Nets, (t, s)-Sequences, and Algebraic Curves

over Finite Fields with Many Rational Points

Harald Niederreiter

Abstract. The current status of the theory of (t, m, s)-nets and (t, s)- sequences is presented in a brief form, with some emphasis on the connections with algebraic geometry. Closely related work on constructions of algebraic curves over finite fields with many rational points and on improving the Gilbert-Varshamov bound in algebraic coding theory is discussed as well.

1991 Mathematics Subject Classification: 05B15, 11G20, 11K38, 11R58, 11T71, 14G15, 14H05, 94B27, 94B65. Keywords and Phrases: quasirandom points, orthogonal arrays, algebraic curves over finite fields, rational points, algebraic-geometry codes, Gilbert-Varshamov bound.

1. Introduction and basic concepts Nets and (t, s)-sequences are finite point sets, respectively infinite sequences, satisfying strong uniformity properties with regard to their distribution in the sdimensional unit cube I^{s} = [0, 1]^{s}. The general theory of these combinatorial objects was first developed in [12]. They have attracted a lot of interest in scientific computing in recent years because of their role as quasirandom points in quasi-Monte Carlo methods, e.g. for numerical integration over I^{s} (see [14] for the details). They also offer a great appeal for theoretical studies in view of the many links with other areas such as classical combinatorial designs, coding theory, algebra, number theory, and algebraic geometry. To set the stage, we first review some basic definitions.

Definition 1. For a given dimension s ≥ 1 and integers b ≥ 2 and 0 ≤ t ≤ m, a (t, m, s)-net in base b is a point set P consisting of b^{m} points in I^{s} such that every subinterval J of I^{s} of the form

J = ∏_{i=1}^{s}[a_{i}b^{−d_{i}}, (a_{i} + 1)b^{−d_{i}})

with integers d_{i} ≥ 0 and 0 ≤ a_{i} < b^{d_{i}} for 1 ≤ i ≤ s and with Vol(J) = b^{t−m} contains exactly b^{t} points of P.

For integers b ≥ 2 and m ≥ 1 and a point x ∈ I^{s}, we obtain [x]_{b,m} ∈ I^{s} by truncating a b-adic expansion of each coordinate of x after m terms. Here expansions with almost all digits equal to b − 1 are allowed – thus, the truncation operates on the expansions of the coordinates of x and not on x itself. The following definition of a (t, s)-sequence is the slightly generalized version described in [20], [21] (see [14, Chapter 4] for the original narrower definition). We assume prescribed b-adic expansions on which the truncations operate.

Definition 2. For a given dimension s ≥ 1 and integers b ≥ 2 and t ≥ 0, a sequence x_{0}, x_{1}, … of points in I^{s} is a (t, s)-sequence in base b if for all integers k ≥ 0 and m > t the points [x_{n}]_{b,m} with kb^{m} ≤ n < (k + 1)b^{m} form a (t, m, s)-net in base b.

The following useful principle shows that if we can construct a (t, s)-sequence, then we can construct infinitely many nets in dimension s + 1 (see [12, Section 5], [20, Section 6]).

Lemma 1. If there exists a (t, s)-sequence in base b, then for every integer m ≥ t there exists a (t, m, s + 1)-net in base b.

The aim in the construction of (t, m, s)-nets and (t, s)-sequences in base b is to make the quality parameter t as small as possible if the other parameters are fixed. Most of the known constructions of nets and (t, s)-sequences are based on the digital method which was introduced in [12, Section 6]. For the sake of brevity, we just sketch the digital method for constructing (t, m, s)-nets in base b. Select a commutative ring R with identity and of finite order b ≥ 2. For given m ≥ 1 and s ≥ 1 choose a system

C = {c_{j}^{(i)} ∈ R^{m} : 1 ≤ i ≤ s, 1 ≤ j ≤ m} .

Now we get the jth b-adic digits of the ith coordinates of the points of the (t, m, s)- net by forming the inner product of c_{j}^{(i)} with all elements of R^{m} and then identifying elements of R with b-adic digits. The value of the quality parameter t depends on the choice of C. The resulting net is called a digital (t, m, s)-net in base b (or constructed over R if we want to emphasize R). Similarly, we speak of a digital (t, s)-sequence in base b (or constructed over R if we want to emphasize R). There is a “digital” analog of Lemma 1, i.e., a digital (t, s)-sequence yields infinitely many digital nets in dimension s + 1 (see [20, Section 2]). For practical purposes it suffices to consider the digital method in the special case where the ring R is a finite field F_{q} of prime-power order q. Digital nets and (t, s)-sequences in an arbitrary base b can be obtained by using rings R that are direct products of finite fields (see [14, Chapter 4], [20, Section 5]).

In this paper we give a brief review of the state-of-the-art in the area of (t, m, s)-nets and (t, s)-sequences, with some emphasis on the connections with algebraic geometry. Section 2 discusses links with classical combinatorial objects such as MOLS and orthogonal arrays. Constructions of nets and (t, s)-sequences, e.g. by methods using algebraic curves over finite fields, are presented in Section 3. This leads to the discussion of algebraic curves over finite fields with many rational points in Section 4. As a by-product we obtain the applications to algebraic coding theory in Section 5, such as improvements on the Gilbert-Varshamov bound. For various aspects, more detailed expository accounts can be found in [14, Chapter 4], [21], [25], [32].

2. Connections with combinatorial designs The fact that there are close links between nets and combinatorial designs was noticed already in [12, Section 5]. For instance, it was shown there that for s ≥ 2 the existence of a (0, 2, s)-net in base b is equivalent to the existence of s−2 MOLS of order b. Later it was proved by Mullen and Whittle [11] that for s ≥ 2 and any t ≥ 0, the existence of a (t, t + 2, s)-net in base b is equivalent to the existence of a certain set of mutually orthogonal hypercubes of order b. In the language of orthogonal arrays, there is the result in [15] that there exists a (t, t + 2, s)-net in base b if and only if there exists an orthogonal array OA(b^{t+2}, s, b, 2) of index b^{t}.

Lawrence [6] and Mullen and Schmid [10] independently established a combinatorial equivalence between arbitrary (t, m, s)-nets in base b and suitable combinatorial designs. Depending on the language that is used, these designs can be generalized orthogonal arrays, ordered orthogonal arrays, or strongly orthogonal hypercubes. The proofs of all these combinatorial results are constructive.

These connections with combinatorial designs imply obstructions to the existence of (t, m, s)-nets for m ≥ t + 2 (nets exist trivially for m − t = 0, 1). Consider e.g. the following simple argument: if there exists a (0, m, s)-net in base b for some m ≥ 2, then there exists a (0, 2, s)-net in base b, hence there are s − 2 MOLS of order b, and so we must have s ≤ b + 1. A more general argument of this type, combined with bounds for the appropriate combinatorial designs, leads to upper bounds on s in terms of b, m, and t, under the assumption that there exists a (t, m, s)-net in base b with m ≥ t + 2. A description of this method, together with tables of bounds, can be found in [2]. More recently, this approach was further refined by Martin and Stinson [7], [8] and improved bounds were obtained. In view of Lemma 1, combinatorial obstructions to the existence of (t, m, s)-nets yield combinatorial obstructions to the existence of (t, s)-sequences, such as the following bound from [21].

Theorem 1. Given b ≥ 2 and s ≥ 1, a (t, s)-sequence in base b can exist only if

t ≥ \frac{s}{b} − log_{b} \frac{(b − 1)s + b + 1}{2}.

3. Constructions of nets and (t, s)-sequences The number of known construction methods for nets and (t, s)-sequences is already quite large and ideas from various areas are employed. The combinatorial approach to the construction of nets uses the equivalences between (t, m, s)-nets and suitable combinatorial designs mentioned in Section 2 and techniques of constructing such combinatorial designs. Surveys of combinatorial methods for the construction of nets are given in [2], [9]. Other important methods for the construction of nets are based on coding theory. This approach goes back to an observation in [12, Section 7] that there is a connection between the digital method over a finite field F_{q} and the construction of parity-check matrices for good linear codes over F_{q}. This connection is conveniently formalized through the notion of a (d, m, s)-system over F_{q} introduced in [32], which is a system {a_{j}^{(i)} ∈ F_{q}^{m} : 1 ≤ i ≤ s, 1 ≤ j ≤ m}

of vectors such that for any integers d_{1}, …, d_{s} ≥ 0 with ∑_{i=1}^{s} d_{i} = d the a_{j}^{(i)}, 1 ≤ j ≤ d_{i}, 1 ≤ i ≤ s, are linearly independent over F_{q}. Finding a digital (t, m, s)- net constructed over F_{q} is then equivalent to finding a (d, m, s)-system over F_{q} with d = m − t. The surveys [2], [9] report on coding-theory methods for the construction of nets and new methods of this type can be found in [32].

Standard constructions of digital (t, s)-sequences in base b are due to Sobol’ [38] for b = 2 and any s, to Faure [3] for prime bases b ≥ s, and to Niederreiter [13] for any b and any s. Generalizations of these sequences are described in Tezuka [39, Chapter 6]. As a by-product, these constructions yield digital (t, m, s+1)-nets in base b.

An important recent development is the use of algebraic curves over finite fields (or, equivalently, of global function fields) for the construction of (t, s)- sequences. The basic idea goes back to Niederreiter [16], [17]. At present, four different construction principles using algebraic curves are available and they all rely on the digital method over F_{q}. We refer to [18], [20], [21], [44] for the detailed description of these constructions and to [25], [32] for further discussions. Three of the methods, and indeed the most effective ones, are based on algebraic curves over F_{q} with many F_{q}-rational points (or, equivalently, on global function fields with many rational places). Given q and a dimension s ≥ 1, the typical procedure is to choose a smooth, projective, absolutely irreducible algebraic curve C over F_{q} containing at least s + 1 F_{q}-rational points, say P_{∞}, P_{1}, …, P_{s}. The point P_{i}, 1 ≤ i ≤ s, is used to produce the data that are needed in the digital method (i.e., certain elements of F_{q}) for generating the ith coordinates of the points of the (t, s)-sequence. These elements of F_{q} are obtained by expansions on the curve C in local coordinates at P_{∞}. The methods in [20] and [44] yield digital (t, s)- sequences constructed over F_{q} with t being the genus of C. If we optimize these constructions, we arrive in a natural way at the following important quantity from algebraic geometry over F_{q} and at the subsequent theorem in [20].

Definition 3. For given g ≥ 0 and q, let N_{q}(g) be the maximum number of F_{q}-rational points that a smooth, projective, absolutely irreducible algebraic curve over F_{q} of genus g can have.

Theorem 2. For every q and s there exists a digital (V_{q}(s), s)-sequence constructed over F_{q}, where V_{q}(s) is the least value of g such that N_{q}(g) ≥ s + 1.

The behavior of V_{q}(s) as s → ∞ can be obtained from class field towers and the asymptotic theory of N_{q}(g) (see Section 5). As stated in Section 1, we can also pass from prime-power bases q to arbitrary bases b in the digital method. Finally, this leads to the following bound (see [20, Section 5]), which in view of Theorem 1 is best possible as far as the order of magnitude in s is concerned.

Theorem 3. For every b ≥ 2 and s ≥ 1 there exists a digital (t, s)-sequence in base b with

t ≤ \frac{c}{log q_{1}} s + 1,

where c > 0 is an absolute constant and q_{1} is the least prime power in the factorization of b into pairwise coprime prime powers.

4. Algebraic curves with many rational points The constructions of (t, s)-sequences in Section 3 based on algebraic curves over F_{q} lead to the requirement of finding good lower bounds for the number N_{q}(g) in Definition 3, or in other words to the problem of constructing algebraic curves over F_{q} of given genus g with many F_{q}-rational points. This problem is also of great importance in the theory of algebraic-geometry codes (see Section 5). Recent surveys of this problem, also in the equivalent language of global function fields, are given in Garcia and Stichtenoth [4], Niederreiter and Xing [26], [30], and van der Geer and van der Vlugt [42].

A well-known technique for establishing the existence of various algebraic curves over F_{q} with many F_{q}-rational points is due to Serre [37] and uses methods of class field theory. This approach was continued by Auer [1] and Lauter [5]. Usually, the curves obtained by this technique are not in an explicit form. On the other hand, constructions in the function field setting that work with Artin-Schreier and Kummer extensions and with subfields of cyclotomic function fields yield explicit generators and defining equations. Such constructions can be found e.g. in [19], [21], [26], [46] for q = 2, in [22], [27] for q = 3, in [22], [23] for q = 4, in [22], [24], [35] for q = 5, in [29] for q = 8, 16, and in [32] for q = 9, 27. Explicit constructions inspired by techniques from coding theory were introduced by van der Geer and van der Vlugt [41] (see also the survey [42]).

In the function field setting, a powerful technique of obtaining global function fields with many rational places is based on Hilbert class fields. The aim is to construct unramified abelian extensions of a given global function field F in which certain selected rational places of F split completely. This method works particularly well if the divisor class number of F is large relative to the genus of F. Applications of this method can be found in [22], [24], [26], [27], [29], [30], [35], [46]. A more general approach, which contains both cyclotomic function fields and Hilbert class fields as special cases, uses the theory of narrow ray class extensions obtained from Drinfeld modules of rank 1 and was introduced in [45]. This method allows great flexibility and produces a large number of families of global function fields with many rational places. We refer to [23], [24], [26], [27], [29], [30], [31], [35], [46] for further results and examples with this method.

Table 1 contains all bounds for N_{q}(g) available to the author for q = 2, 3, 4, 5, 8, 9, 16, 27 and 1 ≤ g ≤ 50 (for g = 0 we trivially have N_{q}(0) = q + 1). In each entry of the table, the first number is a lower bound for N_{q}(g) and the second an upper bound for N_{q}(g). If only one number is given, then this is the exact value of N_{q}(g). A program for calculating upper bounds for N_{q}(g), which is based on Weil’s explicit formula for the number of F_{q}-rational points in terms of the zeta function and on the trigonometric polynomials of Oesterlé, was kindly supplied by Jean-Pierre Serre. The lower bounds in Table 1 are obtained by combining [32, Table 3] with new data in [1], [35]. We refer also to the tables of van der Geer and van der Vlugt [43] which represent the most recent result of an ongoing project to update bounds for N_{q}(g) periodically.

5. Applications to coding theory There is an asymptotic theory of N_{q}(g) which has significant applications to algebraic coding theory. The basic quantity here is

A(q) = lim sup_{g→∞} \frac{N_{q}(g)}{g}.

For values of q for which A(q) is larger than a known comparison function, Goppa’s construction of algebraic-geometry codes leads to improvements on the classical Gilbert-Varshamov bound for the existence of good linear codes over F_{q}.

Let U_{q} be the set of ordered pairs (δ, R) ∈ [0, 1]^{2} for which there exists a sequence of linear codes over F_{q} of increasing lengths such that δ is the limit of the relative minimum distances and R the limit of the information rates. It is known that for some continuous function α_{q} on [0, 1] we have

U_{q} = {(δ, R) : 0 ≤ R ≤ α_{q}(δ), 0 ≤ δ ≤ 1} ,

where α_{q}(0) = 1 and α_{q}(δ) = 0 for δ ∈ [(q − 1)/q, 1]. The function α_{q} is unknown, and it is an important issue in algebraic coding theory to obtain good lower bounds for α_{q} on the interval (0, (q − 1)/q). The Gilbert-Varshamov bound says that

α_{q}(δ) ≥ R_{GV} (q, δ) := 1 − H_{q}(δ) for 0 < δ < (q − 1)/q,

where H_{q} is the q-ary entropy function. Algebraic-geometry codes lead to the bound

α_{q}(δ) ≥ R_{AG}(q, δ) := 1 − \frac{1}{A(q)} − δ for 0 ≤ δ ≤ 1.

By showing that A(q) ≥ q^{1/2} − 1 if q is a square, Tsfasman, Vlǎdut, and Zink [40] proved that R_{AG}(q, δ) > R_{GV} (q, δ) if q is a sufficiently large square and δ belongs to a suitable subinterval of [0, 1].

For nonsquares q only weaker lower bounds for A(q) are known. Serre [37] showed that A(q) is at least of the order of magnitude log q, and an alternative proof and an effective version of this result were recently given in [33]. In many cases the following result in [28] yields a considerable improvement: if q = p^{e} with a prime p and an odd integer e ≥ 3, then A(q) is at least of the order of magnitude q^{1/(2k)}, where k is the least prime factor of e. Further discussions and refinements of this result can be found in [30], [33]. As a consequence we get the following theorem in [28] which improves on the Gilbert-Varshamov bound for sufficiently large composite nonsquares q.

Theorem 4. Let m ≥ 3 be an odd integer and let r be a prime power with r ≥ 100m^{3} for odd r and r ≥ 576m^{3} for even r. Then there exists an open interval (δ_{1}, δ_{2}) ⊆ (0, 1) containing (r^{m} − 1)/(2r^{m} − 1) such that

R_{AG}(r^{m}, δ) > R_{GV} (r^{m}, δ) for all δ ∈ (δ_{1}, δ_{2}).

In connection with lower bounds for A(q) we mention that there is a method of Perret [36] for obtaining such lower bounds which depends, however, on a conjecture that would provide a sufficient condition for the infinitude of certain ramified class field towers. It was recently shown in [34] by a counterexample that this conjecture is wrong. Therefore, the lower bounds for A(q) in Perret [36, Section III] remain unproved.

Table 1: Bounds for N_{q}(g)

\begin{matrix}g\q & 2 & 3 & 4 & 5 & 8 & 9 & 16 & 27 \\ 1 & 5 & 7 & 9 & 10 & 14 & 16 & 25 & 38 \\ 2 & 6 & 8 & 10 & 12 & 18 & 20 & 33 & 48 \\ 3 & 7 & 10 & 14 & 16 & 24 & 28 & 38 & 58 \\ 4 & 8 & 12 & 15 & 18 & 25-29 & 30 & 45-47 & 64-68 \\ 5 & 9 & 12-14 & 17-18 & 20-22 & 29-32 & 32-36 & 49-55 & 55-78 \\ 6 & 10 & 14-15 & 20 & 21-25 & 33-36 & 35-40 & 65 & 76-88 \\ 7 & 10 & 16-17 & 21-22 & 22-27 & 33-39 & 39-43 & 63-70 & 64-98 \\ 8 & 11 & 15-18 & 21-24 & 22-29 & 34-43 & 38-47 & 61-76 & 92-108 \\ 9 & 12 & 19 & 26 & 26-32 & 45-47 & 40-51 & 72-81 & 82-118 \\ 10 & 13 & 19-21 & 27-28 & 27-34 & 38-50 & 54-55 & 81-87 & 91-128 \\ 11 & 14 & 20-22 & 26-30 & 32-36 & 48-54 & 55-59 & 80-92 & 96-138 \\ 12 & 14-15 & 22-24 & 29-31 & 30-38 & 49-57 & 55-63 & 68-97 & 109-148 \\ 13 & 15 & 24-25 & 33 & 36-40 & 50-61 & 60-66 & 97-103 & 136-156 \\ 14 & 15-16 & 24-26 & 32-35 & 39-43 & 65 & 56-70 & 97-108 & 84-164 \\ 15 & 17 & 28 & 33-37 & 35-45 & 54-68 & 64-74 & 98-113 & 136-171 \\ 16 & 17-18 & 27-29 & 36-38 & 40-47 & 56-71 & 74-78 & 93-118 & 136-178 \\ 17 & 17-18 & 24-30 & 40 & 42-49 & 61-74 & 56-82 & 96-124 & 128-185 \\ 18 & 18-19 & 26-31 & 41-42 & 32-51 & 65-77 & 46-85 & 113-129 & 94-192 \\ 19 & 20 & 27-32 & 37-43 & 45-54 & 58-80 & 84-88 & 121-134 & 126-199 \\ 20 & 19-21 & 30-34 & 37-45 & 30-56 & 68-83 & 48-91 & 121-140 & 133-207 \\ 21 & 21 & 32-35 & 41-47 & 50-58 & 72-86 & 82-95 & 129-145 & 163-214 \\ 22 & 21-22 & 28-36 & 40-48 & 51-60 & 66-89 & 78-98 & 129-150 & 112-221 \\ 23 & 22-23 & 26-37 & 41-50 & 55-62 & 68-92 & 92-101 & 126-155 & 114-228 \\ 24 & 20-23 & 28-38 & 42-52 & 46-64 & 66-95 & 91-104 & 129-161 & 166-235 \\ 25 & 24 & 36-40 & 51-53 & 52-66 & 66-97 & 64-108 & 144-166 & 196-242 \\ 26 & 24-25 & 36-41 & 55 & 45-68 & 72-100 & 110-111 & 150-171 & 108-249 \\ 27 & 22-25 & 39-42 & 49-56 & 52-70 & 96-103 & 60-114 & 145-176 & 114-256 \\ 28 & 25-26 & 37-43 & 51-58 & 54-71 & 97-106 & 105-117 & 136-181 & 108-263 \\ 29 & 25-27 & 42-44 & 49-60 & 56-73 & 97-109 & 104-120 & 161-187 & 114-270 \\ 30 & 25-27 & 34-46 & 53-61 & 58-75 & 80-112 & 60-123 & 161-192 & 117-277 \\ 31 & 27-28 & 40-47 & 60-63 & 72-77 & 72-115 & 84-127 & 150-197 & 114-284 \\ 32 & 26-29 & 38-48 & 57-65 & 62-79 & 72-118 & 81-130 & 132-202 & 126-291 \\ 33 & 28-29 & 37-49 & 65-66 & 64-81 & 92-121 & 78-133 & 193-207 & 220-298 \\ 34 & 27-30 & 44-50 & 57-68 & 76-83 & 80-124 & 111-136 & 156-213 & 135-305 \\ 35 & 29-31 & 47-51 & 58-69 & 68-85 & 106-127 & 84-139 & 144-218 & 126-312 \\ 36 & 30-31 & 46-52 & 64-71 & 64-87 & 105-130 & 110-142 & 185-223 & 244-319 \\ 37 & 29-32 & 48-54 & 66-72 & 72-89 & 121-132 & 120-145 & 208-228 & 162-326 \\ 38 & 28-33 & 36-55 & 56-74 & 78-91 & 129-135 & 105-149 & 193-233 & 144-333 \\ 39 & 33 & 46-56 & 65-75 & 76-93 & 117-138 & 84-152 & 160-239 & 271-340 \\ 40 & 32-34 & 54-57 & 75-77 & 65-94 & 100-141 & 90-155 & 162-244 & 244-346 \end{matrix}

\begin{matrix}g\q & 2 & 3 & 4 & 5 & 8 & 9 & 16 & 27 \\ 41 & 33-35 & 50-58 & 65-78 & 80-96 & 112-144 & 84-158 & 216-249 & 153-353 \\ 42 & 33-35 & 39-59 & 66-80 & 60-98 & 129-147 & 90-161 & 209-254 & 280-360 \\ 43 & 33-36 & 55-60 & 72-81 & 84-100 & 100-150 & 120-164 & 226-259 & 196-367 \\ 44 & 33-37 & 42-61 & 68-83 & 60-102 & 129-153 & 90-167 & 162-264 & 153-374 \\ 45 & 32-37 & 48-62 & 80-84 & 88-104 & 144-156 & 112-170 & 242-268 & 171-381 \\ 46 & 34-38 & 55-63 & 81-86 & 75-106 & 129-158 & 138-173 & 243-273 & 162-388 \\ 47 & 36-38 & 47-65 & 73-87 & 92-108 & 120-161 & 154-177 & 176-277 & 174-395 \\ 48 & 34-39 & 55-66 & 77-89 & 82-110 & 126-164 & 163-180 & 184-282 & 325-402 \\ 49 & 36-40 & 63-67 & 81-90 & 96-111 & 130-167 & 168-183 & 192-286 & 268-409 \\ 50 & 40 & 56-68 & 91-92 & 70-113 & 130-170 & 182-186 & 225-291 & 180-416 \end{matrix}

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Harald Niederreiter

Institute of Information Processing Austrian Academy of Sciences Sonnenfelsgasse 19 A-1010 Vienna Austria E-mail: niederreiter@oeaw.ac.at