MinT: A Database for Optimal Net Parameters Rudolf Sch¨ urer1 and Wolfgang Ch. Schmid2 Department of Mathematics, University of Salzburg Hellbrunnerstr. 34, A-5020 Salzburg, Austria Email:1
[email protected] Email:2
[email protected] Summary. An overwhelming variety of different constructions for (t, m, s)-nets and (t, s)-sequences are known today. Propagation rules as well as connections to other mathematical objects make it a difficult task to determine the best net available in a given setting. We present the web-based database system MinT for querying best known (t, m, s)-net and (t, s)-sequence parameters. This new system provides a number of hitherto unavailable services to the research community.
1 Introduction (t, m, s)-nets and (t, s)-sequences [8, 9] are among the best known methods for the construction of low-discrepancy point sets in the s-dimensional unit cube. A problem for the practitioner is that an overwhelming variety of different methods exist. A recent survey of important approaches can be found in [10]. Choosing an optimal net is further complicated by the fact that the existence of nets and sequences is often linked to other mathematical objects, e.g. algebraic function fields (see [12]), linear codes (see [5, 1]), or even other nets. Connections of the last type are usually referred to as “propagation rules”, and a large number of such rules are available. Hence it has become an almost impossible task to determine the best net available in a given setting. This problem led to the publication of tables of net parameters, with [3] and its predecessor [6] being the best-known examples. However, parts of these tables had been outdated before the articles appeared in print. As a more flexible solution the web-based database system MinT for querying best known (t, m, s)-net and (t, s)-sequence parameters has been developed at the Department of Mathematics at the University of Salzburg. It is available on the Internet at the address http://mint.sbg.ac.at/
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MinT is an acronym for “Minimal t”, referring to the common task of finding a (t, m, s)-net with minimal t for given m and s. This new system provides a number of hitherto unavailable services to the scientific community. The following list gives a short overview of its advantages compared with any printed version of such tables: • MinT allows distinction between digital and general constructions • MinT allows distinction between constructive and non-constructive methods • MinT gives bounds on existence • MinT allows different views on the data by appropriately choosing dependent and independent parameters • MinT shows complete construction trees • MinT allows comparing different types • MinT has a flexible viewport • MinT gives extensive literature references • MinT allows fast and dynamic updates • MinT is available to everybody In this article we discuss the unique features and design issues of MinT.
2 Basic Definitions and Results Throughout this article the following notation is used: the integer s ≥ 1 denotes the dimension of the s-dimensional half-open unit cube [0, 1)s , the prime power b is the base of a construction, the integer m ≥ 0 parameterizes the size bm of an object, and the integer t with 0 ≤ t ≤ m is called quality parameter. Finally, the integer k is defined as k := m−t and is called strength. 2.1 Nets and Sequences Following [8] and [9], we have the following definitions: Definition 1. A multi-set of bm points x0 , . . . , xbm −1 ∈ [0, 1)s is a (t, m, s)net in base b if any elementary interval s Y ai ai + 1 [ di , di ) b b i=1
with ai , di ∈ , di ≥ 0, 0 ≤ ai < bdi for i = 1, . . . , s, and volume 1/bk ) contains exactly bt points of the multi-set.
Ps
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di = k (i.e.,
The primary goal of MinT is to determine for which quadruple (b, t, m, s) a (t, m, s)-net in base b can exist and for which it cannot. Closely related to nets are digital nets1 : 1
This definition is different but equivalent to the one in [8] and [9].
MinT: A Database for Optimal Net Parameters
Definition 2. A set of s m × m matrices C(1), . . . , C(s) over (i) c1 (i) c2 C(i) = for i = 1, . . . , s .. . (i) cm
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generates a digital (t, m, s)-net over b if for all 0 ≤ di ≤ m, i = 1, . . . , s and Ps (1) (1) (s) (s) m b are i=1 di = m − t the m − t vectors c1 , . . . , cd1 , . . . , c1 , . . . , cds ∈ linearly independent. The matrices C(1), . . . , C(s) are called generator matrices of the digital net.
In [8, Theorem 6.10] it is shown that every digital (t, m, s)-net over b is a (t, m, s)-net in base b. Therefore digital nets are a special subclass of nets. An important tool for the construction of (digital) nets are (digital) sequences, which can be thought of as an infinite nesting of (digital) nets with increasing size m. The definitions given here are equivalent to the one in [15] for sequences and to the one in [11] for digital sequences, which are more general than the definitions in [8] and [9].
Definition 3. A sequence of points x0 , x1 , . . . ∈ [0, 1]s with a fixed b-adic expansion is a (t, s)-sequence in base b if, for all integers j ≥ 0 and m ≥ t, the point set {[xn ]b,m : jbm ≤ n < (j + 1)bm } is a (t, m, s)-net in base b, where [x]b,m denotes the coordinate-wise m-digit truncation of x in base b.
Definition 4. A set of s ∞ × ∞ matrices C(1) , . . . , C(s) over b generates a digital (t, s)-sequence over b if their s upper left m × m submatrices generate a digital (t, m, s)-net over b for all m ≥ t.
In [8, Theorem 5.15] it is shown that every (t, s)-sequence in base b yields (t, m, s + 1)-nets in base b for all m ≥ t. From [11, Lemma 1 and 2] it follows that every digital (t, s)-sequence over b yields digital (t, m, s + 1)-nets over b for all m ≥ t, and that every digital (t, s)-sequence is also a (t, s)-sequence.
2.2 Linear Codes and Orthogonal Arrays Nets and digital nets are closely related to orthogonal arrays and linear codes. These connections are summarized by the following definitions and results. Definition 5. A set S ⊆ m b of cardinality s is an (s, k)-set in selection of k vectors from S is linearly independent.
m b
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Definition 6. A subspace C ⊆ sb with dimension s − m is a linear [s, s − m, k + 1]-code if the Hamming distance between any two different vectors in C is at least k + 1.
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An (s, k)-set in m b exists if and only if a linear [s, s − m, k + 1]-code over b exists, because the (s, k)-set forms the parity check matrix of the code [2]. For k ≤ s and by choosing di ≤ 1 it follows directly from the definition of (1) (s) digital (m − k, m, s)-nets over b that the first row vectors c1 , . . . , c1 of m the s generator matrices form an (s, k)-set in b . Therefore, the generator matrices of every digital (m − k, m, s)-net over b (with s ≥ m) yield a linear [s, s − m, k + 1]-code over b . Linear codes are again a special case of an even more general structure, namely orthogonal arrays:
Definition 7. An array with s columns and elements from a set with cardinality b is an orthogonal array OAλ (k, s, b) if in the projection onto any set of k columns each k-tuple of entries occurs exactly λ times. Let H be the parity check matrix of a linear [s, s − m, k + 1]-code over i.e., an s × m matrix with its rows forming an (s, k)-set in m b . Then it is easy to see that the bm × s matrix produced by listing all vectors of the subspace spanned by the columns of H is an OAbt (k, s, b). Therefore, (s, k)sets and linear codes can be identified with a special class of (namely linear) orthogonal arrays. In addition to that, for every (t, m, s)-net in base b with points x n = (1) (s) (xn , . . . , xn ) for n = 0, . . . , bm − 1, the bm × s matrix A = (ani ) defined by j k ani = bx(i) for n = 0, . . . , bm − 1 and i = 1, . . . , s n
b,
is an orthogonal array OAbt (k, s, b), i.e., the OA is constructed using the leading digit of the b-adic expansions of the coordinates of the points [13]. 2.3 Ordered Orthogonal Arrays
The connection between nets, digital nets, linear codes, and orthogonal arrays is even more explicit in the terminology of ordered orthogonal arrays (OOA), introduced in [4] and [7]. In this setting, nets and digital nets are OOAs with depth T = k whereas linear codes and OAs have depth T = 1. Digital nets and linear codes are linear OOAs whereas nets and OAs do not necessarily have vector space structure. Figure 1 sums up the dependencies between these four classes of objects. The existence of a digital net implies the existence of a net as well as the existence of a linear code. Each of them implies the existence of an orthogonal array. To establish bounds on the existence of these objects, implications run in the opposite direction using proofs by contradiction.
MinT: A Database for Optimal Net Parameters
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Digital (t, m, s)-Net Linear OOA with T = m – t
(t, m, s)-Net OOA with T = m – t
Linear [s, s – m, d]-Code Linear Orthogonal Array Linear OOA with T = 1
Orthogonal Array OOA with T = 1 Fig. 1. The four classes of objects tracked by MinT
3 Constructions, Existence, and Bounds For all four classes of mathematical objects introduced in the previous section MinT tracks three different sets of parameters, leading to a total of 12 sets of parameters: • Constructions are explicit and effective methods for creating the object, i.e., the generator matrices for digital nets and linear codes, the point sets for nets, and the runs of an orthogonal array. A method is considered constructive if a computer implementation or at least an algorithm that allows such an implementation is available. If generator matrices are available explicitly (tabulated in print or electronically), the method is also considered constructive. • Existence Results provide proof of the existence of an object with certain parameters without giving an explicit method for its construction. A brute force search through the finite space of possible matrices for given parameters is not considered an effective method. Of course, a construction implies an existence result. • Bounds are proofs ruling out the existence of objects with certain parameters.
4 Sets of Possible Parameters and Their Defining Functions ∗ We are interested in Htype for type ∈ {net, dignet, code, OA}, the set of all (b, t, m, s) such that a type-object with these parameters exists. Even though ∗ Htype is mathematically well defined, there is no efficient routine known for ∗ ∗ determining whether a given quadruple (b, t, m, s) is in Htype . Therefore, Htype has to be bounded by more concrete sets.
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mode Let Htype for mode ∈ {constructive, existent, potential} and type as above be the set of all quadruples (b, t, m, s) ∈ × 0 × ( 0 ∪ {seq}) × with b a prime power and t ≤ m, such that an object of type type can “exist”, where the mode of existence is further specified by mode in the following way: potential Htype is defined as the set of all (b, t, m, s) such that no bound rules existent out the existence of a type-object with these parameters; Htype is the set of constructive all parameters for which a type-object is known to exist; finally, Htype is the set of all (b, t, m, s) such that a type-object with these parameters can be constructed explicitly. (Digital) (t, s)-sequences in base b can be represented in this framework by the tuple (b, t, seq, s). This notation allows a consistent handling of (digital) sequences as special cases of (digital) nets without introducing additional types. Obviously, we have certain relations between these 16 sets. To be specific, mode mode mode mode mode mode we have Hdignet ⊆ Hnet ⊆ HOA and Hdignet ⊆ Hcode ⊆ HOA for all potential constructive existent ∗ modes, and Htype ⊆ Htype ⊆ Htype ⊆ Htype for all types. mode Providing information about the shape of Htype is the main goal of MinT.
4.1 Projections of Hmode type mode Each H = Htype has a very regular structure. For instance, if (b, t, m, s) is in 0 H, so is (b, t , m, s) for t ≤ t0 ≤ m. For all four types this result follows directly from the definition of the particular object. Therefore, the exact shape of H is determined completely by a function yielding the minimum t for given b, m, and s. Maps of this type are usually referred to as t-tables. Furthermore, if (b, t, m, s) is in H, so is (b, t, m, s0 ) for 1 ≤ s0 ≤ s. This result follows from reducing the dimension s for nets and digital nets [8, Lemma 2.7], from shortening a linear code, and from dropping arbitrary columns from an OA. Therefore, the shape of H is also determined by a function yielding the maximum s for given b, m, and t, which is usually referred to as an s-table. Note that s may be unbounded for certain b, m, and t. In this case an s-table returns ∞. Some other tables are possible: for instance, a minimum-m-table yielding the smallest possible m for given b, k = m − t, and s is well defined because (b, t, m, s) ∈ H implies (b, t+u, m+u, s) ∈ H for all u ≥ 0, i.e., (b, m−k, m, s) ∈ H implies (b, m0 − k, m0 , s) ∈ H for all m0 ≥ m. This follows from replicating a net bu times, from adding u arbitrary rows and columns to a digital net, from taking a subcode of a linear code, and from duplicating each row of an orthogonal array bu times. Tables for arbitrary projections are not possible in general. For instance, a maximum-m-table yielding the largest possible m for given b, t, and s makes perfect sense for nets and digital nets because (b, t, m, s) ∈ H implies (b, t, m0 , s) ∈ H for all t ≤ m0 ≤ m due to the propagation rule for m-reduction for nets [8, Lemma 2.8] and digital nets [14, Lemma 3]. On the other hand, for
MinT: A Database for Optimal Net Parameters
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Table 1. MinT creates tables for the following projections Display . . . Depending on . . . minimal t s, m maximal k s, m maximal s m, t maximal s m, k maximal s k, t maximal m t, s maximal k t, s minimal m k, s minimal t k, s
linear codes and orthogonal arrays no such propagation rule exists. It should also be noted that a maximum-m-table may return two values in addition to integers: ∞ if m is unbounded and “seq” if a (t, s)-sequence also exists. 4.2 An Efficient Representation of Hmode type mode As outlined in the previous section the shape of Htype is completely determined by certain functions, for instance a t-table, an s-table, or a minimumm-table. These functions, however, have quite different properties and are not equally well suited for tabulation or for a computer implementation. A t-table is a comparatively smooth function, because its entries start with t = 0 for small s and m and are bounded by
t ≤ min{m, tb (s − 1)}, where tb (s − 1) is a function growing linearly in s (tb (s − 1) is the t-parameter of an optimal (t, s − 1)-sequence, see [11]). This is contrary to an s-table, which approaches t−1 b (t) + 1 for m → ∞, but is unbounded for m = t. It is easy to see that the information contained in a finite number of t-table entries can always be stored in a finite number of s-table entries, while the opposite may not be possible. For this reason, MinT uses only s-tables internally and generates all other requested tables based thereupon. 4.3 Parameter Range Considered by MinT mode Htype is an infinite set, and also its s-table representation has an unbounded domain and image. Thus truncation must occur when the table is represented in the finite memory of a computer system. As far as the s-values are concerned, finite numbers exceeding about 224 are truncated and stored as a special entry signalizing overflow. This bound seems reasonable because a successful evaluation of integrals with larger dimensions is very unlikely.
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Maximal s-table for base 2 — Arbitrary m=
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The entry “∞” marks unbounded s, while “##” stands for a bounded, but very large s (exceeding 8388602) where MinT does not know the exact value. Values above 1000 are given in exponential notation: an entry of the form “xey” denotes a value of s with x 10y ≤ s < (x + 1) 10y. Created by MinT, Dept. of Mathematics, University of Salzburg Please send comments to mint|at|sbg|dot|ac|dot|at
Last change 24.01.05
Fig. 2. Screen shot of MinT: an s-table showing largest possible s for given m and t in base 2 for Hexistent . net
The domain of the s-tables suffers more severe restrictions, because the number of tuples (b, t, m) taken into account directly affects the required amount of memory. At the moment, MinT considers all prime powers b = pr with p ≤ 7 and b ≤ 32, i.e., b = 2, 3, 4, 5, 7, 8, 9, 16, 25, 27, and 32. The size m is restricted to 0 ≤ m ≤ 80, which (even for base b = 2) exceeds the size of nets that can be enumerated by any computer imaginable today. Finally, t is between 0 and m for nets, so the full range is taken into account. (t, s)-sequences are only considered for t ≤ 150. 4.4 Tables Generated by MinT Even though only an s-table (storing the largest possible s for given b, t, and m) is used internally, the MinT front-end can produce tables for all nine reasonable projections based on the variables t, k, m, and s (see Table 1). Providing different projections is an important feature because it allows the user to see the data in the way that is most natural to her or him. While earlier tables of net parameters appeared as t-tables, [6, 3] already contain stables in addition to t-tables. Depending on the methods used for the creation of new nets, researchers may prefer different formats (for instance, [1] contains tables listing s depending on m and k).
MinT: A Database for Optimal Net Parameters
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Minimal t-table for base 2 — Lower bound on t (digital) s=
m=9
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 ∞ 1 2 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 8
m = 10 1 2 3 3 3 m = 11 1 2 3 3 4
m = 12 1 2 3 3 4 m = 13 1 2 3 3 4
m = 14 1 2 3 3 4 m = 15 1 2 3 3 4 m = 16 1 2 3 3 4
m = 17 1 2 3 3 4 m = 18 1 2 3 3 4 m = 19 1 2 3 3 4
m = 20 1 2 3 3 4 m = 21 1 2 3 3 4 m = 22 1 2 3 3 4
m = 23 1 2 3 3 4 m = 24 1 2 3 3 4 m = 25 1 2 3 3 4
m=∞ 1 2 3 3 4
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Move table to m = 9
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Type Lower bound on t (digital) 6 Base b = 2
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Created by MinT, Dept. of Mathematics, University of Salzburg Please send comments to mint|at|sbg|dot|ac|dot|at
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Fig. 3. Screen shot of MinT: a t-table showing smallest possible t for given s and m in base 2 for Hpotential . dignet
Tables are produced for a fixed base b. They can contain data about any mode Htype with type ∈ {net, dignet} and mode ∈ {constructive, existent, potential}. It is also possible to produce a table containing data from two sets H and H 0 , making it easy to investigate the differences between both sets. Thus, e.g., potential entries that can possibly be improved can be found by comparing Htype existent with Htype . existent Figure 2 contains a screen shot of an s-table for Hnet and b = 2. Note that there are no entries for t > m. For t = m and t = m − 1, s is unbounded (entry “∞”), while values exceeding 224 are denoted by “##”. Other large values are written in exponential notation. Entries for sequences as well as the limits for m → ∞ can be found at the right side of the table. Using the arrow buttons or the input fields in the first row below the table, the viewport can be moved to any desired position or its size can be changed. Additional controls allow transposing the table and changing the base b as mode well as the set Htype that is currently displayed. potential Figure 3 contains a screen shot of a t-table for Hdignet in base b = 2. In this case entries for sequences are displayed at the bottom of the table. By default each cell has a colored background depending on its value, making it easy to identify parameters yielding identical t-values. Unfortunately, coloring cannot be reproduced here.
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Best known (t, 52, 120)-nets in base 3 (37, 52, 120)-net over F3 — Constructive and digital Digital (37, 52, 120)-net over F3, using s-reduction based on digital (37, 52, 144)-net over F3, using net duplication based on digital (36, 51, 144)-net over F3, using base reduction of the second kind based on digital (2, 17, 48)-net over F27, using net from sequence based on digital (2, 47)-sequence over F27, using Niederreiter/Xing sequence construction II/III based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using fields by Garica/Quoos
(35, 52, 120)-net over F3 — Digital Digital (35, 52, 120)-net over F3, using s-reduction based on digital (35, 52, 123)-net over F3, using Gilbert–Varshamov bound for OOAs
(21, 52, 120)-net in base 3 — Lower bound on t There is no (20, 52, 120)-net in base 3, because s-reduction would yield (20, 52, 107)-net in base 3, but generalized Rao bound for nets Go to b = 3
, m = 52
, s = 120
. Show quality t
Created by MinT, Dept. of Mathematics, University of Salzburg Please send comments to mint|at|sbg|dot|ac|dot|at
6.
Submit
Last change 24.01.05
Fig. 4. Screen shot of MinT: Details about the existence of (digital) (t, 52, 120)-nets in base b = 3.
5 Construction Trees MinT provides detailed information about each table entry by simply clicking on it. Figure 4 gives an example showing the result of a query for optimal (t, 52, 120)-nets in base b = 3. The upper part of the screen lists a chain of explicit constructions leading to a digital (37, 52, 120)-net over 3 : starting with a digital net over 27 derived from a Niederreiter/Xing sequence based on a certain global function field, the final digital net over 3 is obtained using a basis-reduction method and two trivial propagation rules. The next part of the result shows that a digital (35, 52, 120)-net over 3 does also exist. However, this net is not constructive and its existence is only guaranteed by the Gilbert–Varshamov bound for ordered orthogonal arrays.
MinT: A Database for Optimal Net Parameters
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MinT
Best known (12, 12+7, s)-nets in base 3 (12, 12+7, 54)-net over F3 — Constructive and digital Digital (12, 19, 54)-net over F3, using (u, u+v)-construction for OOAs based on 1. digital (2, 5, 20)-net over F3, using net-embeddable linear codes with strength k = 3 based on linear [20, 15, 4]-code over F3, using (u, u+v)-construction with parity-check code based on linear [10, 6, 4]-code over F3, using ovoid 2. digital (7, 14, 34)-net over F3, using linear code embeddings found in a computer search
(12, 12+7, 64)-net over F3 — Digital Digital (12, 19, 64)-net over F3, using net from net-embeddable linear code based on linear [90, 71, 8]-code over F3, using extended or lengthened BCH codes
(12, 12+7, 659)-net in base 3 — Upper bound on s There is no (12, 19, 660)-net in base 3, because m-reduction would yield (12, 18, 660)-net in base 3, but generalized Rao bound for nets Go to b = 3
,k= 7
, t = 12
.
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Created by MinT, Dept. of Mathematics, University of Salzburg Please send comments to mint|at|sbg|dot|ac|dot|at
Last change 24.01.05
Fig. 5. Screen shot of MinT: Details about the existence of (digital) (12, 12 + 7, s)nets in base b = 3.
As far as optimality is concerned MinT shows that even a (21, 52, 120)net cannot be ruled out by any bounds2 . A (20, 52, 120)-net, however, cannot exist, because s-reduction would yield a (20, 52, 107)-net, which cannot exist due to the generalized Rao bound for nets. Figure 5 contains another example, demonstrating the close relationship between linear codes and digital nets, especially if s is large compared to m. The constructive as well as the non-constructive existence result for (12, 12 + 7, s)-nets in base b = 3 is based on a linear code: in the constructive case on a [20, 15, 4]-code, which is obtained from the [10, 6, 4]-ovoid code; in the nonconstructive case on a [90, 71, 8]-code, which is an extended or lengthened BCH code. Figure 5 serves also as an example for a construction method with two 2
For bases b > 2 differences between bounds and existence results are in general much larger than for b = 2.
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Rudolf Sch¨ urer and Wolfgang Ch. Schmid
parents. The (12, 19, 54)-net is constructed using the (u, u + v)-construction for nets, which produces a new net based on two other nets. In these examples, all existence results yield digital nets, and the bounds are applicable to general nets. If better constructions for non-digital nets or sharper bounds for digital nets were available, these results would also be displayed on this page. While printed versions of net parameters can usually only show a single key referring to the construction method, MinT lists the full construction chain – or, in the case of propagation rules depending on several nets, the complete construction tree. For each node in the tree the parameters of the object as well as the construction method are given. Each node is a link to a page providing detailed information about the particular construction, including formulas for the resulting parameters and literature references. A form at the bottom of the page facilitates a quick change of the parameters.
6 Outlook MinT has reached a stage of maturity that makes it useful for a broad audience. The user interface is full-featured and easy to use. All important construction methods are included, which guarantees up-to-date data in the tables. However, MinT is still an active project. Our main goal is to fill the remaining gaps in the data, even though most of these gaps are for large values of s or for uncommon bases b. In this quest we hope for help from the scientific community: information about missing constructions, bounds, or other results is highly appreciated and can be sent to the e-mail address
[email protected]. In addition to getting and keeping the data up-to-date, some improvements to the user interface are planned. In the long term MinT will be able to provide generator matrices for selected classes of digital nets. This feature will be implemented based on HIntLib, the high-dimensional integration library3 .
Acknowledgments The development of MinT and this work were supported by the Austrian Science Foundation (FWF), project no. S 8311-MAT. 3
HIntLib is available at http://www.cosy.sbg.ac.at/~rschuer/hintlib/
MinT: A Database for Optimal Net Parameters
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References 1. J. Bierbrauer, Y. Edel, and W. Ch. Schmid. Coding-theoretic constructions for (t, m, s)-nets and ordered orthogonal arrays. J. Combin. Designs, 10:403–418, 2002. 2. R. C. Bose. On some connections between the design of experiments and information theory. Bull. Internat. Statist. Inst., 48:257–271, 1961. 3. A. T. Clayman, K. M. Lawrence, G. L. Mullen, H. Niederreiter, and N. J. A. Sloane. Updated tables of parameters of (t, m, s)-nets. J. Combin. Designs, 7:381–393, 1999. 4. K. M. Lawrence. A combinatorial characterization of (t, m, s)-nets in base b. J. Combin. Designs, 4:275–293, 1996. 5. K. M. Lawrence, A. Mahalanabis, G. L. Mullen, and W. Ch. Schmid. Construction of digital (t, m, s)-nets from linear codes. In S. D. Cohen and H. Niederreiter, editors, Finite Fields and Applications, volume 233 of Lect. Note Series of the London Math. Soc., pages 189–208, Cambridge, UK, 1996. Cambridge University Press. 6. G. L. Mullen, A. Mahalanabis, and H. Niederreiter. Tables of (t, m, s)-net and (t, s)-sequence parameters. In H. Niederreiter and P. J.-S. Shiue, editors, Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, volume 106 of Lecture Notes Statistics, pages 58–86. Springer-Verlag, 1995. 7. G. L. Mullen and W. Ch. Schmid. An equivalence between (t, m, s)-nets and strongly orthogonal hypercubes. J. Combin. Theory A, 76:164–174, 1996. 8. H. Niederreiter. Point sets and sequences with small discrepancy. Monatsh. Math., 104:273–337, 1987. 9. H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods, volume 63 of CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1992. 10. H. Niederreiter. Constructions of (t, m, s)-nets. In H. Niederreiter and J. Spanier, editors, Monte Carlo and Quasi-Monte Carlo Methods 1998, pages 70–85. Springer-Verlag, 2000. 11. H. Niederreiter and C. P. Xing. Low-discrepancy sequences and global function fields with many rational places. Finite Fields Appl., 2:241–273, 1996. 12. H. Niederreiter and C. P. Xing. The algebraic-geometry approach to lowdiscrepancy sequences. In H. Niederreiter et al., editors, Monte Carlo and Quasi-Monte Carlo Methods 1996, volume 127 of Lecture Notes Statistics, pages 139–160. Springer-Verlag, 1998. 13. A. B. Owen. Orthogonal arrays for computer experiments, integration and visualization. Statist. Sinica, 2:438–452, 1992. 14. W. Ch. Schmid and R. Wolf. Bounds for digital nets and sequences. Acta Arith., 78:377–399, 1997. 15. S. Tezuka and T. Tokuyama. A note on polynomial arithmetic analogue of Halton sequences. ACM Trans. Modeling and Computer Simulation, 4:279–284, 1994.
