RUUD PELLIKAAN, HENNING STICHTENOTH, AND FERNANDO TORRES

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1 Appeared in: Finite Fields and their Applications, vol. 4, pp , 998. WEIERSTRASS SEMIGROUPS IN AN ASYMPTOTICALLY GOOD TOWER OF FUNCTION FIELDS RUUD PELLIKAAN, HENNING STICHTENOTH, AND FERNANDO TORRES Abstract. The Weierstrass semigroups of some places in an asymptotically good tower of function fields are computed. 0. Introduction A tower F F 2 F 3... of algebraic function fields over a finite field F l is said to be asymptotically good if number of rational places of F m /F l lim > 0. m genus of F m Recently an explicit description was obtained of several asymptotically good towers {}, {2}. The motivation to consider these came from coding theory: such towers give rise to asymptotically good sequences of codes. Although the existence of good codes on or above the Tsfasman-Vladut-Zink bound was guaranteed {7} and even a polynomial construction was given {5}, the methods used (namely, modular curves) and the degree of the complexity of the construction were such that hardly any of the resulting codes were known explicitly. Now that asymptotically good towers (F m ) m of function fields are known explicitly, the next step would be to give an explicit description of the vector spaces L(G (m) ) resp. L(rP (m) ), where G (m) is a divisor (resp. P (m) is a rational place) of F m. The latter space L(rP (m) ) is the F l -vector space of all rational functions in F m that have no poles outside P (m) and pole order at most r at P (m). The first attempts have been made in this direction: these vector spaces were explicitly determined for the fields F, F 2 and F 3, by {8}, and for F 4 over F 6 by {3}, in the tower F = (F m ) m over F q 2 which is given {} by F = F q 2(x ) and F i+ = F i (z i+ ) with z q i+ + z i+ = x q+ i, where x i = z i /x i. In this paper we consider another tower T = (T m ) m over F q 2; this tower was introduced in {2} and seems to be easier to handle than the tower F above. It is defined as follows: T = F q 2(x ) and T i+ = T i (x i+ ) with x q i+ + x x q i i+ = x q i +.. The first and third author were supported by grants of Deutsche Forschungsgemeinschaft DFG.

2 2 R. PELLIKAAN, H. STICHTENOTH, AND F. TORRES By the form of the defining equations it is readily seen that N(T m ), the number of rational places of T m, is at least (q 2 q)q m. The genus g(t m ) is computed by using the theory of Artin-Schreier extensions, and one finds that N(T m ) lim m g(t m ) = q. Hence the tower T is asymptotically good and in fact optimal {2}. This implies that geometric Goppa codes which are constructed by means of this tower T lie on or above the Tsfasman-Vladut-Zink bound, which is better than the Gilbert- Varshamov bound for all q 2 > 25 {7}. The element x T T m has in T m a unique pole that we denote by P (m), and in fact P (m) is a rational place. Hence it is natural to consider the spaces L(rP (m) ) for all m and r. The main result of our paper is Theorem 3. where the dimension of all these spaces is determined. In other words, we describe explicitly the Weierstrass semigroup of P (m), that is H(P (m) ) = {i N 0 there is some f T m having a pole of order i at P (m) and no pole outside P (m) }. We remark that the minimum distance of some geometric Goppa codes is related to Weierstrass semigroups (see {4} and the references therein).. Preliminaries and Notation Throughout this paper, we will use the following notation: K = F q 2 - the finite field of cardinality q 2. F - an algebraic function field of one variable over K. g(f ) P(F ) - the genus of F/K. - the set of all places of F/K. (x) F 0 - the zero divisor of an element x 0 in F. (x) F - the pole divisor of x. (x) F = (x) F 0 (x) F - the principal divisor of x. supp A - the support of the divisor A in F. deg A - the degree of the divisor A. L(A) H(P ) - the K-vector space of all elements x F with (x) F A. - the Weierstrass semigroup of a place P P(F ), i.e. H(P ) = {i N there is some x F with (x) F = ip }.

3 WEIERSTRASS SEMIGROUPS 3 If E/F is a finite extension of F/K and A is a divisor of F/K; con E F (A) - the conorm of A in E/F. We will consider the following tower T = (T m ) m of of function fields T m /K: T m = K(x,..., x m ) with x q i+ + x i+ = x q i x q i + for i =,..., m. This tower was studied in {2}; we need some results from that paper: Proposition.. i) For all m 2, the extension T m /T m is a Galois extension of degree q. ii) The pole of x in T is totally ramified in T m /T, i.e. (x ) Tm = q m P (m) with a place P (m) P(T m ) of degree one. iii) The genus g(t m ) is (q m/2 ) 2 if m 0 mod 2 g(t m ) = (q m+ 2 )(q m 2 ) if m mod 2 Proof. i), ii) see {2, Lemma 3.3}. iii) see {2, Remark 3.8}. 2. The semigroups S m A numerical semigroup is a subset S N 0 having the following properties: i) 0 S; ii) a, b S a + b S; iii) N 0 \ S is finite. The numbers c N 0 \ S are called gaps of S. As an example, consider an algebraic function field F/K and a place P P(F ) of degree one. Then H(P ), the Weierstrass semigroup of P, is a numerical semigroup, and the number of gaps of H(P ) is equal to the genus g(f ) (this is the Weierstrass gap theorem; see {6, p. 32}). In this Section we study certain numerical semigroups S m N 0 which are defined recursively as follows. Definition 2.. ii) S = N 0 and, for m, i) For m, let q m q m 2 if m 0 mod 2, c m = q m q m+ 2 if m mod 2. S m+ = q S m {x N 0 x c m+ }.

4 4 R. PELLIKAAN, H. STICHTENOTH, AND F. TORRES We will prove in Section 3 that S m is in fact the Weierstrass semigroup of the place P (m) P(T m ). Recall that g(t m ) denotes the genus of the function field T m /F. Proposition 2.2. The number of gaps of S m is g(t m ). Proof. Let g m be the number of gaps of S m. Define for a subset S N 0 and c N 0 the set S(c) = {x S x c}. The integer c m is the largest gap of S m (for m 2). So if c c m, then Therefore If m >, then g m = #(N 0 \ S m ) = #({0,,..., c} \ S m (c)). #S m (c) = c + g m if c c m. S m = q S m {x N 0 x c m }. Since c m c m /q and c m q S m, it follows that #S m (c m ) = #S m ( c m q ). Hence c m + g m = #S m (c m ) = #S m ( c m q ) = c m q + g m. This gives the recursion formula g m = q c m + g m. q Now we proceed by induction on m. If m =, then S = N 0. So g = 0 = g(t ). Assume now that m > and g m = g(t m ) as induction hypothesis. Then g m = q c m + g(t m ). q a) If m 0 mod 2 then we obtain from Definition 3. and Proposition 2. g m = q q (qm q m 2 ) + (q m 2 )(q m 2 2 ) = (q m q m 2 ) (q m q m 2 2 ) + (q m q m 2 q m ) = q m 2q m 2 + = (q m 2 ) 2 = g(t m ). b) If m mod 2, then g m = q q (qm q m+ 2 ) + (q m 2 ) 2 = (q m q m+ 2 ) (q m q m 2 ) + (q m 2q m 2 + ) = q m q m+ 2 q m 2 + = (q m+ 2 )(q m 2 ) = g(t m ).

5 WEIERSTRASS SEMIGROUPS 5 3. The Main Result We consider again the tower of function fields T = (T m ) m over the field of constants K = F q 2; i.e., T = K(x ) and T i+ = T i (x i+ ) with x q i+ + x i+ = x q i x q i +. Recall that H(P (m) ) denotes the Weierstrass semigroup of the unique pole P (m) of x in T m, and that the numerical semigroup S m and the number c m are given by Definition 3.. Our main result is the following: Theorem 3.. H(P (m) ) = S m. The proof will be given in this Section. Proposition 3.2. Suppose that for all m there exists a divisor A (m) of T m with the following properties: i) A (m) 0 and deg A (m) = c m g(t m ); ii) dim L(c m P (m) ) =. Then we have H(P (m) ) = S m, i.e., Theorem 3. holds. Proof. The assertion is trivial for m =, since H(P () ) = N 0 = S. We proceed by induction. Assume that m > and that H(P (m ) ) = S m holds, as induction hypothesis. We have from i) and ii) that deg (c m P (m) ) = g(t m ) and dim L(c m P (m) ) =. This means that c m P (m) is a non-special divisor of T m (see {6, p.33}). Hence for any divisor B c m P (m) one has In particular we obtain for c c m + So c is a non-gap of P (m) the extension T m /T m, As c m q S m we conclude that dim L(B) = deg B + g(t m ). dim L((c )P (m) ) = c g(t m ), dim L(cP (m) ) = c + g(t m ). for all c > c m. Moreover, since P (m) q S m = q H(P (m ) ) H(P (m) ). S m = q S m {x N 0 x c m } H(P (m) ). is totally ramified in

6 6 R. PELLIKAAN, H. STICHTENOTH, AND F. TORRES By Proposition 3.2 and the Weierstrass gap theorem, both semigroups S m ) have the same number of gaps, namely g(t m ). Hence H(P (m) H(P (m) ) = S m. and It remains to prove the existence of divisors A (m) as in Proposition 3.2. The following elements π T m will play a crucial role. Definition 3.3. For m we define π = (x q i + ) and Z (m) Lemma 3.4. by i= = {P P(T m ) P is a zero of x q i where B (m) +, for some i {,..., }}. i) Let m. Then the principal divisor of π in T m is given (π ) Tm = B (m) 0 is a divisor of T m with (q m q m )P (m), supp (B (m) ) = Z (m). ii) Let m and 0 e q. Then the principal divisor of π x e + in T m is given by where C (m),e (π x e +) Tm = C (m),e 0 is a divisor of T m with (q m q m + eq m )P (m), supp (C (m),e ) Z(m). Proof. i) We proceed by induction on m. The case m = is trivial. Now let m 2 and assume that the assertions hold for m and all =,..., m. a) m. Then π T m T m, and by the induction hypothesis, (π ) T m = B (m ) Observing that P (m ) where B (m) Z (m) (π ) Tm = B (m) (q m q m )P (m ). is totally ramified in T m /T m, we obtain = B (m) is the conorm of B (m ) (q m q m ) qp (m) (q m q m )P (m), are exactly those of T m lying above Z (m ). in T m /T m. Note that the places of

7 WEIERSTRASS SEMIGROUPS 7 b) = m. The field H m = K(x 2,..., x n ) is isomorphic to T m, and we write m π m = (x q + ) ρ with ρ = (x q i + ) H m. By induction hypothesis, the principal divisor of ρ in H m is i=2 (ρ) H m = C (q m )Q (m ), where Q (m ) P(H m ) is the unique pole of x 2 in H m and C 0 is a divisor of H m whose support is the set of all zeroes of x q 2 +,..., x q m + in H m. By {2, Lemma 3.2 and 3.3}, Q (m ) splits in T m /H m as follows: con Tm ) = P (m) + Q (m). Hence we obtain H m (Q (m ) Q (m) Z (m) (π m ) Tm = (x q + ) Tm + con Tm H m (C (q m )Q (m ) ) = q m Q (m) Z (m) Q (m) (q )q m P (m) + con Tm H m C (q m )(P (m) + = con Tm H m C + Q (m) Z (m) Note that the support of the divisor B (m) m = con Tm H m C + Q (m) Z (m) Q (m) (q m )P (m). Q (m) Z (m) is Z m (m), as claimed. The proof of ii) is similar; we leave it to the reader. Definition 3.5. For m, let Remark. π L((q m q m )P (m) 3.4. Proposition 3.6. For m, A (m) = L((q m q m )P (m) i.e., the space L((q m q m )P (m) P Z (m) P. Q (m) Q (m) ) ). This follows immediately from Lemma ) =< π > ; ) is one-dimensional.

8 8 R. PELLIKAAN, H. STICHTENOTH, AND F. TORRES Proof. The assertion is trivial for m = since deg ((q )P () A () ) = 0. Suppose now that m 2 and that the Proposition holds for m. We have to show that any z L((q m q m )P (m) a) m. Observe that L((q m q m )P (m) ) can be written as z = α π, with α K. ) T m = L((q m q m )P (m ) A (m ) ). This follows easily from the definition of the divisors A (m) and A (m ) and the fact that P (m) is totally ramified in T m /T m. The latter space is generated by π, by induction hypothesis, hence we have L((q m q m )P (m) Assume now that L((q m q m )P (m) an element in L((q m q m )P (m) ) T m =< π >. ) < π >. Then there exists ) \ T m, and we choose such an element z of minimal pole order at P (m), say v (m) (z) = r (where v (m) denotes the discrete valuation of T m corresponding to the place P (m) ). Let σ Gal (T m /T m ). Then σp (m) = P (m) and σa (m) = A (m), so σz L((q m q m )P (m) one, hence there is some α K such that ) and v (m) (σz) = r. The place P (m) v (m) (σz αz) > r. has degree Since r was chosen to be minimal, we conclude that σz αz T m, so σz αz = β π with β K. But so β = 0 and v (m) (σz αz) > r (q m q m ) = v (m) (π ), σz = αz (with α K ). The order of Gal (T m /T m ) is q, so σ q is the identity and z = σ q z = α q z. As α q = α = it follows that σz = z for all σ Gal (T m /T m ), therefore z T m. This is a contradiction because z L((q m q m )P A (m) ) \ T m. b) = m. We know from a) that dim L((q m q)p (m) m ) =, so dim L((q m )P (m) m ) q. The elements π m x e m (0 e q ) are in L((q m )P (m) Lemma 3.4 ii), and they are linearly independent. Since L((q m )P (m) any element y L((q m )P (m) m ) L((q m )P (m) m ), m ) can be written as y = π m h(x m ) m ), by

9 WEIERSTRASS SEMIGROUPS 9 with a polynomial h(x m ) K[x m ] of degree q. The divisor A (m) m contains all zeroes of x q m + in T m, and these places are not zeroes of π m. So h(x m ) = γ (x q m + ) with γ K and therefore y = π m γ (x q m + ) = γ π m < π m >. Lemma 3.7. Let m/2. Then deg A (m) = q. Proof. Let A (m) i = {P P(T m ) P is a zero of x q i + }. It follows from {2, Lemma 3.6} that for i m/2, deg ( P ) = (q )q i. Since we obtain P A (m) i A (m) = deg A (m) = i= P A (m) i P, (q )q i = q. i= Definition 3.8. We define a divisor A (m) of T m as follows: A () = 0 and, for m 2, m for m 0 mod 2 2 A (m) = A (m) with = m for m mod 2. 2 By Proposition 3.2, the proof of Theorem 3. will be finished when we prove the following Lemma: Lemma 3.9. i) deg A (m) = c m g(t m ). ii) dim L(c m P (m) ) =. Proof. For m =, all assertions are obvious since c = g(t ) = 0 and A () = 0. Now let m 2. a) m 0 mod 2. Then c m = q m q m/2 and g m = (q m/2 ) 2 (see Definition 2. and Proposition.). Hence c m g m = q m/2 = deg A (m), by Lemma 3.7. On the other hand, we have L(c m P (m) by Proposition 3.6. ) = L((q m q m/2 )P (m) m/2 ) = < π m/2 >,

10 0 R. PELLIKAAN, H. STICHTENOTH, AND F. TORRES b) m mod 2. The proof is similar. References. Garcia, A.; Stichtenoth, H.: A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound, Invent. Math. 2 (995), Garcia, A.; Stichtenoth H.: On the asymptotic behaviour of some towers of function fields over finite fields, J. Number Theory 6 (996), Haché, G.: Construction effective des codes géométriques, Thèse, Paris VII Kirfel, C.; Pellikaan, R.: The minimum distance of codes in an array coming from telescopic semigroups, IEEE Trans. Inform. Theory 4 (995), Manin, Y.I.; Vlǎdut, S.G.: Linear codes and modular curves, J. Soviet. Math. 30 (985), Stichtenoth H., Algebraic Function Fields and Codes, Springer Universitext, Berlin-Heidelberg-New York, Springer, Ţsfasman, M. A.; Vlǎdut, S. G.; Zink, T.: Modular Curves, Shimura Curves and Goppa Codes, better than the Varshamov-Gilbert Bound, Math. Nachr. 09 (982), Voss, C.; Høholdt, T.: An explicit construction of a sequence of codes attaining the Tsfasman-Vladut-Zink bound. The first steps, IEEE Trans. Inform. Theory 43 (997), Department of Mathematics and Computing Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands address: ruudp@win.tue.nl Universität GH Essen, FB 6 Mathematik und Informatik, D-457 Essen, Germany address: stichtenoth@uni-essen.de address: fernando.torres@uni-essen.de