Math. Nachr. 196 (1998), 171-186

Towers of Global Function Fields with Asymptotically Many Rational Places and an Improvement on the Gilbert - Varshamov Bound

BY HARALD NIEDERREITER of Vienna and CHAOPING XING of Hefei

(Received March 12, 1997)

(Revised Version July 31, 1997)

Abstract. We construct infinite class field towers of global function fields with asymptotically many rational places. In this way, we improve on asymptotic bounds of SERRE, PERRET, SCHOOF, and XING. The results can be interpreted equivalently as asymptotic bounds on the number of rational points of smooth algebraic curves over finite fields. As an application, we show an improvement on the Gilbert - Varshamov bound for linear codes over finite fields of a sufficiently large composite nonsquare order.

1. Introduction

Let F, denote the finite field of order q, where q is an arbitrary prime power, and let K be a global function field with full constant field F,, that is, with F, algebraically closed in K . By a rational place of K we mean a place of K of degree 1. We write N ( K ) for the number of rational places of K and g ( K ) for the genus of K . According to the Weil- Serre bound (see [14], (16, Theorem V.3.11) we have

where LtJ is the greatest integer not exceeding the real number I!.

Definition 1.1. For any prime power q and any integer g 2 0 put

Nq(9) = m = N ( K ) 1

where the maximum is extended over all global function fields K of genus g with full constant field F,.

1991 Mathematics Subject Classification. l lG20 , l lR58 , l l T 7 1 , 14G15, 14H05, 94B27,94B65. Keywords and phrases. Global function fields, rational places, algebraic - geometry codes,

Gilbert - Varshamov bound.

172 Math. Nachr. 196 (1998)

It is well known that N,(g) is also the maximum number of F, -rational points that a smooth, projective, absolutely irreducible algebraic curve over F, of genus g can have. The following quantity was introduced by IHARA (71.

Definition 1.2. For any prime power q let

N9 (9) A(q) = limsup - , g+m 9

where g runs through positive values.

It follows from (1.1) that A ( q ) 5 [2q'/2], and some improvements on this upper bound were obtained by IHARA [7] and MANIN [lo]. Furthermore, IHARA [7] showed that A(q) 2 q1/2 - 1 if q is a square. In the special cases q = p2 and q = p4, this lower bound was also proved by TSFASMAN, VLADUT, and ZINK [18]. Here and in the following, p always denotes a prime number. Soon afterwards, VLXDUT and DRINFEL'D [19] established the bound

(1.2) A(q) 5 q' /2 - 1 for a11 q .

In particular, this yields

(1.3) A(q) = q1j2 - 1 if q is a square.

Recently, GARCIA and STICHTENOTH [3], [5] proved that if q is a square, then A(q) = q1j2 - 1 can be achieved by an explicitly constructed tower of global function fields.

In the case where q is not a square, no exact values of A(q) are known, but lower bounds are available which complement the general upper bound (1.2). According to a result of SERRE [14], [15] based on class field towers, we have

A(q) 1 clogq

with an absolute constant c > 0. ZINK [23] showed that

Later, PERRET [12] proved that if 1 is a prime and if q > 42 + 1 and q 1 mod 1 , then

More recently, GARCIA and STICHTENOTH [4] gave a simple proof that A(q) > 0 if q is not a prime number, by constructing a suitable explicit tower of global function fields.

For small values of q, the method of class field towers was used by SERRE [15] and SCHOOF [13] to show that A(2) 1 (the weaker bound A(2) 2 $ was noted earlier by SERRE (141) and by XING [21] to show that A(3) 1 $ and A ( 5 ) 2 i.

Njederreiter/Xing, Towers of Global Function Fields 173

In this paper we employ class field towers to generalize and improve (1.5) and to get better lower bounds for A(2), A(3), and A(5) than those listed above. These results can be found in Sections 4 and 5. The proofs depend crucially on new lower bounds for the 1 -rank of S - divisor class groups given in Section 3. The relevant background on class field theory is recalled in Section 2. As an application of the results in Section 4, we show in Section 6 an improvement on the Gilbert - Varshamov bound for linear codes over F, for sufficiently large composite nonsquares q.

2. Background on class field theory

Let K be a global function field with full constant field IF,. We denote by JK the id& group of K and by CK the idkle class group of K .

Let S be a finite nonempty set of places of K and US the subgroup of JK consisting of all idkles that are units at all places of K outside S. Let 0 s be the S-integral ring of K , i. e., 0 s consists of all elements of K that have no poles outside S. We denote by 0; the subgroup 0 s CI U s of K'. Let Cls be the S- divisor class group of K , i. e., Cis is the quotient of the group of all divisors of K of degree 0 with support outside S by its subgroup of principal divisors.

The S-Hilbert class field H s of K is the class field of K corresponding to the subgroup K*Us of JK. Thus we have

where the second isomorphism follows from [20, Proposition 5 - 2 - 11. By global class field theory, Hs is the maximal unramified abelian extension of K (in a fixed separable closure of K ) in which all places in S split completely.

Now we define the S-class field tower of K . Let K1 be the S-Hilbert class field Hs of K and S1 the set of places of K1 lying over those in S. Recursively, we define Ki to be the Si-1 -Hilbert class field of Ki-1 for i 2 2 and Si to be the set of places of K, lying over those in Si-1. Then we get the tower

The S- class field tower of K is called infinite if Ki # Ki-1 for all i 2 1. The following proposition, which is the basic tool for our work, provides a sufficient condition for this tower to be infinite, and this leads to the stated lower bound for A(q). The proposition is obtained by combining results in (13, Section 21. The method goes back to SERRE [15] and is based on a condition of the Golod-Shafarevich type. For a prime 1 and an abelian group B we write dlB for the 1-rank of B.

Proposition 2.1. Let K be a global function field of genus g ( K ) > 1 with full constant field IF, and let S be a nonempty set of rational places of K. Suppose that there exists Q prime 1 such that

(2.2) diCls 2 2 + 2(drOi +

174 Math. Nachr. 195 (1998)

Then we have

The 1 -rank of 0; can be determined. By Dirichlet's unit theorem we have

0;. N IF; x zlSl-1,

and so

3. Lower bounds for the 1 -rank of S- divisor class groups

In view of the condition (2.2) in Proposition 2.1 it is important that we have good lower bounds for the 1 -rank of the S-divisor class group Cls. The following result, which can be viewed as a refinement of [13, Proposition 2.41, uses Tate cohomology (see [2, Chapter IV]) to obtain such a bound.

Propositjon 3.1. Let k be a global junction field and K / k a finite abelaan extension. Let T be a finite nonempty set of places of k and S the set of places of K lying over those in T . Then fo r any prime 1 we have

diCls 2 C dlGp - diO; - dlG,

where G = Gal(K/k) and G p is the inertia group of the place P an K/k. The sum is extended over all places P of k.

Proof. An exact sequence of abelian groups leads to a general bound for 1-ranks as in [12, Proposition 61. First of all, since 8 - ' ( G , Cls) is a subgroup of a quotient group of Cls, we get

dlCls 2 d1H-l (G, C l s ) . The exact sequence

P

yields the exact sequence

k ' ( G , C l s ) + f i o ( G , U s / O ; ) --t ( G,CK)

and hence

The trivial exact sequence

dlfi-'(G, C l s ) 2 dlEio(G, Us/O;) - diBo(G, CK) .

0 --t 0; + us + us/o; ---t 0

175

fields the exact sequence

B~ combining these inequalities, we get

diCZs 2 difio(G, U s ) - difio(G, 0:) - difio(G, CK) . B~ global class field theory,

f i o ( G , C ~ ) N k 2 ( G , Z ) N G .

Since H o (G, 0:) is a quotient group of 0>, we have

diHo(G,O:) 5 dlO>.

Therefore

(3.1) diCls 2 diHo(G, U s ) - diO> - diG.

Now let R be a finite set of places of k such that R is disjoint from T and R contains d l ramified places in K / k outside T. By global class field theory (compare with [l, pp. xxiv-xxv] and [2, Section VI1.71) we have

H ( O G,Us) N fi O ( G " , U ~ ) x PER PET

f i O ( ~ ~ , ( K ~ ) ' ) ,

where K P is the completion of K at some place lying over P, U p is the unit group of K P , and GP = Gal(KP/kP) , with k p being the completion of k at P. Note that GP is isomorphic to the decomposition group of P in K / k by [2, Chapter VII, Proposition 1.21. By local class field theory (compare with [8, Section XI.41) we have

f io (GP,UP) . N G p , f i o (GP, (KP) ' ) N GP.

Since clearly dlGP 2 diGp, we obtain

and together with (3.1) this yields the desired result. 0

Proposi t ion 3.2. Let k be a global function field and let PI, . . . , P,, be n 2 1 distinct places of k. Let T be a finite nonempty set of places of k. Suppose that there exist a prime 1 and n Galois extensions K l l k , . . . , K,/k such that the following four conditaons are satisfied:

(i) [Ki : k ] = 1 for 1 5 i 5 n; (ii) for 1 5 i, j 5 n, the place Pi is ramified in K,/k if and only ifi = j ;

Math. Nachr. 195 (1998)' 176

(iii) any place of k can be ramified in at most one of the extensions K1 /kl . . . , Kn/k; (iv) all places an T split completely in K,/k for 1 5 i 5 n .

Then there exists a subfield K of the extension K1 . . . Kn/k such that [K : k] = I , the extension K1 . . . Kn/K i s unramified, and

dlCls 2 n - 1 ,

where S is the set of places of K lying over those in T .

Proof . For any fixed i, the place Pi is ramified in Ki/k and unramified in K1 . . . Ki-1 K,+l . . . K,/k, hence Ki n K1 . . . Kiwi Ki+l . . . Kn = k. Thus

n

Gal(K1.. . Kn/k) N n Gal(Ki/k) N (Z/lZ)n.

We prove the proposition by induction on n. The case n = 1 is trivial. Suppose the proposition is correct for n - 1, where n 2 2. Let the subfield F of K1 . . . Kn-l/k be such that [F : k] = 1 and K1 . . . Kn-l/F is unramified. Since P, is ramified in Kn/L and unramified in F / k , we have F n K, = k. Thus

Gal(FK,/k) N Gal(FK,/F) x Gal(FK,/K,) 2: (Z/IZ)2.

a= 1

Let CT and 0 be generators of Gal(FKn/F) and Gal(FK,/K,), respectively. Then (d) N Z/lZ, (d) n Gal(FK,/F) = {I}, and (06) n Gal(FKn/Kn) = {I}. Let K be the subfield of FK,/k fixed by (aB), then [K : k] = I . For any place R of K, let Q be the place of k lying under R. We now distinguish three cases.

Then Q is unramified in K1 . . . K,-1 /k since K1 . . . Kn-l/F is an unramified extension. Thus, Q is unram- ified in K1 . . . K,/k, and so R is unramified in K1 . . . K,/K.

Case 2: Q is ramified in F / k . Then Q is ramified in Ki /k for some 1 5 i 5 n - 1. By condition (iii), Q is unramified in KJk. Hence the inertia group of Q in FKn/k is Gal(FKn/Kn) = (6 ) . Since (6)n Gal(FK,/K) = {l}, R is unramified in FKn/K, and so Q is ramified in K/k. By condition (iii), the ramification index of Q in K1 . . . Kn/k is at most 1, hence R is unramified in K1 . . . Kn/K.

Case 3: Q is ramified in K,/k. Then Q is unramified in F /k by condition (iii). Now the arguments in Case 2 can be applied mutatis mutandis to show that R is unramified in K1 . . . K,/K.

Thus, we have proved in all cases that R is unramified in K1 . . . Kn/K, and SO the extension K1 . . . K,/K is unramified. Since GaI(K1 . . . Kn/k) N (Z/lZ)" and [K : k] = 1, we have

By condition (iv) it is obvious that all places in S split completely in Ki . . . Kn/K. It follows that K1 . , . K, is contained in the S- Hilbert class field Hs of K. By [12, Proposition 61 the 1 -rank of an abelian group is at least as large as that of any of its quotient groups. Thus together with (2.1),

dlCls = dlGal(Hs/K) 2 dlGd(K l . . . K , / K ) = n-1,

Case 1: Q is unramified in both F / k and K,/k.

Gal(K1 . . . Kn/K) N (Z/ lZ)n-l .

and the induction is complete. 0

Niederreiter/Xing, Towers of Global Function Fields 177

4. General lower bounds for A(qm)

We need some facts from the theory of cyclotomic function fields as developed by HAYES IS]. Let A = F,[z] be the polynomial ring over F, and k = lF9(x) be the rational function field. We will often use the convention that a monic irreducible

P in A is identified with the place of k which is the unique zero of P , we will denote this place also by P. It will also be convenient to write 00 for the

“infinite place” of k , that is, for the place of k which is the unique pole of x. For a monic polynomial M E A let A M be the A -module of M - division points of

the Carlitz A-module. Then AM is a cyclic A-module which is A-isomorphic to := A/(M). The cyclotomic function field kM is obtained by adjoining all elements

of AM to k . Then kM/k is a finite abelian extension and G a l ( k M / k ) N A h , the group of units of the ring A M . If f is the image of f under the canonical homomorphism A 3 A M , then the Galois automorphism uf E G a l ( k ~ / k ) associated with f E A;, is determined by u j ( X ) = Af for X E A M , where A f denotes the action of the Carlitz A -module. If P E A is a monic irreducible polynomial not dividing M, then the Artin symbol of the place P relative to k M / k is equal to u p . Furthermore, if M and N are two monk coprime polynomials in A, then kMN is the composite field of kM and kN. If M = Q” for some monic irreducible Q E A and some integer n 2 1, then the only places of k that can be ramified in k M / k are 00 and &, and Q is totally ramified, whereas 00 splits into rational places of kM, each with ramification index q - 1. For any nonconstant monk M , the ramification index of 00 in k M / k is also q - 1.

In the following, for a real number t we write It1 for the least integer greater than or equal to 1.

Theorem 4.1. If q as odd and rn 2 3 is an integer, then

Proof . Put n = [2(2q + 1)1/21 + 3, then n does not exceed the number of monk irreducible polynomials of degree m in F,[x], except for q = m = 3, but in this case the result of the theorem is implied by (1.4).

Let f l , . . . , fn be n distinct monic irreducible polynomials of degree m in F,[z]. Then each fi decomposes into a product

m

j = 1

of m distinct monic linear factors Pij in F q m [z]. Put k = F q m (z). For each 1 5 i 5 n, consider the cyclotomic function field Ei = k f i , which is the composite of the m cyclotomic function fields over k defined by the Pij, 1 5 j 5 m. Then

Let Ii be the inertia group of 00 in E , / k and Fi the subfield of E i / k fixed by the subgroup Ii.(F,[z]/(ji))* of G a l ( E i / k ) . From lIil = qm-1 it follows that (qm-l )m-2

178 Math. Nachr. 195 (1998)

divides [Fi : k ] . The place 00 is unramified in F!/k and all places in the set

T = { z - c : c € F , )

of places of k split completely in Fi/k. The order of the inertia group of Pij in Fi/k divides qm - 1 for each 1 5 j 5 m, and therefore the inertia groups of Pi4, . . . ,Pi, generate a subgroup Gi of Gal(Fi/k) of order dividing (qm - l)m-3. Let Li be the subfield of F,/k fixed by Gi, then qm - 1 divides [Li : k ] . The only possible ramified places in Li/k are Pil, Piz, Pis.

Let K , be a subfield of L,/k with [Ki : k] = 2. The only possible ramified places in K i / k are Pil, Piz, Pi3, and there is a t least one place Pj E {El , Piz, Pi3) that is ramified in K i / k since k is the rational function field. It is impossible that all three places Pil ,Piz, Pi3 are ramified in K i / k , for otherwise the Hurwitz genus formula would yield

which is a contradiction to g(Ki) being an integer. Let T be as above, then all places in T split completely in Ki/k. NOW all four

conditions in Proposition 3.2 are satisfied for 1 = 2, therefore there exists a subfield K of K 1 . . . K n / k such that [K : k] = 2, the extension K1 . . . K n / K is unramified, and dzCls 2 n - 1, where S is the set of places of K lying over those in T. Thus by the definition of n,

2 g ( K i ) - 2 = - 2 . 2 + ( 2 - 1 ) . 3 ,

dzCls 2 2 + 2(2q + 1)’” = 2 + 2(lSl + 1)112 = 2 + 2(dzO> + l)’”,

where we used (2.3) in the last step. Since it is clear that g ( K ) > 1, we get from ProoDosition 2.1 that

As we have shown above, in each Ki/k the only possible ramified places are two rational places of k, and so the only possible ramified places in K / k are 2n rational places of k. Hence by the Hurwitz genus formula

2g(K) - 2 5 - 2 . 2 + (2 - 1) a 2n ,

therefore

and so the desired result follows.

g ( K ) - 1 _< n - 2 = J2(2q + 1)’I2] + 1 , 0

Corollary 4.2. If q = p e with an odd prime p and an odd integer e 2 3, then

where m is the least prime dividing e.

In view of the formula (1.3) it suffices to consider odd integers m in the folloWiag result.

Njederreiter/Xing, Towers of Global Function Fields 179

Theorem 4.3. 1 .q 2 4 1s even and m 2 3 is an odd integer, then

Proof . In the case q = 4 the result is implied by (1.3), and so we can assume q 2 8. put n = [2(2q + 2)1/2] + 3, then n does not exceed the number of monic irreducible polynomials of degree 2 in IF, [XI.

k t PI , . . . , P n be n distinct monic irreducible polynomials of degree 2 in Fg[2]. Since rn is odd, each Pi is irreducible over F g m . Put k = IFgm (z). For each 1 5 i _< n, consider the function field k(yi) defined by

(4.1) 1 yq"-' + y p m - 2 + . . . + Y p + Y i = _ . Pi

From the fact that the trace function 'll from IFgm to IF, is a surjective IF, -linear transformation (see [9, Theorem 2.23]), it follows that all places in the set

T = { x - c c : c E F ~ } U { W )

of places of k split completely in k(yi)/k. Furthermore, by [16, Proposition III.7.10] the only ramified place in k(y i ) / lc is Pi, and it is totally ramified.

Let Kj be a subfield of k(yi)/k with [Kj : k] = 2. Then all four conditions in Proposition 3.2 are satisfied for 1 = 2, hence there exists a subfield K of K1 . . . Kn/k such that [K : k] = 2, the extension K 1 . . . K,/K is unramified, and d2Cls 2 n - 1, where S is the set, of places of K lying over those in T. Thus by the definition of n,

d2Cls 2 2 + 2(2q + 2)'12 = 2 + 21S1'/2 = 2 + 2(dzO; + 1)1'2,

where we used (2.3) in the last step. Since it is clear that g ( K ) > 1, we get from Proposition 2.1,

IS1 - 29 + 2 - A ( q m ) g ( K ) - 1 g ( K ) - 1 '

It remains to calculate the genus of K . The only possible ramified places in K / k are PI , . . . , P,. For each 1 5 i 5 n, it follows from (4.1) that y f l is a prime element for the place &i of k(yi) lying over Pi. Furthermore, Gal(k(yi)/k) N ker(Tr) by [16, Proposition II1.7.10], and for each u E ker(7.k) there exists a unique (T,, E Gal(k(yi)/k) determined by

Let Hj be the subgroup of ker(Tr) corresponding to Gal(k(yi)/Ki), then by [16, Propo- sition 111.5.121 the different exponent of Qi in k(yi)/K, is given by

O,(Yi) = Yi + u .

m- 1

= 2(+ l ) ,

180 Math. Nachr. 195 ( i g g ~ j ,

where V Q ~ is the normalized discrete valuation corresponding to Qi. In a similar way we get

If Ri is the place of Ki lying over Pi, then by the tower formula for different exponents we have

dQ, ( k ( % ) / k ) = [ k ( y i ) : KiIdR, (Ki/k) + dQ, (k(%)/Ki) 3

that is, dR,(K,/k) = 2. Since K1 . . . Kn/K is unramified, the place Vi of K lying over Pi has different exponent dv, ( K / k ) = 2, and so

dQ,(k(&)/k) = 2(qm-' - 1 ) .

n 2g(K) - 2 = - 2 . 2 + 1 dv, ( K / k ) deg(l4) = 4n - 4 .

i= 1

Therefore

and the desired result follows. g ( K ) - 1 = 2n - 2 = 2[2(2q + 2)'/'1 + 4 ,

0

We note that for the only even q that is excluded in Theorem 4.3, namely q = 2, it is trivial that A(2m) 2 A(2) by considering constant field extensions, and so a lower bound for A(2m) follows from Theorem 5.1 in the next section. An obvious analog of Corollary 4.2 can be deduced from Theorem 4.3.

5 . Lower bounds for A(2) , A(3 ) , and A(5) We derive lower bounds for A(q) in the cases q = 2, 3, and 5 on the basis of

Propositions 2.1 and 3.1. These bounds improve on previous results mentioned in Section 1.

Theorem 5.1. We have

= 0.2555 . . . , 81 4 2 ) L 317

Proof. Let k be the rational function field Fz(x). Put M = (z2+x+1) (z6+x3+l) E Fz[z] and let F be the subfield of the cyclotomic function field k,u fixed by the cyclic subgroup (Z) of G a l ( k ~ / k ) N (Fz(z]/(M))'. Then (F : k] = 21 and the places 00 and I split completely in F/k. hrthermore, z2 + x + 1 is tamely ramified in F/k with ramification index 3 and z6 + z3 + 1 is totally ramified in F/k, hence g(F) = 54. The function field F was also considered in [22, Example 91. .

Put N = x4 E Fz[x] and let K be the subfield of the cyclotomic function field kiv fixed by the cyclic subgroup (z2 + 1) of G a l ( k ~ / k ) N (Fz[z]/(N))'. Then [K : k] = 4 and Gal(K/k) 11 (Z/2Z)2. Let P be the place of k~ lying over x and X a generator Of the cyclic F2[z] -module AN, If vp is the normalized discrete valuation corresponding to P, then it is shown in the proof of [6, Proposition 2.41 that vp(X) = 1. Thus, by [16, Proposition 111.5.121 the different exponent of P in k N / K is given by

dp(kIV/K) = v p ( x - - x ~ 2 + ' ) = 4 .

1 Ni&rreiter/Xing, Towers of Global Function Fields 181

Furthermore, dp(kN/k) = 24 by [6, Theorem 4.11. If Q is the place of K lying over z, then the tower formula for different exponents shows that

1 dQ(K/k) = ( d ~ ( k ~ / k ) - d p ( k ~ / K ) ) = 10.

NOW we consider the composite field FK of F and K. It is clear that

Gal(FK/F) = Gal(K/k) N (Z/22)2

since F n K = k. The only ramified places in FK/F are those places of F lying over x. Any such place is totally ramified in FK/F, and so its inertia group in FK/F is the whole group G = Gal(FK/F). Let T be a set of rational places of F consisting of 20 places lying over 00 and one place lying over z. Let S be the set of places of FK lying over those in T , then IS1 = 4 . 20 + 1 = 81. By Proposition 3.1, applied to FK/F,

dzCls 2 C d z G ~ - (IT1 - 1) - d2G = 2 1 . 2 - 20 - 2 = 20, R

where the sum is over all places R of F . Thus, the condition (2.2) in Proposition 2.1 is satisfied. I t remains to calculate the genus of F K . The ramified places in FK/F are exactly the 21 places of F lying over 2. For any such place, the different exponent in F K / F is equal to dQ(K/k) = 10. Hence by the Hurwitz genus formula,

2g(FK) - 2 = 4(2g(F) - 2) + 21 * 10 = 634,

and so 81 317’

= - 1st A(2) ’ g ( F K ) - 1 0

Theorem 5.2. We have

62 163 A(3) 1 - = 0.3803 . . . .

Proof . Let k be the rational function field F3(z). Put M = x8 E F3[2] and let F be the subfield of the cyclotomic function field kM fixed by the subgroup of Gd(kM/k) N (Fs[z]/(M))* generated by and m. Then [F : Ic ] = 27 and the places z + 1, z + 2, and 00 split completely in F/k. The only ramified place in F / k is x, and it is totally ramified. Then, by proceeding as in [l l , Example 3.151 or by using the result of [22, Example 41, we get g ( F ) = 69.

Let K = k(y) be the function field defined by

y2 = ( z+ 1)(z + 2 ) .

Then K/k is a Kummer extension in which the only ramified places are z + 1 and x + 2. Now we consider the composite field FK of F and K. The only ramified places in FK/F are those places of F lying over z + 1 or z + 2, and the places of F lying over 0;) split completely in FK/F. Let T be a set of rational places of F consisting of

Math. Nachr. 195 (1998) 182

the 27 places lying over 00 and of 8 places lying over z + 1. Let S be the set of places of F K lying over those in T , then IS1 = 2 . 2 7 + 8 = 62. By Proposition 3.1,

d2Cls 2 C d 2 G p - ( T I - d 2 G = 5 4 - 3 5 - 1 = 18, P

where G = Gal(FK/F) N 2 / 2 2 and the sum runs over all places P of F . Thus, the condition (2.2) in Proposition 2.1 is satisfied. By the Hurwitz genus formula we have

2 9 ( F K ) - 2 = 2(2g(F) - 2) + 5 4 . (2 - 1) = 326,

and so IS1 - 62 - -

A(3) g ( F K ) - 1 163’ 0

Theorem 5.3. We have 2

4 5 ) L 5 . Proof . Let k be the rational function field Fs(z). Let F = k(y) be the function

field defined by y4 = 4 x 4 + 2 .

Then F/k is a Kummer extension in which the places z - 1, z - 2, z - 3, and z - 4 split completely and 00 splits into two places of degree 2. The function field F was considered in [ll, Example 5.31, and we have g ( F ) = 3.

Let K = k(z) be the function field defined by

z2 = (z - 1)(x - 2)(z - 3)

Then K / k is a Kummer extension in which the only ramified places are x - 1, z - 2, z - 3, and w. Now we consider the composite field FK of F and K. The only ramified places in F K / F are those places of F lying over z - 1, z - 2, x - 3, or 00. Let T be the set of places of F lying over x - 4 and S the set of places of FK lying over 2 - 4. Then IT[ = 4 and IS[ = 8. By Proposition 3.1,

d2Cls 2 Cd2Gp- ITI -d2G = 1 2 + 2 - 4 - 1 = 9 , P

where G = Gal(FK/F) N 2 / 2 2 and the sum runs over all places P of F . Thus, the condition (2.2) in Proposition 2.1 is satisfied. By the Hurwitz genus formula we have

2g(FK) - 2 = 2(2g(F) - 2) + 12 a (2 - 1) + 4 . (2 - 1) = 24,

and so 2 IS1 = -

A ( 5 ) ’ g ( F K ) - 1 3 ’

Niederreiter/Xing, Towers of Global Function Fields 183

6.

The Gilbert - Varshamov bound is the classical existence result for good linear codes Over F,. As an application of the theorems in Section 4, we show that algebraic- geometry codes over F, lead to an improvement on the Gilbert - Varshamov bound for sufficiently large composite nonsquares q. For squares q an analogous improvement

For a linear code C over F, we denote by n(C), k(C) , and d(C) the length, the dimension, and the minimum distance of C, respectively. Let Uin be the set of ordered D&s (6, R) E R2 for which there exists an infinite sequence Cl , Cz, . . . of linear codes

An improvement on the Gilbert - Varshamov bound

established in the well- known paper [18].

over IF, with n(Ci) + 00

6

The following description

and

of Ufi'" can be found in (17, Section 1.3.11.

Proposit ion 6.1. There exists a continuous function atn(6), 6 E (0,1], such that

up = { (6, R) E R2 : 0 5 R 5 c$"(6), 0 5 6 5 1 } .

Moreover, afii"(0) = 1, afii"(6) = 0 for 6 E [ ( q - l ) / q , 11, and aj;'"(6) decreases on the intervai [O, ( q - l)/q].

For 0 < 6 < 1 define the q - ary entropy function

H,(6) = dlog,(q - 1) - 6log,6 - (1 - 6)log,(l- 6),

where log, is the logarithm to the base q, and put

RGV (91 6) = 1 - Hq(6) .

Then the Gilbert - Varshamov bound says that

alin(6) 2 RGv(q,6) for all 6 E 4

F'rom [17, Corollary 3.4.21 we see that Goppa's construction of algebraic -geometry codes yields

1 a!"(6) 2 1 - - - 6 for all 6 E [ O , l ] .

4 s ) (6.2)

Now let q = pn with n > 1 and p a prime, and assume that n is composite if p = 2. Let 1 be the least prime dividing n. Then for 0 5 6 5 1 we define the function

- 6 if q is odd , P(2q'I' + 1)lI2

29'11 R(q,6) = 1 -

[2(2q'/' + 2)'/2] + 2 gill + 1

R(q,b) = 1 - - 6 if q is even.

Math. Nachr. 195 (1998) 184

From (6.2) and the results in Section 4 we obtain

(6.3) crf;'"(6) 2 R(q,6) for all 6 E [0, I].

The following theorem shows that (6.3) improves on the Gilbert - Varshamov bound (6.1) in a certain range of parameters.

Theorem 6 . 2 . Let m 2 3 be an odd integer and let r be a prime power with T 2 100m3 for odd r and T 2 576m3 for even r. Then there exists an open interval (61 ,bz) G (0 , l ) containing ( rm - 1)/(2rm - 1) such that

R(Tm,6) > &'V(Tmld) for dl 6 E (61,62) *

Proof. It suffices to prove the inequality for 60 = ( r m - 1)/(2rm - 1). Note that with q = rm we have

H,(SO) -6, = log,

Thus, for odd T it is enough to show that

(27- + 1 y + 1 9 log,(2- f) 5 T

or equivalently

(6.4)

We have loge 1 2 iogt 5 F t 1 for t 2 c 2 e 2 .

Hence, for T 2 100m3 and m 5 3 we get

4.61 + 3M(m)m1/' 10 m3/2 I

with M(m) = m-'/'logm for m = 3, for m 5 9 that

( (2T + 1)1/2 + 1) logr 5

5

< -

5, 7 and M(m) = 2e-I for m 2 9. It follows

(1.44)~ (0.77) m1j2 + 3 e-Im1/' 5m3/2

T (0.6) - m

m log(2 - f).

Njederreiter/Xing, Towers of Global Function Fields 185

The last bound is easily checked for m = 3, 5, 7 , and so (6.4) is established. For even T it suffices to show that

or equivalently,

For T 2 576m3 and m 2 3 we obtain

lOgT 5 M 5 7 6 m 3 ) T1/2

24 m312 6.36 + 3 M ( m ) m 1 I 2 T112

24 m3I2 2.12 + M(m)rn'l2

8 m3I2

5

< (0 .36) (2 (2~ + 2)'12 - 3) . -

It follows for m 2 9 that

(0.71) m112 + 2e-'m'12 8 m3I2

( 2 . 8 8 ) ~ (2(2T + 2)'12 -k 3 ) lOgT 5 T 5 (0.6) - m

T

- m

and the last bound is easily checked for m = 3, 5 , 7 . 0

For some small m, for instance m = 3, 5, we have

R ( T ~ , ~ ) > R G V ( T ~ , ~ )

for all 6 in some open interval containing ( T ~ - l ) / ( 2 r 3 - 1) if (i) T is odd and T > 2294 or (ii) T is even and T > 13822; and

R(T5,6) > R G V ( T ~ , ~ )

for all 6 in some open interval containing ( r 5 - 1 ) / ( 2 ~ ~ - 1) if (i) T is odd and T > 8666 or (ii) T is even and T > 48999.

References [I] ARTIN, E., and TATE, J . : Class Field Theory, Addison- Wesley, Reading, MA, 1990 [2] CASSELS, J. W . S . , and FROHLICH, A. (eds.): Algebraic Number Theory, Academic Press, Lon-

[3] GARCIA, A , , and STICHTENOTH, H . : A Tower of Artin-Schreier Extensions of Function Fields don, 1967

Attaining the Drinfeld - Vladut Bound, Invent. Math. 121 (1995), 211 -222

Math. Nachr. 195 (199@, 186

[4] GARCIA, A., and STICHTENOTH, H.: Asymptotically Good Towers of Function Fields over Finite

[5] GARCIA, A., and STICHTENOTH, H.: On the Asymptotic Behaviour of Some Towers of Function

[6] HAYES, D. R.: Explicit Class Field Theory for Rational Function Fields, Tkans. Amer. Math,

[7] IHARA, Y.: Some Remarks on the Number of Rational Points of Algebraic Curves over Finite

[8] LANG, S.: Algebraic Number Theory, Addison- Wesley, Reading, MA, 1970 191 LIDL, R., and NIEDERREITER, H.: Introduction to Finite Fields and Their Applications, revied

[lo] M A N I N , Y u . 1.: What is the Maximum Number of Points on a Curve over F2?, J . Fac. Sci. Univ.

[11] NIEDERREITER, H . , and XING, C.P. : Cyclotomic Function Fields, Hilbert Class Fields, a d

[12] PERRET, M.: Tours RamifiBes Infinies de Corps de Classes, J. Number Theory 38 (1991), 300-

(131 SCHOOF, R.: Algebraic Curves over Fa with Many Rational Points, J . Number Theory 41 (1992),

[14] SERRE, J . -P . : Sur le Nombre des Points Rationnels d'une Courbe AlgBbrique sur un Corps

[15] SERRE, J . - P.: Rational Points on Curves over Finite Fields, Lecture Notes, Harvard University,

[l6] STICHTENOTH, H.: Algebraic Function Fields and Codes, Springer, Berlin, 1993

(171 TSFASMAN, M. A , , and V L ~ D U T , S. G.: Algebraic-Geometric Codes, Kluwer, Dordrecht, 1991 [la] TSFASMAN, M. A., V L ~ D U T , S. G., and ZINK, T.: Modular Curves, Shimura Curves, and Goppa

[19] V L ~ D U T , S. G., and DRINFEL'D, V. G.: Number of Points of an Algebraic Curve, Funct. Anal.

I201 WEISS, E.. Algebraic Number Theory, McGraw-Hill, New York, 1963 [21] XING, C. P.: Multiple Kummer Extension and the Number of Prime Divisors of Degree One in

Function Fields, J . Pure Appl. Algebra 84 (1993), 85-93

[22] XING, C.P. , and NIEDERREITER, H.: Drinfeld Modules of Rank 1 and Algebraic Curves with Many Rational Points, Monatsh. Math., to appear

[23] ZINK, T.: Degeneration of Shimura Surfaces and a Problem in Coding Theory, in Fundamen- tals of Computation Theory, L. BUDACH (ed.), Lecture Notes in Computer Science, Vol. 199, Springer, Berlin, p. 503-511, 1985

Fields, C. R. Acad. Sci. Paris SBr. I Math. 322 (1996), 1067- 1070

Fields over Finite Fields, J. Number Theory 63 (1996), 248-273

SOC. 189 (1974), 77-91

Fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 721-724

ed., Cambridge University Press, Cambridge, 1994

Tokyo Sect. IA Math. 28 (1981), 715-720

Global Function Fields with Many Rational Places, Acta Arith. 79 (1997), 59-76

322

6- 14

Fini, C.R. Acad. Sci. Paris SBr. 1 Math. 296 (1983), 397-402

1985

Codes, Better than Varshamov- Gilbert Bound, Math. Nachr. 109 (1982), 21 -28

Appl. 17 (1983), 53-54

Institute of Information Processing Department of Mathematics Austrian Academy of Sciences University of Science and Sonnenfelsgasse 19 Technology of China A - 101 0 Vienna Hefei, Anhui 230026 Austria P . R . China e - mail niederreiter@oeaw. m a t