Low-discrepancy sequences using duality and global function fields

Transcription

1 ACTA ARITHMETICA (2007) Low-dscrepancy sequences usng dualty and global functon felds by Harald Nederreter (Sngapore) and FerruhÖzbudak (Sngapore and Ankara) 1. Introducton. Let s 1 be an nteger and I s be the s-dmensonal unt cube [0, 1] s. We consder (fnte) pont sets and (nfnte) sequences of ponts n I s, where the term pont set s used n the sense of the combnatoral noton of multset, that s, a set n whch multplcty of elements s allowed and taken nto account. Constructng sequences wth good equdstrbuton propertes s an mportant problem n number theory and has applcatons to quas-monte Carlo methods n numercal analyss (see [6], [7], [8], [15]). The precse formulaton of the problem leads to the concept of star dscrepancy and the requrement of constructng low-dscrepancy sequences. A very powerful method for constructng low-dscrepancy sequences s the constructon of (t, s)-sequences usng global functon felds n [12] and [19] (see also [15, Chapter 8]). A relevant method for constructng low-dscrepancy pont sets s the constructon of (t, m, s)-nets and dgtal nets. The concept of dualty was ntroduced n [11] and used n [10] for the constructon of dgtal nets from global functon felds. We refer the reader to [8] and [9] for recent surveys on constructons of (t, m, s)-nets and (t, s)-sequences. Recently Krtzer [2] mproved the star dscrepancy bounds for (t, m, s)-nets and (t, s)-sequences. In ths paper we construct low-dscrepancy sequences usng the concept of dualty and global functon felds. For certan parameters these sequences gve asymptotcally better star dscrepancy bounds than (t, s)-sequences. An mportant role n our constructon s played by dfferentals of global functon felds. We note that a completely dfferent constructon of lowdscrepancy sequences usng dfferentals of global functon felds was recently gven n [4] Mathematcs Subject Classfcaton: 11K38, 11K45, 11R58. Key words and phrases: low-dscrepancy sequences, (t, s)-sequences, global functon felds, dfferentals. [79] c Instytut Matematyczny PAN, 2007

2 80 H. Nederreter and F. Özbudak The paper s organzed as follows. We gve some basc defntons n the remander of ths secton. Secton 2 contans some prelmnares and auxlary results. In Secton 3 we present our constructon of low-dscrepancy sequences. In Secton 4 we obtan a star dscrepancy bound for a class of sequences ncludng those constructed n Secton 3. We gve concrete examples and llustrate our mprovements by numercal results n Secton 5. Now we present some basc defntons. For a subnterval J of I s and for a pont set P of N 1 ponts x 0, x 1,..., x N 1 I s, we wrte A(J; P) for the number of ntegers n wth 0 n N 1 for whch x n J. We put A(J; P) (1.1) R(J; P) = λ s (J), N where λ s s the s-dmensonal Lebesgue measure. Defnton 1.1. The star dscrepancy DN (P) of the pont set P of N 1 elements of I s s defned by DN(P) = sup R(J; P), J where the supremum s extended over all subntervals J of I s wth one vertex at the orgn. For a sequence S of ponts n I s and N 1, the star dscrepancy DN (S) s meant to be the star dscrepancy of the frst N terms of S. Gven an nteger b 2, an nterval of the form s J = [a b d, (a + 1)b d ) I s =1 wth ntegers d 0 and 0 a < b d for 1 s s called an elementary nterval n base b. Defnton 1.2. For ntegers b 2, s 1, and 0 t m, a (t, m, s)-net n base b s a pont set P consstng of b m ponts n I s such that R(J; P) = 0 for every elementary nterval J I s n base b wth λ s (J) = b t m. 2. Prelmnares. We ntroduce some notaton whch wll be used n what follows. Let b 2 be an nteger and Z b = {0, 1,...,b 1} be the least resdue system modulo b. For a real number x [0, 1], let (2.1) x = y j b j wth all y j Z b j=1 be a b-adc expanson of x, where the case n whch y j = b 1 for all but fntely many j s allowed. Usng the expanson of x n (2.1), for an nteger

3 m 1 we defne the truncaton Low-dscrepancy sequences 81 [x] b,m = m y j b j. j=1 Note that the truncaton operates on the expanson of x and t may yeld dfferent results dependng on whch b-adc expanson of x s used. If x = (x (1),..., x (s) ) I s and the x (), 1 s, are gven by prescrbed b-adc expansons, then we defne (2.2) [x] b,m = ([x (1) ] b,m,...,[x (s) ] b,m ). The concept of a (T, s)-sequence n base b was ntroduced by Larcher and Nederreter [3]. We use a slght varant of ths concept whch, at the same tme, generalzes the verson of the defnton of a (t, s)-sequence n base b used n [12] and [15, Chapter 8]. We wrte N for the set of postve ntegers and N 0 for the set of nonnegatve ntegers. Defnton 2.1. Let b 2 and s 1 be ntegers and let T : N N 0 be a functon wth T(m) m for all m N. Then a sequence x 0, x 1,... of ponts n I s s a (T, s)-sequence n base b f for all k N 0 and m N, the ponts [x n ] b,m wth kb m n < (k + 1)b m form a (T(m), m, s)-net n base b. Remark 2.2. The orgnal defnton of a (T, s)-sequence n base b n [3] requred that for all k N 0 and m N, the ponts x n wth kb m n < (k + 1)b m form a (T(m), m, s)-net n base b. For ths earler defnton, all ponts x n need to be n the half-open unt cube [0, 1) s, whereas Defnton 2.1 allows ponts from the closed unt cube I s. The devce of truncaton n (2.2) and n Defnton 2.1 guarantees that even though all the ponts x n are n I s, all the ponts [x n ] b,m are n [0, 1) s. Note that t s a necessary condton for a (t, m, s)-net P n base b that all ponts of P be n [0, 1) s. Remark 2.3. If T s such that T(m) t for some nteger t 0 and all ntegers m > t, then Defnton 2.1 yelds the concept of a (t, s)-sequence n base b. The smaller the value of t, the better the equdstrbuton propertes of a (t, s)-sequence n base b. Next we recall the dgtal method for the constructon of sequences of ponts n I s. Ths method goes back to [5]. For our purposes, t s convenent to follow the presentaton n [15, Secton 8.2]. We fx a base b 2 and a dmenson s 1. Let R be a fnte commutatve rng wth dentty and of order b. We set up a map φ : R [0, 1] by selectng a bjecton η : R Z b and puttng φ (r 1, r 2,...) = η(r j )b j for (r 1, r 2,...) R. j=1

4 82 H. Nederreter and F. Özbudak Furthermore, we choose matrces C (1),..., C (s) over R whch are called generatng matrces. For n = 0, 1,..., let n = a j (n)b j j=0 be the dgt expanson of n n base b, where a j (n) Z b for j 0 and a j (n) = 0 for all suffcently large j. Choose a bjecton ψ : Z b R wth ψ(0) = 0 and assocate wth n the sequence n = (ψ(a 0 (n)), ψ(a 1 (n)),...) R. Now we defne the sequence x 0, x 1,... of ponts n I s by (2.3) x n = (φ (nc (1) ),...,φ (nc (s) )) for n = 0, 1,.... Note that the products nc () are well defned snce n contans only fntely many nonzero terms. For each = 1,...,s and m N, let C m () be the m m submatrx of C () obtaned from the frst m rows and columns of C (). For j = 1,...,m, let c () m,j be the jth column vector of C() m. For any d = (d 1,...,d s ) N s 0 wth d m for 1 s and d := s =1 d > 0, we defne the m d matrx (2.4) C m,d = [c (1) m,1...c(1) m,d 1...c (s) m,1...c(s) m,d s ] whose columns are obtaned from the ndcated columns of C (1) m,...,c (s) m. Proposton 2.4. The sequence (2.3) s a (T, s)-sequence n base b f and only f for any m N wth T(m) < m and any d = (d 1,...,d s ) N s 0 wth s =1 d = m T(m) the system of homogeneous lnear equatons kc m,d = 0 R m T(m) has exactly b T(m) solutons k R m, where C m,d s the matrx n (2.4). Proof. Ths s shown by the same argument as n the proof of [15, Theorem 8.2.9]. Note that we need not check the condton n Defnton 2.1 when T(m) = m snce any pont set consstng of b m ponts n [0, 1) s s an (m, m, s)-net n base b. We now consder the specal case where the rng R s the fnte feld F q of order q, wth q beng an arbtrary prme power. As above, let c () m,1,...,c() m,m denote the column vectors of the matrx C m (). For ntegers 0 < d m, we call {c () m,j Fm q : 1 j m, 1 s} a (d, m, s)-system over F q f for any (d 1,...,d s ) N s 0 wth s =1 d = d the vectors c () m,j, 1 j d, 1 s, are lnearly ndependent over F q (see [11, Defnton 3]).

5 Low-dscrepancy sequences 83 Corollary 2.5. Suppose that for any m N wth T(m) < m, {c () m,j F m q : 1 j m, 1 s} s an (m T(m), m, s)-system over F q. Then (2.3) s a (T, s)-sequence n base q. Proof. The gven hypothess guarantees that any matrx C m,d n Proposton 2.4 has rank m T(m), and so the result follows mmedately from Proposton 2.4. We need some notaton and concepts from the dualty theory developed by Nederreter and Prsc [11]. For m N and a = (a 1,...,a m ) F m q, we put v(a) = 0 f a = 0, and otherwse v(a) = max{j : a j 0}. For ntegers s 2, we extend ths defnton to F ms q by wrtng a vector A F ms q as the concatenaton of s vectors of length m,.e., A = (a (1),...,a (s) ) F ms q wth a () F m q for 1 s, and puttng s V m (A) = v(a () ). =1 Defnton 2.6. For any nonzero F q -lnear subspace N of F ms q, we defne the mnmum dstance δ m (N) = mn V m(a). A N \{0} For any F q -lnear subspace M of F ms q, we defne ts dual space M by M = {A F ms q : A M = 0 for all M M}, where denotes the standard nner product on F ms q. Note that (2.5) dm(m ) = ms dm(m), where here and subsequently we wrte dm(w) for the F q -dmenson of a fnte-dmensonal vector space W over F q. Let C (1),...,C (s) agan be the generatng matrces over F q n (2.3). For each = 1,...,s and m N, let C m () be the m m submatrx of C () defned above. Then we set up the m ms matrx (2.6) C m = [C m (1) C m (2)... C m (s) ] over F q and let C m be the row space of C m. It s trval that dm(c m ) m, and so (2.5) shows that Cm has postve dmenson whenever s 2. Therefore the mnmum dstance δ m (Cm) s defned n ths case. Proposton 2.7. Let s 2 and suppose that for any m N wth T(m) < m, the dual space C m of C m satsfes δ m (C m) m T(m) + 1.

6 84 H. Nederreter and F. Özbudak Then the sequence (2.3) wth generatng matrces C (1),...,C (s) over F q s a (T, s)-sequence n base q. Proof. For any m N wth T(m) < m, consder the system {c () m,j F m q : 1 j m, 1 s} of column vectors of the matrx C m n (2.6). In vew of [11, Theorem 1 and Defnton 3], the gven hypothess mples that {c () m,j Fm q : 1 j m, 1 s} s an (m T(m), m, s)-system over F q. The desred result now follows from Corollary A constructon from global functon felds. Throughout ths secton we assume the exstence of a global functon feld F satsfyng the followng assumpton. Assumpton 3.1. Let s 2 and g 0 be ntegers and let q be a prme power. Assume that there exsts a global functon feld F wth full constant feld F q and wth the followng propertes: () the genus of F s g; () there exst s dstnct places P 1,...,P s of F of degree 1; () there exsts a place Q of F of degree 2. Usng places of F of suffcently large degree, we can fnd a dvsor G of F of degree g 1 such that the support of G s dsjont from {Q, P 1,...,P s }. For even ntegers 2m g, let A 2m, G 2m, and G 2m+1 be the dvsors of F gven by A 2m := G mq, G 2m := G mq + 2m(P P s ), G 2m+1 := G mq + (2m + 1)(P P s ). For a dvsor A of F, let Ω(A) denote the F q -lnear subspace of the space Ω of dfferentals of F gven by Ω(A) = {ω Ω : (ω) A} {0}. We refer to the book of Stchtenoth [18] for the theory of dfferentals of global functon felds and for other background on global functon felds. Lemma 3.2. For all even ntegers 2m g, we have Ω(A 2m ) Ω(A 2m+2 ) and dm(ω(a 2m )) = 2m. Proof. As A 2m+2 A 2m, t s clear that Ω(A 2m ) Ω(A 2m+2 ). Note that deg(a 2m ) = g 1 2m < 0 snce 2m g. Then we have and the proof s complete. dm(ω(a 2m )) = deg(a 2m ) + g 1 = 2m,

7 Low-dscrepancy sequences 85 Usng Lemma 3.2, let ω 1, ω 2,... be a sequence of dfferentals of F such that for all ntegers 2m g we have ω 1,..., ω 2m = Ω(A 2m ). For a dvsor A of F, let L(A) denote the Remann Roch space L(A) = {x F : (x) A} {0}. Lemma 3.3. For all even ntegers 2m g we have the followng: () If ω Ω(A 2m ) and x L(G 2m ) are nonzero, then (xω) 2m(P P s ). () If ω Ω(A 2m ) and x L(G 2m+1 ) are nonzero, then Proof. Note that (xω) (2m + 1)(P P s ). A 2m G 2m = 2m(P P s ). For nonzero ω Ω(A 2m ) and x L(G 2m ), we have (ω) A 2m and (x) G 2m. Usng also the fact that (xω) = (x) + (ω), we complete the proof of (). The proof of () s smlar. For a dfferental δ Ω and a place P of F of degree 1, let res P (δ) F q denote the resdue of the dfferental δ at P. For = 1,...,s, let t be a local parameter of F at P. We wll construct our low-dscrepancy sequences n Theorem 3.7 below by usng the mages of F q -lnear spaces ω 1,...,ω m of dfferentals under sutable F q -lnear maps formed from resdues of some dfferentals at the places P 1,..., P s for m g + 1. If m = 2m g + 1, then ω 1,...,ω m = Ω(A 2m ) and the mage wll be the mage of Ω(A 2m ) under a sutable F q - lnear map dependng on m. If m = 2m + 1 g + 1, then ω 1,...,ω m Ω(A 2m+2 ) and the mage wll be the mage of the proper subspace ω 1,...,ω 2m+1 of Ω(A 2m+2 ) under a sutable F q -lnear map dependng on m. Now we defne these F q -lnear maps for m g+1. The defntons depend heavly on the party of m. For even ntegers 2m g+1 and for = 1,...,s, let ϕ 2m, and ϕ 2m+1, be the F q -lnear maps defned by ϕ 2m, : Ω(A 2m ) Fq 2m, ω (res P (t 1 ω), res P (t 2 ω),...,res P (t 2m ω)), and ϕ 2m+1, : Ω(A 2m+2 ) F 2m+1 q, ω (res P (t 1 ω), res P (t 2 ω),...,res P (t 2m 1 ω)).

8 86 H. Nederreter and F. Özbudak Moreover, let Φ 2m and Φ 2m+1 be the F q -lnear maps (3.1) and (3.2) Φ 2m : Ω(A 2m ) F 2ms q, Φ 2m+1 : Ω(A 2m+2 ) F (2m+1)s Furthermore, we put ω (ϕ 2m,1 (ω), ϕ 2m,2 (ω),...,ϕ 2m,s (ω)), q, ω (ϕ 2m+1,1 (ω), ϕ 2m+1,2 (ω),...,ϕ 2m+1,s (ω)). M 2m := Φ 2m (Ω(A 2m )), M 2m+1 := Φ 2m+1 ( ω 1,...,ω 2m+1 ). Lemma 3.4. For even ntegers 2m g +1, the F q -lnear maps Φ 2m and Φ 2m+1 are njectve and dm(m 2m ) = 2m, dm(m 2m+1 ) = 2m + 1. Proof. It s well known that for a dvsor A of F wth deg(a) 2g 1 we have dm(ω(a)) = 0. Moreover, for = 1,...,s and l N, f ν P ((ω)) 0 and res P (t 1 ω) = res P (t 2 ω) = = res P (t l ω) = 0, then ν P ((ω)) l. Assume that ω Ω(A 2m ) s nonzero and Φ 2m (ω) = 0 F 2ms q. Then (ω) A 2m + 2m(P P s ) = G mq + 2m(P P s ). Thus, ω Ω(A 2m +2m(P 1 + +P s )) and deg(a 2m +2m(P 1 + +P s )) = g 1 + 2m(s 1) g 1 + 2m 2g, where we have used the facts that s 2 and 2m g + 1. Hence dm(ω(a 2m + 2m(P P s ))) = 0, a contradcton. Ths shows that Φ 2m s njectve, and so dm(m 2m ) = 2m by Lemma 3.2. Smlarly, the njectvty of Φ 2m+1 follows from the observaton that deg(a 2m+2 + (2m + 1)(P P s )) = g 1 + (s 2) + 2m(s 1) 2g. It s then obvous that dm(m 2m+1 ) = 2m + 1. For even ntegers 2m g + 1, we defne further F q -lnear maps. For = 1,...,s and x L(G 2m ), let x ( 1), x ( 2),...,x ( 2m) be the elements of F q whch are the coeffcents n the local expanson x = x ( 2m) t 2m + x ( 2m+1) t 2m+1 + of x at P. Smlarly, for = 1,...,s and x L(G 2m+1 ), we defne x ( 1), x ( 2),...,x ( 2m 1) F q. Let ψ 2m, : L(G 2m ) F 2m q, x (x ( 1), x ( 2),..., x ( 2m) ), ψ 2m+1, : L(G 2m+1 ) F 2m+1 q, x (x ( 1), x ( 2),...,x ( 2m 1) ).

9 Low-dscrepancy sequences 87 Moreover, let Ψ 2m and Ψ 2m+1 be the F q -lnear maps and Ψ 2m : L(G 2m ) F 2ms q, x (ψ 2m,1 (x), ψ 2m,2 (x),...,ψ 2m,s (x)), Ψ 2m+1 : L(G 2m+1 ) F (2m+1)s q, x (ψ 2m+1,1 (x), ψ 2m+1,2 (x),...,ψ 2m+1,s (x)). Furthermore, we put N 2m := Ψ 2m (L(G 2m )), N 2m+1 := Ψ 2m+1 (L(G 2m+1 )). Lemma 3.5. For even ntegers 2m g +1, the F q -lnear maps Ψ 2m and Ψ 2m+1 are njectve and dm(n 2m ) = 2ms 2m, dm(n 2m+1 ) = (2m + 1)s 2m. Proof. Assume that x L(G 2m ) and Ψ 2m (x) = 0 Fq 2ms. Then ν P (x) 0 for 1 s and hence x L(G 2m 2m(P P s )). Note that deg(g 2m 2m(P P s )) = g 1 2m < 0 as 2m g + 1. Hence x = 0 and Ψ 2m s njectve. We also have deg(g 2m ) = g 1 2m + 2ms > 2g 1 as 2m g + 1 and s 2. Therefore by the Remann Roch theorem, dm(n 2m ) = dm(l(g 2m )) = deg(g 2m ) + 1 g = 2ms 2m. Next assume that x L(G 2m+1 ) and Ψ 2m+1 (x) = 0 F (2m+1)s q. Smlarly, we have x L(G 2m+1 (2m + 1)(P P s )) and hence x = 0 and Ψ 2m+1 s njectve. Also deg(g 2m+1 ) = g 1 2m+(2m+1)s s+2g 1 > 2g 1 and then dm(n 2m+1 ) = dm(l(g 2m+1 )) = deg(g 2m+1 ) + 1 g = (2m + 1)s 2m. Proposton 3.6. For even ntegers 2m g + 1 we have: () M 2m = N 2m. () M 2m+1 N 2m+1. Proof. Frst we prove (). We wll show that for ω Ω(A 2m ) and x L(G 2m ), we have Φ 2m (ω) Ψ 2m (x) = 0, where the nner product s the standard nner product on Fq 2ms. Ths mples that M 2m N 2m n Fq 2ms. Moreover, by Lemmas 3.4 and 3.5, dm(m 2m )+dm(n 2m ) = 2ms and hence we get M 2m = N 2m by (2.5). Now we prove that for ω Ω(A 2m ) and x L(G 2m ), we have Φ 2m (ω) Ψ 2m (x) = 0. For = 1,..., s, the local expanson of x L(G 2m ) at P s (3.3) x = x ( 2m) t 2m + x ( 2m+1) t 2m x ( 1) t 1 + y, where ν P (y ) 0. For ω Ω(A 2m ), usng the F q -lnearty of the resdue map res P we get res P (xω) = x ( 2m) res P (t 2m ω) + + x ( 1) res P (t ( 1) ω) + res P (y ω).

10 88 H. Nederreter and F. Özbudak As ν P (y ) 0 and ν P (ω) ν P (A 2m ) = 0, we have res P (y ω) = 0 and hence (3.4) res P (xω) = ϕ 2m, (ω) ψ 2m, (x), where the nner product s the standard nner product on Fq 2m. Usng the Resdue Theorem (cf. [1, Secton III.5, Theorems 2 and 3]), Lemma 3.3(), and (3.4), we obtan 0 = s res P (xω) = ϕ 2m, (ω) ψ 2m, (x) = Φ 2m (ω) Ψ 2m (x), P =1 where the frst sum s over all places P of F. Ths fnshes the proof of (). Now we consder (). Let W be the F q -lnear subspace of F q (2m+1)s gven by W = Φ 2m+1 (Ω(A 2m )). By Lemmas 3.2 and 3.4, we have dm(w) = 2m and M 2m+1 W. It suffces to prove that W = N 2m+1. Indeed, ths mples that M 2m+1 W = N 2m+1. Usng Lemma 3.5, we deduce that dm(w) + dm(n 2m+1 ) = 2m + (2m + 1)s 2m = (2m + 1)s. Therefore t remans to show that f ω Ω(A 2m ) and x L(G 2m+1 ), then Φ 2m+1 (ω) Ψ 2m+1 (x) = 0, where the nner product s the standard nner product on F (2m+1)s q. We follow smlar arguments to those n the proof of (). For = 1,..., s, for x L(G 2m+1 ) and ω Ω(A 2m ), usng the local expanson of x at P, we obtan (3.5) res P (xω) = ϕ 2m+1, (ω) ψ 2m+1, (x), where the nner product s the standard nner product on Fq 2m+1. Note that n the local expanson of x L(G 2m+1 ) at P, we have the extra term x ( 2m 1) t 2m 1, n addton to the terms n (3.3). Then, smlarly to the case (), usng the Resdue Theorem, Lemma 3.3(), and (3.5), we complete the proof of (). For an nteger m g +1, let C m be the m ms matrx over F q gven by Φ m (ω 1 ) (3.6) Φ C m = m (ω 2 ).. Φ m (ω m ) Note that for an nteger m g + 1, f m s even (resp. odd), then Φ m s defned by (3.1) (resp. (3.2)). Let C m (1), C m (2),...,C m (s) be the m m matrces over F q defned by (3.7) C m = [C (1) m C (2) m... C (s) m ].

11 Low-dscrepancy sequences 89 We observe that for each = 1,...,s, C m () s the m m submatrx of the (m + 1) (m + 1) matrx C () m+1 formed from the frst m rows and columns of C () m+1. Hence, for each = 1,...,s, we can buld an matrx C() over F q such that for any nteger m g + 1 the m m submatrx of C () formed from the frst m rows and columns of C () s equal to C () m. Our constructon of low-dscrepancy sequences now proceeds by the dgtal method descrbed n Secton 2. We use the matrces C (1),...,C (s) over F q defned n the prevous paragraph as the generatng matrces n (2.3). The resultng sequence s a (T, s)-sequence n base q n the sense of Defnton 2.1, wth the functon T : N N 0 gven n the followng theorem. Theorem 3.7. Under Assumpton 3.1, let S be the sequence of ponts n I s whch s constructed n (2.3) usng the generatng matrces C (1),...,C (s) over F q defned after (3.7). Then S s a (T, s)-sequence n base q wth T(m) = m for 1 m g, T(m) = g for even m g +1, and T(m) = g +1 for odd m g + 1. Proof. We proceed by Proposton 2.7. Frst let m = 2m be even. We can assume that m = 2m g + 1. Then by constructon, the row space C m of the matrx C m n (3.6) s gven by C m = M 2m. Hence t follows from Proposton 3.6 that C m = N 2m. Now we apply [10, Theorem 3.1] wth N = C m (P 1,...,P s ; G 2m ) n the notaton of that theorem and we observe that N = N 2m. Ths yelds, agan n the notaton of [10, Theorem 3.1], δ m (C m) = δ m (N 2m ) δ m(1,...,1; ms m + g 1). Next we use [10, Lemma 2.1] to obtan and so δ m(1,...,1; ms m + g 1) m g + 1, δ m (C m) m g + 1. Now let m = 2m + 1 be odd. We can assume that m = 2m + 1 g + 2. So we have C m = M 2m+1, and hence Proposton 3.6 yelds C m N 2m+1. We apply [10, Theorem 3.1] wth N = C m (P 1,...,P s ; G 2m+1 ) = N 2m+1 and obtan δ m (C m) δ m (N 2m+1 ) δ m(1,...,1; ms m + g). By [10, Lemma 2.1] we get and so δ m(1,...,1; ms m + g) m g, δ m (C m) m (g + 1) + 1. Thus, the theorem s proved n all cases.

12 90 H. Nederreter and F. Özbudak Remark 3.8. Prevous constructons of low-dscrepancy sequences usng global functon felds over F q led to (t, s)-sequences n base q (see Remark 2.3). For fxed q and s 2, the best prevous constructons of ths type usng a global functon feld F satsfyng Assumpton 3.1 yeld (t, s)- sequences n base q wth t = g + 1 (see [4], [15, Theorem 8.4.1], [19]). Theorem 3.7 mproves on these constructons under Assumpton 3.1. Ths mprovement s also reflected n better bounds on the star dscrepancy of the new sequences, as wll be shown n Secton 4. There are combnatons of values of q and s for whch the global functon felds satsfyng Assumpton 3.1 have gven the best prevous constructons of (t, s)-sequences n base q, for nstance when s = q + 1. Examples of ths type wll be presented n Secton 5. Remark 3.9. Our constructon of low-dscrepancy sequences starts from sequences of certan F q -lnear spaces of dfferentals of F. In order to construct such low-dscrepancy sequences, t s possble to use a dual approach startng from sequences of certan Remann Roch spaces of F. Snce we start from dfferentals of F, n the proof of Theorem 3.7 we can estmate the T-parameters of the low-dscrepancy sequences by usng results of [10], whch would not have been possble n a dual approach. Thus, the essental ponts of our approach are usng the Resdue Theorem and reducng the estmaton of T-parameters to the results of [10]. 4. Bounds on the star dscrepancy. In ths secton we obtan bounds on the star dscrepancy of a class of sequences of ponts n I s, ncludng those constructed n Theorem 3.7. Ths wll mply that the sequences n Theorem 3.7 have asymptotcally better bounds on the star dscrepancy than (t, s)-sequences for certan parameters. We wll also llustrate our mprovements by some concrete examples n Secton 5. For ntegers b 2, m 1, 0 t m, and s 2, let b (t, m, s) be a number for whch b m D b m(p) b(t, m, s) holds for any (t, m, s)-net P n base b. We quote the followng result n [2, Corollary 4] n a smplfed form. Proposton 4.1. If b s even, then we can take b (t, m, s) = and f b s odd, then we can take b t+s (b + 1)2 s (s 1)! ms 1 + O(b t m s 2 ), b (t, m, s) = bt (b 1) s 1 2 s (s 1)! m s 1 + O(b t m s 2 ).

13 Low-dscrepancy sequences 91 In both cases, the mpled constants n the Landau symbols depend only on b and s. The followng lemma allows us to use a star dscrepancy bound for the orgnal concept of (T, s)-sequences n base b (see Remark 2.2) just as well for the concept of (T, s)-sequences n base b ntroduced n ths paper (see Defnton 2.1). Lemma 4.2. Let P be the pont set consstng of the ponts y n, n = 0, 1,...,b m 1, n I s. Suppose that the ponts [y n ] b,m, n = 0, 1,...,b m 1, form a (t, m, s)-net n base b. Then b m D b m(p) b(t, m, s). Proof. For n = 0, 1,...,b m 1, we can wrte y n = [y n ] b,m + z n wth z n [0, b m ] s. Let 0 < ε 1 be gven and let P(ε) be the pont set consstng of y n (ε) = [y n ] b,m + (1 ε)z n, n = 0, 1,...,b m 1. By Defnton 1.2 and the assumpton that the ponts [y n ] b,m, n = 0, 1,..., b m 1, form a (t, m, s)-net n base b, t s clear that P(ε) s a (t, m, s)-net n base b. Therefore b m D b m(p(ε)) b(t, m, s). Furthermore, for each n = 0, 1,...,b m 1, correspondng coordnates of y n and y n (ε) dffer by at most b m ε. Therefore, by a well-known prncple (see e.g. [6, Lemma 2.5] for the one-dmensonal case, whch can be mmedately extended to the multdmensonal case), and so b m Db m(p) bm Db m(p(ε)) sε, b m D b m(p) b(t, m, s) + sε. Lettng ε 0+, we get the desred result. Theorem 4.3. Let s 2, b 2, and t 0 be ntegers. Assume that S s a (T, s)-sequence n base b wth T(m) = m for 1 m t, T(m) = t for even m t + 1, and T(m) = t + 1 for odd m t + 1. Then for N 2, the star dscrepancy DN (S) of the frst N terms of S satsfes ( DN(S) (log N)s (log N) s 1 B s (b, t) + O N N where the mpled constant n the Landau symbol does not depend on N. ),

14 92 H. Nederreter and F. Özbudak Here (b 1)b t+s 2 B s (b, t) = s+2 s!(log b) s f b s even, (b 1) s (b + 1)b t 2 s+2 s!(log b) s f b s odd. Proof. For a gven N 2, let k N 0 be such that b k N < b k+1 and let r N 0 be maxmal such that b r dvdes N. Note that r k. In vew of Lemma 4.2, we can apply [3, Lemma 2]. Puttng T(0) = 0 and b (0, 0, s) = 1, ths yelds ND N(S) b 1 2 k b (T(m), m, s) m=r b(t(r), r, s) b(t(k + 1), k + 1, s). Now we use the values of b (t, m, s) n Proposton 4.1. The case k = 0 s trval, and so we can assume k 1. Then we obtan NDN(S) b 1 k b (T(m), m, s) + O(b t k s 1 ), 2 m=1 where the mpled constant n the Landau symbol depends only on b and s. If b s even, then we get NDN(S) (b 1)b t+s k (b + 1)2 s+1 m s 1 (s 1)! + (b 1)bt+s+1 (b + 1)2 s+1 (s 1)! m=1 m even k m=1 modd m s 1 + O(b t k s 1 ) (b 1)b t+s (b + 1)2 s+1 (s 1)! ks 2s + (b 1)bt+s+1 (b + 1)2 s+1 (s 1)! ks 2s + O(bt k s 1 ) (b 1)bt+s = 2 s+2 k s + O(b t k s 1 ). s! If b s odd, then we smlarly get NDN(S) (b 1)s (b + 1)b t 2 s+2 k s + O(b t k s 1 ). s! Usng k (log N)/(log b), we arrve at the desred result. Usng Theorems 3.7 and 4.3, we obtan the followng corollary. Corollary 4.4. Let s 2 be an nteger and q be a prme power. Suppose that there exsts a global functon feld F of genus g satsfyng Assump-

15 Low-dscrepancy sequences 93 ton 3.1. Let S be the (T, s)-sequence n base q constructed n Theorem 3.7 usng the global functon feld F. Then, for N 2, the star dscrepancy DN (S) of the frst N terms of S satsfes ( ) DN(S) (log N)s (log N) s 1 B s (q, g) + O, N N where the mpled constant n the Landau symbol does not depend on N. Here (q 1)q g+s 2 B s (q, g) = s+2 s!(log q) s f q s even, (q 1) s (q + 1)q g 2 s+2 s!(log q) s f q s odd. Remark 4.5. Accordng to the currently best bound (see [2, Corollary 11]), the star dscrepancy DN (S) of the frst N 2 terms of a (t, s)- sequence S n base b satsfes ( ) DN(S) (log N)s (log N) s 1 C s (b, t) + O, N N where the mpled constant n the Landau symbol does not depend on N and where (b 1)b t+s (b + 1)2 C s (b, t) = s+1 s!(log b) s f b s even, (b 1) s b t 2 s+1 s!(log b) s f b s odd. 5. Examples. In ths secton we gve some concrete examples and we llustrate our mprovements by numercal results. Frst we gve some examples of global functon felds satsfyng Assumpton 3.1. For d = 1, 2, we wrte N d (F) for the number of places of F of degree d. Example 5.1. Let q be any prme power, g = 0, s = q + 1, and F = F q (x) be the ratonal functon feld over F q. Then F s a functon feld wth full constant feld F q and the genus of F s 0. Moreover, N 1 (F) = q + 1 and N 2 (F) = (q 2 q)/2. Therefore F satsfes Assumpton 3.1. By Corollary 4.4, we obtan the coeffcent B q+1 (q, 0) of the leadng term n the star dscrepancy bound. On the other hand, for s = q + 1 the smallest possble t-value of a (t, s)-sequence n base q s t = 1 (see [6, Corollary 4.24] and Remark 3.8). By Remark 4.5, ths yelds the coeffcent C q+1 (q, 1) of the leadng term n the star dscrepancy bound. It s now easly seen that B q+1 (q, 0) < C q+1 (q, 1) for any prme power q. Thus, for any prme power q and s = q + 1, we always get an asymptotc mprovement on the prevously best star dscrepancy bound for a (T, s)-sequence n base q by usng the constructon n Theorem 3.7.

16 94 H. Nederreter and F. Özbudak Example 5.2. Let q = 3, g = 2, s = 8, and F = F 3 (x, y) wth y 2 = x 6 x (cf. [13, Example 3.2] and [15, Table 4.2.1, F.13]). Then F s a global functon feld wth full constant feld F 3 and the genus of F s 2. Moreover, N 1 (F) = 8 and N 2 (F) = 2. Therefore F satsfes Assumpton 3.1. By Corollary 4.4, we obtan the coeffcent B 8 (3, 2) of the leadng term n the star dscrepancy bound. On the other hand, the smallest known t-value of a (t, 8)-sequence n base 3 s t = 3 (see [8, Table 1] and [16]). By Remark 4.5, ths yelds the coeffcent C 8 (3, 3) of the leadng term n the star dscrepancy bound. We have B 8 (3, 2) < C 8 (3, 3). Example 5.3. Let q = 3, g = 4, s = 12, and F = F 3 (x, y) wth y 3 y = x3 x (x 2 + 1) 2 (cf. [13, Example 3.4]). Then F s a global functon feld wth full constant feld F 3 such that the genus of F s 4 and N 1 (F) = 12. Moreover, N 2 (F) 1 snce x 2 +1 s totally ramfed n the extenson F/F 3 (x). Therefore F satsfes Assumpton 3.1. By Corollary 4.4, we obtan the coeffcent B 12 (3, 4) of the leadng term n the star dscrepancy bound. On the other hand, the smallest known t-value of a (t, 12)-sequence n base 3 s t = 5 (see [8, Table 1] and [16]). By Remark 4.5, ths yelds the coeffcent C 12 (3, 5) of the leadng term n the star dscrepancy bound. We have B 12 (3, 4) < C 12 (3, 5). Example 5.4. Let q = 5, g = 1, s = 10, and F = F 5 (x, y) wth y 2 = 3(x 4 + 2) (cf. [13, Example 5.1]). Then F s a global functon feld wth full constant feld F 5 such that the genus of F s 1 and N 1 (F) = 10. Moreover, N 2 (F) 1 snce there s a place of F of degree 2 lyng over the nfnte place of the ratonal functon feld F 5 (x). Therefore F satsfes Assumpton 3.1. By Corollary 4.4, we obtan the coeffcent B 10 (5, 1) of the leadng term n the star dscrepancy bound. On the other hand, the smallest known t-value of a (t, 10)-sequence n base 5 s t = 2 (see [8, Table 1] and [16]). By Remark 4.5, ths yelds the coeffcent C 10 (5, 2) of the leadng term n the star dscrepancy bound. We have B 10 (5, 1) < C 10 (5, 2). Example 5.5. Let q = 8, g = 3, s = 24. Then t s shown n [14, Example 4.2] that there exsts a global functon feld F wth full constant feld F 8 such that the genus of F s 3 and N 1 (F) = 24. Moreover, N 2 (F) 1 snce t s noted n [14, Example 4.2] that x 2 +x+1 s totally ramfed n the extenson F/F 8 (x). Therefore F satsfes Assumpton 3.1. By Corollary 4.4, we obtan the coeffcent B 24 (8, 3) of the leadng term n the star dscrepancy bound. On the other hand, the smallest known t-value of a (t, 24)-sequence n base 8

17 Low-dscrepancy sequences 95 s t = 4 accordng to [16]. By Remark 4.5, ths yelds the coeffcent C 24 (8, 4) of the leadng term n the star dscrepancy bound. We have B 24 (8, 3) < C 24 (8, 4). Example 5.6. Let q = 8, g = 7, s = 34, and F = F 8 (x, y 1, y 2 ) wth y1 2 + y 1 = 1 x + w(x + w3 ) x 2 + w 5 x + w, y2 2 + y 2 = 1 x + w2 (x + w 6 ) x 2 + w 3 x + w 2, where w F 8 wth w 3 +w+1 = 0 (cf. [17]). Then F s a global functon feld wth full constant feld F 8 and the genus of F s 7. Moreover, N 1 (F) = 34 and N 2 (F) = 14. Therefore F satsfes Assumpton 3.1. By Corollary 4.4, we obtan the coeffcent B 34 (8, 7) of the leadng term n the star dscrepancy bound. On the other hand, the smallest known t-value of a (t, 34)-sequence n base 8 s t = 8 accordng to [16]. By Remark 4.5, ths yelds the coeffcent C 34 (8, 8) of the leadng term n the star dscrepancy bound. We have B 34 (8, 7) < C 34 (8, 8). Example 5.7. Let q = 9, g = 5, s = 32, and F = F 9 (x, y 1, y 2, y 3 ) wth y 2 1 = x(x + w), y 2 2 = (x + 1)(x + w 3 ), y 2 3 = (x + w 6 )(x + w 7 ), where w F 9 wth w 2 +2w+2 = 0 (cf. [17]). Then F s a global functon feld wth full constant feld F 9 and the genus of F s 5. Moreover, N 1 (F) = 32 and N 2 (F) = 12. Therefore F satsfes Assumpton 3.1. By Corollary 4.4, we obtan the coeffcent B 32 (9, 5) of the leadng term n the star dscrepancy bound. On the other hand, the smallest known t-value of a (t, 32)-sequence n base 9 s t = 6 accordng to [16]. By Remark 4.5, ths yelds the coeffcent C 32 (9, 6) of the leadng term n the star dscrepancy bound. We have B 32 (9, 5) < C 32 (9, 6). Table 1. Numercal comparson of our mprovements for some values s q C s (q) B s (q)

18 96 H. Nederreter and F. Özbudak For a prme power q and an nteger s 2, let C s (q) = C s (q, t 0 ) wth the smallest currently known t-value t 0 of a (t, s)-sequence n base q accordng to [16]. Let B s (q) = B s (q, g) wth g as n the examples above. In Table 1, we llustrate our mprovements by comparng C s (q) and B s (q) numercally usng Examples Acknowledgments. Ths research was supported by the DSTA grant R wth Temasek Laboratores n Sngapore. The second author would lke to express hs thanks to Temasek Laboratores and the Department of Mathematcs at the Natonal Unversty of Sngapore for the hosptalty. References [1] C. Chevalley, Introducton to the Theory of Algebrac Functons of One Varable, Amer. Math. Soc., Provdence, RI, [2] P. Krtzer, Improved upper bounds on the star dscrepancy of (t, m, s)-nets and (t, s)- sequences, J. Complexty 22 (2006), [3] G. Larcher and H. Nederreter, Generalzed (t, s)-sequences, Kronecker-type sequences, and dophantne approxmatons of formal Laurent seres, Trans. Amer. Math. Soc. 347 (1995), [4] D. J. S. Mayor and H. Nederreter, A new constructon of (t, s)-sequences and some mproved bounds on ther qualty parameter, Acta Arth. 128 (2007), [5] H. Nederreter, Pont sets and sequences wth small dscrepancy, Monatsh. Math. 104 (1987), [6], Random Number Generaton and Quas-Monte Carlo Methods, SIAM, Phladelpha, [7], Hgh-dmensonal numercal ntegraton, n: Appled Mathematcs Enterng the 21st Century: Invted Talks from the ICIAM 2003 Congress, J. M. Hll and R. Moore (eds.), SIAM, Phladelpha, 2004, [8], Constructons of (t, m, s)-nets and (t, s)-sequences, Fnte Felds Appl. 11 (2005), [9], Nets, (t, s)-sequences, and codes, n: Monte Carlo and Quas-Monte Carlo Methods 2006, A. Keller, S. Henrch, and H. Nederreter (eds.), Sprnger, Berln, to appear. [10] H. Nederreter and F. Özbudak, Constructons of dgtal nets usng global functon felds, Acta Arth. 105 (2002), [11] H. Nederreter and G. Prsc, Dualty for dgtal nets and ts applcatons, bd. 97 (2001), [12] H. Nederreter and C. P. Xng, Low-dscrepancy sequences and global functon felds wth many ratonal places, Fnte Felds Appl. 2 (1996), [13],, Cyclotomc functon felds, Hlbert class felds, and global functon felds wth many ratonal places, Acta Arth. 79 (1997), [14],, Algebrac curves wth many ratonal ponts over fnte felds of characterstc 2, n: Number Theory n Progress, K. Győry, H. Iwanec, and J. Urbanowcz (eds.), de Gruyter, Berln, 1999,

19 Low-dscrepancy sequences 97 [15] H. Nederreter and C. P. Xng, Ratonal Ponts on Curves over Fnte Felds: Theory and Applcatons, Cambrdge Unv. Press, Cambrdge, [16] R. Schürer and W. Ch. Schmd, MnT: A database for optmal net parameters, n: Monte Carlo and Quas-Monte Carlo Methods 2004, H. Nederreter and D. Talay (eds.), Sprnger, Berln, 2006, ; updated onlne at [17] S. Sémrat, 2-extensons wth many ponts, arxv:math.nt/ v1. [18] H. Stchtenoth, Algebrac Functon Felds and Codes, Sprnger, Berln, [19] C. P. Xng and H. Nederreter, A constructon of low-dscrepancy sequences usng global functon felds, Acta Arth. 73 (1995), Department of Mathematcs Natonal Unversty of Sngapore 2 Scence Drve 2 Sngapore , Republc of Sngapore E-mal: ned@math.nus.edu.sg Temasek Laboratores Natonal Unversty of Sngapore 5 Sports Drve 2 Sngapore , Republc of Sngapore and Department of Mathematcs Mddle East Techncal Unversty Ankara 06531, Turkey E-mal: ozbudak@metu.edu.tr Receved on and n revsed form on (5415)