lq t + 1 + x Thus, it follows fom Remak 9 that thee exists a subspace patition P 0 of V (n, q) of type [t nt, 1 n 1 ], with (9) n t = lq t + 1 + x, and n 1 = q t Θ xθ t = q t (Θ q t xθ t ) + (q t q t xθ t xθ t ) = q t (Θ h) + δ t, whee h is given by (8) and δ t is given by (7) We will pove by induction that fo each intege j with 0 j t, thee exists a subspace patition P j of H j = V (n j, q) of type (10) [t m j,t, (t ) m j,t 1,, (t j) m j,t j, 1 m j,1 ], whee m j,t,, m j,t j ae nonnegative integes such that t (11) m j,i = n t = lq t + 1 + x, i=t j and whee m j,1 and c j ae integes such that (1) m j,1 = c j q t j + δ t j, and 0 c j max{θ h j, 0} The base case, j = 0, holds since P 0 is a subspace patition of H 0 = V (n, q) with type [t nt, 1 n 1 ], and letting m 0,t = n t and m 0,1 = n 1, P 0 is of type given in (10), and it satisfies the popeties given in (11) and (1) Fo the inductive step, suppose that fo some j, with 0 j < t, we have constucted a subspace patition P j of H j = V (n j, q) of the type given in (10), and with the popeties given in (11) and (1) We then use Lemma 1 to detemine the aveage, b avg,1, of the values b H,1 ove all hypeplanes H of H j We have (13) b avg,1 := m j,1θ n 1 j Θ n j = ( c j q t j + δ t j ) < (c j q t j + δ t j )q 1 = c j q t j 1 + q 1 δ t j It follows fom (13) that thee exists a hypeplane H j+1 of H j with (14) b Hj+1,1 b avg,1 < c j q t j 1 + q 1 δ t j ( q n 1 j ) q n j Next, we apply Lemma 11 to the subspace patition P j and the hypeplane H j+1 of H j to obtain: t 1 + b Hj+1,1 q + b Hj+1,i q i = P j (15) i=t j = n t + m j,1 = lq t + 1 + x + c j q t j + δ t j, whee 0 c j max{θ h j, 0} Simplifying (15) yields t (16) b Hj+1,1 + b Hj+1,i q i 1 = lq t 1 + c j q t j 1 + q 1 (x + δ t j ) i=t j

THE MAXIMUM SIZE OF A PARTIAL SPREAD II: UPPER BOUNDS 7 Then, it follows fom Lemma 13(ii) and (16) that (17) b Hj+1,1 q 1 (x + δ t j ) δ t j 1 (mod q t j 1 ) Since 0 q 1 δ t j < q t j 1 by Lemma 13(ii), it follows fom (14) and (17) that thee exists a nonnegative intege c j+1 such that (18) b Hj+1,1 = c j+1 q t j 1 + δ t j 1 and 0 c j+1 max{c j, 0} max{θ h j, 0} Let P j+1 be the subspace patition of H j+1 defined by: P j+1 = {W H j+1 : W P j }, and by the definition made in (18), let m j+1,1 = b Hj+1,1 Since t j > and dim(w H j+1 ) {dim W, dim W } fo each W P j, it follows that P j+1 is a subspace patition of H j+1 of type (19) [t m j+1,t, (t ) m j+1,t 1,, (t j ) m j+1,t j 1, 1 m j+1,1 ], whee m j+1,t, m j+1,t 1,, m j+1,t j 1 ae nonnegative integes such that t t (0) m j+1,i = m j,i = n t i=t j 1 The inductive step follows since P j+1 is a subspace patition of H j+1 = V (n j, q) of the type given in (19), which satisfies the conditions in (18) and (0) Thus fa, we have shown that the desied subspace patition P j of H j exists fo any intege j such that 0 j t Since q x by hypothesis, Lemma 13(iii) implies that δ t j 0 fo j [0, t ] Thus, m j,1 = c j q t j + δ t j 0 fo j [0, t ] If j [Θ h, t ], then it follows fom (1) that c j = 0, and thus, m j,1 = δ j 0 In paticula, since t Θ h +, we have c t = 0 and m t,1 = δ 0 Fo the final pat of the poof, we set j = t, and then show that the existence of the subspace patition P t of H t leads to a contadiction It follows fom the above obsevations and Lemma 13(ii) that i=t j (1) m t,1 = δ = q xq Θ xθ and 0 < δ < q Since m t 1, > 0, the smallest dimension of a subspace in P t is 1 So let s be the second smallest dimension of a subspace in P t (Note that the existence of s follows fom (11)) To deive the final contadiction, we conside the following cases Case 1: s 3 Then by applying Theoem 10(ii)&(iv) to the subspace patition P t with d = s and d 1 = 1, we obtain m t,1 min{(q s )/(q ), q s 1, q s } > q, which contadicts the fact that m t,1 < q given by (1) Case : s = Since q x by hypothesis, it follows fom (1) that q m t,1 Thus, by applying Theoem 10(iv) to P t with d = s = and d 1 = 1, we obtain m t,1 q, which contadicts the fact that m t,1 < q given by (1) We ae now eady to pove Theoem 6 and Coollay 8

8 ESMERALDA NĂSTASE AND PAPA SISSOKHO Poof of Theoem 6 Recall that () c 1 t (mod q), 0 c 1 < q, and c = Define (3) x := q (q )(t ) c 1 + c { q if q ((q )(t ) + c 1 ), 0 if q ((q )(t ) + c 1 ) Since, it follows fom () and (3) that: (a) If q ((q )(t ) + c 1 ), then c = q, and also, q (q (q )(t ) c 1 ) Thus, x q 0 (mod q ) (b) If q ((q )(t ) + c 1 ), then c = 0, and also, q (q (q )(t ) c 1 ) Thus, x = q (q )(t ) c 1 0 (mod q ) Thus, q x holds in all cases Also, since c 1 t (mod q) by (), we have t = αq + c 1 fo some nonnegative intege α Thus, it follows fom (3) that (4) x = q αq(q ) c 1 q + c Since c {0, q} by (), it follows fom (4) that q x Moeove, since 0 c 1 q and c {0, q}, we obtain (5) x = q (q )(t ) c 1 + c q (q )(t ) (q ) = x q q q + 1 q t + 1 x = q q q t + x = t Θ + q Since the hypothesis holds fom the above obsevations, Lemma 14 yields µ q (n, t) lq t + x = qn q t+ q t + q (q )(t ) c 1 + c Moeove, since q + 1 c 1 + c q, it follows that µ q (n, t) qn q t+ qn q t+ = qn q t+ which concludes the poof of Theoem 6 + q (q )(t ) c 1 + c + q (q )(t ) + q + q (q )(t 3) + 1, Poof of Coollay 8 Let f q (n, t) and g q (n, t) be as defined in the statement of the coollay Then (6) g q (n, t) = qn q t+ q t + q (q )(t ) c 1 + c,

whee c 1 and c ae as in (), and THE MAXIMUM SIZE OF A PARTIAL SPREAD II: UPPER BOUNDS 9 (7) f q (n, t) = qn q t+ q t + q ω, whee ω = 4q t (q t q ) + 1 (q t q + 1) If 1 and t, then it is staightfowad to show that (eg,see [19, Lemma ]) q q (8) ω = = Now it follows fom (6) (8) that if t, then q (9) g q (n, t) f q (n, p) = (q )(t ) c 1 + c We now pove the second pat of the coollay fo q > If Θ + 4 t Θ, then by applying (9) with 0 c 1 < q and c {0, q}, we obtain q g q (n, t) f q (n, p) (q )(t ) + q ( ) q Θ (q ) + + q q q = (q ) q + (q ) ( q q ) (q ) q + (q ) = 5/ q < 0 (since q > ) If q =, then by doing the same analysis as above with t Θ + 5 instead of t Θ + 4, we obtain g q (n, t) f q (n, p) < 0 This completes the poof of the coollay Acknowledgement: We thank the efeees fo thei detailed comments, suggestions, and coections which have geatly impoved the pape Refeences [1] J Andé, Übe nicht-desaguessche Ebenen mit tansitive Tanslationsguppe, Math Zeit 60 (1954), 156 186 [] A Beutelspache, Patial speads in finite pojective spaces and patial designs, Math Zeit 145 (1975), 11 9 [3] P Dembowski, Finite Geometies, Spinge Classics in Mathematics, 1997 [4] D Dake and J Feeman, Patial t-speads and goup constuctible (s,, µ)-nets, J Geom 13 (1979), 11 16 [5] J Eisfeld and L Stome, (Patial) t-speads and minimal t-coves in finite spaces, Lectue notes fom the Socates Intensive Couse in Finite Geomety and its Applications, Ghent, Apil 000, Published electonically at http://wwwmathsqmulacuk/ leonad/patialspeads/eisfeldstomeps [6] J Eisfeld, L Stome, and P Sziklai, On the spectum of the sizes of maximal patial line speads in P G(n, q), n 3, Des Codes Cyptog 36 (005), 101 110 [7] T Etzion, Pefect byte-coecting codes, IEEE Tans Inf Theoy 44 (1998), 3140 3146 [8] T Etzion A Vady, Eo-coecting codes in pojective space, IEEE Tans Inf Theoy 57 (1998), 1165 1173 [9] S El-Zanati, H Jodon, G Seelinge, P Sissokho, and L Spence, The maximum size of a patial 3-spead in a finite vecto space ove GF(), Des Codes Cyptog 54 (010), 101 107

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