Codes on fibre products of hyperelliptic curves

Discrete Math. Appl., Vol. 7, No. I, pp. 77-88 (1997) @ VSP 1997.

Codes on fibre products of hyperelliptic curves*

S. A. STEPANOV

Abstract- The purpose of this paper is to construct a new family of smooth projective curves over a finite field Fq with a lot of Fg-rational points. The genus in this family is considerably less than the number of rational points, so that the corresponding geometric Goppa codes have rather good parameters.

The work was supported by Bilkent University, 06533 Bilkent, Ankara, Turkey.

1. INTRODUCTION

Let X be a smooth projective curve of genus g = g(X) defined over a finite field k' =

Fq. Recall the basic ideas of the Goppa construction [3] of the linear [n, k, d]q -codes associated with the curve X. Let { x1, ••• , xn} be the set of k' -rational points of X and

Do = X1

+ ... +

Xn.

Let D be a k' -rational divisor on X. We assume that D has the support disjoint from D0 , i.e., the points x;, i = I, ... , n, occur with multiplicity zero in D. Denote by k'(X) the field of rational functions on X and consider the vector space over k'

L(D) =

{!

E k'(X)*

I

(f)

+

D ~

O}

u

{O}.

The linear [n, k; d]q-code C = C(D0 , D) associated with the pair (D0 , D) is the image of the linear evaluation map

Ev: L(D) ~

F;,

Such a q-ary linear code is called a geometric Goppa code. If deg D < n, then the map Ev is an injection, so that C -::: L(D).

Dually, denote by Q(X) the k' (X)-vector space of rational differential forms on X and consider the linear space over k'

Q(D0 - D) = { w E Q(X)*

I

(w)

+

D0 - D ~

O}

u

{O}.

The linear map

Res: Q(D0 - D) ~

F;,

W ~

(ResxJ

w ), ... , Resx.( w))

• UDC 519. 72. Originally published in Diskretnaya Matematika (1997) 9, No. I, 83-94 (in Russian). Received May 12, 1996. Translated by the author.

defines the linear [n,k,d]q-code C* = C*(D0,D) associated with the pair (D0,D). If

deg D > 2g - 2, then the map Res is injective, so that C* '.:::' D.(D0 - D) '.:::' L(K

+

D0 - D),

where K is a canonical divisor on X.

Each linear [n, k, d]q-code C defines a pair of its relative parameters ( 8, R), where

8 = din is the relative minimum distance and R is the transmission rate of C. The points ( 8, R) form the set of code points V!in i;;;; [O, 1 ]2 . Let

u~n

denote the subset of limit points of V!in. In other terms, ( 8, R) E

u~n

if and only if there exists an infinite sequence of

different linear codes C; with relative parameters 8; = 8( C;) and R; = R( C;) such that lim(8;,R;) = (8,R).

i~oo

If 8 > 0 and R > 0, then such a family of codes C; is called asymptotically good. The structure of

u~n

can be described as follows (see [1, 4]): there exists a continuous function cx~n( 8) such that

u~n

= {(8,R)

IO~

R ~ (X~n(8)},

moreover, cx~n(O)

=

1, cx~n(8)

=

0 for (q - l)lq ~ 8 ~ 1, and cx~n(8) decreases on the interval [O, (q - l)lq].

It follows from the Riemann-Roch theorem that the relative parameters R = kin and

8

=

din both for L-and Q.-constructions satisfy (see [8, 10]) the inequality

g - 1

R~ 1- 8- - - .

n (1)

In order to produce a family of asymptotically good geometric Goppa codes for which R + 8 comes above the Gilbcrt-Varshamov bound

where

Hq(8) = 8 log/q - 1) - 8 logq 8 - (1 - 8) log/1 - 8),

one needs a family of smooth projective curves with a lot of k' -rational points compared to the genus. Examples of such families are provided by classical modular curves X0(N) and X(N) (see [5, 9]), or by Drinfeld modular curves (see [10], Chapters 4.1 and 4.2). Thus, if q = pv is an even power of a prime number p, then there exists an infinite

sequence of geometric Goppa codes C; which gives the lower bound

The line R

=

1 - 8 -

(..fij -

0-

1 _{intersects the curve R}

=

_{1 - Hq(8) for q} ~ _49. Much easier proof of this result based on consideration of a sequence of (modified) Artin-Schreier coverings of the projective line P1 _{(k.') was recently proposed by Garcia} and Stichtenoth [2].

Codes on fibre products of hyperelliptic curves 79

In this paper we consider a new family of smooth projective curves X.,· given over

k' = Fq by equations

z~ = f;(u), I $; i $; s, (2)

wheref;(u) are relatively prime square-free polynomials in k'[u] of a special form. Every such curve is actually a fibre product of hyperelliptic curves. The main point of the paper is to calculate the genus g(X.,) (Lemma I) and determine the number Nq(X.,) of k'-rational points (Lemma4) on the curveX., .. We show that the ratio g(X.,)!Nq(X.,) is small enough and deduce from (I) that the corresponding geometric Goppa codes C(D,D0 )

and C*(D,D0 ) have rather good parameters. For small values of

s,

these parameters are comparable with the parameters of codes on Artin-Schreier coverings introduced by Garcia and Stichtenoth [2]. In particular, ifs= 1, then the codes C(D, D0 ) and C*(D, D0 )

have the same parameters as the codes on Hermitian curves (see [8], Section VIl.3). Unfortunately, the parameter

s

in our construction is bounded by q112 _{and as a result the}

genus g(X.,) is bounded by

(q - 3)iq'12 - 2>

+

I.

However, since the above upper bound is large enough for q ;;;>; q0 , the curves

X.,

provide sufficiently long geometric Goppa codes.

A similar construction of non-singular projective curves with a lot of k' -rational points based on the use of fibre products of some Artin-Schreier curves was independently considered by van der Geer and van der Vlugt [11].

The genus g(X,.) can be easily calculated using the Hurwitz genus formula. However, we prefer to use a slightly more complicated argument which allows us to find explicitly a basis of the space Q(D0 - D). This provides an easy way to write out the generator matrices for codes in the family and to find a fast decoding algorithm.

Applying to curves X.,· the Goppa constructions, we obtain the following results.

Theorem 1. Let p

>

2 be a prime number, v

>

1 be an even integer, and let Fq be a finite field consisting of q = pv elements. For any positive integers s $; q111 and l

>

(sq 1'1 - 3)2"-2 _{there exists a geometric Goppa [n, k, d]q-code C = C(D0 , D) with} parameters

l < n $; (2q 111 - s)q1112·'- 1, k;;;,; l - (sq1,1 - 3)2·'·-2,

d;;;>;

n-

l.

Theorem 2. For p, q and s as in Theorem 1, and for any positive integer l

>

(sq1' 2 - 3)2"-1 _{there exists a geometric Goppa [n,k,d]q-code C*}

=

_{C*(D0,D) with} parameters

l - (sq111 _ 3)2.,·-2

<

n $; (_{2q1,1 _ s)q1,12.,·-1,}

k ~ n - l

+

(sq1'1 - 3)2"-2, d ~ l - (sq 111 - 3)2.1"-1•

Corollary 1. The relative parameters R

=

kin and

o

=

din of the above codes satisfy the inequality (sq112 - 3)2·1- 2 R ~ l o -n In particular; for n = (2q112 - s)q1'22s-I sq1,2 - 3 R > 1 o -- 2(2q1,2 _ s)q1,1

By a suitable concatenation one gets reasonably good codes over

FP.

Indeed, let

ko > 1 be an even number. Applying a linear [no, ko, do]p-code C0 to an [n, k, d]q-code

C = C(D0 , D) over Fq, where q = pko, we obtain an [n', k', d']p-code C' with parameters

n'

=

non, k'

=

k0k, d'

=

d0d.

Let us denote by R0 = kolno and

Co

= d0ln0 the relative parameters of the code C0•

Corollary 2. For any positive integers n0 > 1, s ::;; q112 and l > (sq 111 - 3)2s-2 there exists a linear [n', k', d']p-code C' with parameters

nol < n' =non::;; no(2ql/2 - s)q1112s-1. k' ~ ko(l - (sq 112 - 3)2"'-2),

d' ~ do(n - l).

The relative parameters R'

=

k' In' and

o'

=

d' In' of the code C' satisfy the inequality

/ / ( l

(sq111 - 3)2·1

· -2) (

l)

R +

o

~ Ro - - +

Co

1 - - .

n n n

Applying a linear [no, k0 , do]p-code C0 to a linear [n, k, d]q-code C* = C*(D0 , D), we

obtain the following result.

Corollary 3. For any positive integers n0

>

1, s ::;; q1' 2 and l

>

(sq 111 - 3)2"'-2 there exists a linear [n", k", d"]p-code C" with parameters

no(l - (sql/2 - 3)

<

n" =non::;; no(2ql/2 - s)q1,22s-1. k" ~ ko(n - l + (sq 1'1 - 3)2·1_{· -2),}

d" ::;; d0(l - (sq112 - 3)2·'-1).

The relative parameters R11

=

_{k"ln" and}

o" =

_{d"ln" of the code C" satisfy the inequality} II II ( l (sq1,2 - 3)2·•·-2) ( l (sq112 - 3)2s-l)

R

+o

~Ro 1--+ +Co - - .

n n n n

The results of this paper can be extended to the case of fibre products of more general form over an arbitrary finite field.

Codes on.fibre products of hyperelliptic curves 81

2. NOTATION AND LEMMAS

Let k" be the algebraic closure of k' = Fq and A.<+1 _{be an (s + 1 )-dimensional affine space} over k". Assume that char k' > 2.

Lemma 1. Letf1, ••• ,f, be pairwise coprime square-free manic polynomials in k'[u] of the same odd degree m ?: 1 and Y be the fibre product in A.<+1 _{given over k' by the} equations

zf = f;(u), 1 ~ i ~ s.

Then the genus g = g(X) of a smooth projective model X of the curve Y is equal to

g = (ms - 3)2.,._2 _{+ 1.}

(3)

Proof LetXbe a smooth projective model of the curve Y. Denote by Vx the canonical valuation of the function field k" (X), and by .Q[X] the space of regular differential forms on X. The affine curve Y is easily seen to be smooth. If

Y

is its projective closure, then

X is a normalization of Y and we have the map 1/f: X ~

Y

which is an isomorphism between Y and 1J!-1_{(Y). Hence it follows that g = g(X) = g(Y).}

The rational map (u, z1, .•. , zs) ~ u of the curve Y in A I determines a morphism

<p: X ~ P1 _{of degree 2.,., so that for u0 E A} I _{either cp-}1 _{(u0 ) consists of 2·' points of the} form x' = (u0 ,

±z

1 , •.• ,

±z,)

at each of which vxi(t) = l for the local parameter t at u0 , or

else cp-1 _{(u0 ) consists of 2·}1·-1 _{points of the form x;'}

=

_{(u0 ,}

_±z

_{1, ... ± Z;- 1, 0,}

_±z;+

_{1, ... ,}

_±z.,),

and Vx" (t) = 2.

L~t us consider the point at infinity u~ E P1• If the coordinate on A 1 is denoted by u, then t = u-1 _{is the local parameter at u~. If cp-}1_{(u~) consisted of 2.,. points}

x~>,

_then

at each

x~

=

x'.:>

the function t would be the local parameter. Hence it would follow that

vx_(t)

=

l and vxJf;(t))

=

-m. But since m is odd, this contradicts the condition that

vxJf;(u))

=

2vx_(z;). Thus, cp-1_{(u~) consists of r}

=

_{2s-I points}

x~>,

₁ ~ _r ~ _{r, with the}

projectivecoordinatesx::> = (0, 1,±1, ... ,±1,0). ItfollowsthatX = Yu{x~l}u ... u{x~l}. At any such point

x~

=

x::>

we have vx_(u) = -2 and Vx_(z;) = -m.

Let us now find a basis of the space .Q[X] over the field k". Any element ro E .Q[ Y]

can be written as a k" -linear combination of the differential forms % = P0(u)du and (/). . __ P;, .. ;0(u)du

11, ... ,lo - - - - ,

Z;, .. -Z;0

where i1, ... , (, are integers such that 1 ~ i1 < , .. < ia ~sand P;,, .. ,;0 are polynomials in

k"[u]. Indeed, the differential form

I du wii, .. ,ic, =

Z;, .. -Z;0

is regular at any point u0 E A 1 with the condition z;(u0 ) '# _{0 for i E { i1, ... , ia}.} Now if z;(u0 ) = 0 for an unique i E { i1, ... , ia}, then Z; is the local parameter at

x:'

=

(uo, ±zi, ... , ±z;-1, 0, ±z;+J, ... , ±z.,.), so that Vx:,(z;)

=

1 and Vx:r(U - Uo)

=

2. Therefore

vx1,(du) = 1 and again

o( ... ,;

0 is regular at u0 • The form% = du is also regular at any

point u0 E A 1. Thus, the differential forms % = du and

o( ... ,;

0 form a basis of the

k"[u]-module Q[Y].

It remains to clarify which of the forms 0-0 and

w;, ... ;

0 are regular at the points

x~l, ... ,x~l. Let x_ be one of these points. If t is the local parameter at x_, then

u

=

t-2_{u', z;}

=

_{t-mz;, where u' and z; are units in the local ring Ox_· Therefore}

w:, .... ,;

0

=

tme1-31li.. ... ,;0dt, where 71;1,. .. ,;0 is a unit in Ox_, hence

(w:, .... ,;) =

(mcr - 3)x_.

Thus, the differential form

P;,, ... ,;)u)du

Wj,, ... ,ia =

-Z;1 ... z;a is regular at x_ if and only if

vx_(P;,, ... ,;)u)) ~ -(mcr - 3). This means that deg P;,, ... ,;)u) ~ (mcr - 3)/2 and hence

e · · u <

d P _{ (mcr - 4)/2, if cr

=

0 (mod 2), g ,, ... ,) ) - (mcr - 3)/2, if cr

=

I (mod 2).

The differential form 0-0 = P0du is not regular at

x_

for any non-zero polynomial

P0 E k"[u], therefore the regular differential forms

w' .

l(, .. ,,la'

uw' . ,

IJ, .. ,,lu

,u"w' . ,

11, ... ,lq

where I ~ i1

< ... <

ia ~sand

n = {(mcr - 4)/2, if cr

=

0 (mod 2) (mcr - 3)/2, if cr

=

1 (mod 2), form a basis of the space Q[X] over k". Therefore

dimk" Q[X]

=

1

L

L

(mcr-2)

+

~

L

L

(mcr- 1)

C1=<l (mod 2) ]Si1 < ... <i0 s.1· 2 a,;;J (mod 2) ISi1 < ... <i0S.r

and hence

=mt cr(s) _

L

(s) _

~

L

(s)

2 a=I (1 a;e() (mod 2) (1 2 a,;;J (mod 2) (1

1

= -(ms2"-1 - 2" - 2"-1

₊

₂₎

2

g

=

g(X)

=

dimk" Q[X]

=

(ms - 3)2"-2

₊

_I.

Codes on.fibre products of hyperelliptic curves 83

Let p be a prime number, v be a positive integer and let Fq be a finite field with

q = pv elements. The field Fq is a Galois extension of the prime finite field FP of degree v with the cyclic Galois group of order v. The action of a generator

e

of this group on an element x E Fq is given by the rule El(x) = ;,!'. The map

I • - I

normv(x) = xlJ(x) ... ev- (x) = d ... x'

of Fq onto Fp is the norm of the element x.

Let

x

be a multiplicative character of the field FP and x an element of Fq. Set

xvCx) = x(normv(x))

and call Xv a multiplicative character of the field Fq induced by the character X·

Now let/ be a square-free polynomial in the ring Fq[u] of degree m and let

x

be a non-trivial quadratic character of FP. Consider the character sum

SJJ) =

L

xv<f(u)) =

L

x(normv(f(u)))

ueFq ueFq

and recall the well-known Weil bound [12] (see also [7], Chapters 1 and 5)

[m -

1]

I

Sv(f)

1~

2 - 2 - q1,2_

The following result of the author (see [6], Theorem 3) shows us that the Weil bound cannot be sharpened essentially in any extension Fq of the field Fp.

Lemma 2. Let Fq be a finite field with q = pv elements of characteristic p > 2 and let Xv be the character of Fq induced by a non-trivial quadratic character X of the field FP.

If

v > I, then for the square-free polynomial/ E Fp[u],

we have { .12 u+ uP f(u) = p<•'-1v2 p<-+1)12) (u

+ u

)(u + u , {(ql/2 _ l)ql/2 LXvCf(u)) = q - 1 ueFq

if

v = 0 (mod 2),

if

v

=

1 (mod 2),

if

v = 0 (mod 2),

if

v

=

1 (mod 2).

Proof Let v

>

1 be an even number. Since uP" = u in Fq, for any u E Fq we have

V V

normvetcun= I1cu+uP·'2

i-l

= ITcuP·-I +uP·

12··-

1)

i=l i=l

v/2 v/2

= Il(uPi-1 + uPvf2+i-l) ITcuP•f2+j-l + uP-1)

i=l j=l

v/2

: Il(uPi-1 +uP•l2+i-1)2. i=I

Therefore

L

Xvlf(u))

=

L

x(normvlf(u)))

=

q - N,

ueFq ueFq

where N is the number of elements of the set A = {u e Fq

I

f(u) = O}. Since

f(u) = u(l

+

uP''2

- 1), we have A= {O} u B, where

B

=

{u

e Fq

I

I+ uP''2-1

=

_O} is the set of roots of the polynomial 1 + uP"2

- 1 in Fq. Taking into account that the greatest common divisor of (pv12 - 1 and pv - 1) is equal to pv12 - 1, we obtain from the Euler criterion that the number of roots of the polynomial 1

+

uP·"-t is equal to pv12 - 1. In that case

N

=

IAI

=

1

+

IBI

=

1

+

(pv12 - 1)

=

ql/2,

and hence

L

Xvlf(u)) = (ql/2 - l)q112 •

ueFq

This proves the lemma for

v

an even positive integer.

Let now

v

> 1 be an odd number. In this case for any

u

e Fq

V

normvlf(u)): II<uPi-1 + uP(v-lV2+/-l)(uP/-I + ,/(v+IV2+H)

i=I

(v-1)!2 V

: II

(iii-I +uP(,-IV2+i-l)

II

(uPi-1 +uP(v+IV2+i-l)

i=I i=(v+l)/2

(v-Jn V

X

II (

uPH + uP(v+IV2+/- I)

II (

uPi-1 + uP(v+IV2+i-I)

i=I i=(v+l)/2

(Min (v-Jn

: II

(uPi-1 + uP(,-IV2+i-l)

II

(uP(v+IV2+j-l + ,1-1)

~I ~I

(v-tn (Min

X

II

(uP1-1 + uPcv+1v2+1-1)

II

(uPc,-1v2+;-1 + ,1-1)

i=l j=l

(v+l)/2 (v-1)/2

=

II {

uPi-I + uP(v- lV2+i-I

)2

II {

uP/-1 + uP(>+IV2+/-I

)2

i=l i=l

and hence

LXvlf(u))

=

I:x<normvlf(u)))

=

q - N',

ueFq ueFq

where N' is the cardinality of the set A

= {

u e Fq

I

f(u)

=

O}. Clearly, N'

=

1 and therefore

L

Xvlf(u))

=

q - l.

ueFq

Codes on fibre products of hyperelliptic curves 85

Lemma 3. Let FP be a prime finite field of characteristic p > 2, Fq

=

FP, be an extension of FP of even degree v > land let A be the set of roots in Fq of the polynomial

'"

f(u) = u +

uP .

Then

(i) A is a subgroup of the additive group F; of the field Fq;

(ii) if {A1 = A,A2, ••• ,A,} is the set of all cosets in F;1A and

{a

1

,a

2 , .••

,a,}

are

distinct representatives of the cosets, then the polynomials ,12

f;(u) = (u

+

a;)+ (u

+

a;)P , 1 :;; i:;; r,

(4)

are pairwise coprime in Fq[u];

Proof The main point is (i). First of all we note thatf(O) = 0. Now if a and

/3

are

zeros off(u), then

v/2 v/2 v/2

f(a

+

/3)

=(a+

/3)

+(a+ /3Y =a+ aP

+

(/3

+ f3P )

=

f(a) + f(/3)

=

0,

so that a+

/3

is also a root of the polynomialf(u). Thus, A is a subgroup of F;.

To prove (ii), let us suppose thatf;(u) andJ;(u) for i =t: j have a common root in Fq,

say u =

e.

In that case

and therefore

v/2 vf2 v/2 v/2

e+a;+if +af =8+<XJ+eP +af.

This yields

,12

a; -(X_j+(a; - ajf =0,

and we find that a; - aj is a root off(u), hence a; - aj E A. But a; - aj ¢ A according

to the choice of a1, ••• ,

a,,

and we arrive at a contradiction.

Finally, since

JAi

= pv12 , we find that

Lemma 4. Let Fp be a prime finite field of characteristic p > 2, Fq be an extension of Fp of even degree v

>

1 and lets ~ q112 be a positive integer. Let Nq be the number of Fq-rational points of the affine curve Y given by equations (2) with the polynomials

,12 f;(x)

=

(u +a;)+ (u + a;f , 1 ~

i

~

s,

defined by (3). Then Proof. We have Nq =

1)1

+ xv(fi(u))) ...

(1

+

xv<fs(u))) ueFq and hence

L

xv<fi1 (u)) ... xv<f;.(u)).

It follows from Lemmas 2 and 3 that

{O, ifu EA;, xv<f;(u))= 1, 'f _{l U E} F\A _{q ;,}

and since any two distinct sets A; and Ai have no common element, we obtain

Nq

=

pv

+

t (:)

(pv _ <1pv12)

=

pv

+ (2

s _ l)pv _ sz"-1pv12

a=I

=

(2pvt2 - s)pvt2zs-l

=

(2ql/2 - s)q112zs-l.

This proves the lemma.

3. PROOF OF THE THEOREMS

Let p

>

2 be a prime number, k'

= Fq

be an extension of a prime finite field Fp of an

even degree v

>

1, and lets~ q112 be a positive integer. Letfi, .. .

,f,

be pairwise coprime polynomials in k' [u] of the same degree q112 defined by (3), and let Y c A'+1 _{be the affine} curve defined over k' by equations (2). Let

Y

c ps+I be the projective closure of Y, and

X be a non-singular projective model of

Y

over the algebraic closure k" of the field k'.

Since the curves

Y

and X are birationally isomorphic, we have g

=

g(Y)

=

g(X), and by Lemma 1

Codes on fibre products of hyperelliptic curves 87

Next, let Nq be the number of k' -rational points of Y and Mq be the number of k' -rational points of X. We have Mq 2'. Nq + 1, and by Lemma 4

Let n :5; Nq be a positive integer, let x1, ••• , Xn be k' -rational points of the curve X at

the finite part of X, and let

x_

be the point of X at infinity. Set

Do =xi+··· +xn, D = lx_.

Applying to X the L-construction for l

>

(sq 112 - 3)2"-2 _{and n}

_>

_{l, we obtain the}

geometric Goppa [n, k, d]q-code C = C(D0 , D) with parameters

This proves Theorem 1.

l < n :5; (2q112 _ s)q1t22 .. -1,

k 2'. [ - g

+

1 = l - (sq 112 - 3)2·1·-2, d:2'. n - l.

Now, applying to X the n-construction for

n

> l -

(sq 112 - 3)2"-2 ,

we obtain the geometric Goppa [n, k, d]q-code C* = C*(D0 , D) with parameters

l - (sq1,2 - 3)2"-2

<

n :5; (2q1,2 - s)q1t22.,-1, k 2'. n - l

+

(::q 112 - 3)2·1_{· -}2 ,' d 2'. l - (sq 112 - 3)2·1- 1.

This gives the result of Theorem 2.

Finally, it follows from (I) that the relative parameters R = kin and 8 = din of the

codes C

=

C(D0 , D) and C*

=

C* (D0 , D) satisfy the inequality

This proves Corollary 1.

(sq1,2 - 3)2·'-2 R : 2 ' . 1 8

REFERENCES

I. M. J. Aaltonen, Notes on the asymptotic behavior of the information rate of block codes. IEEE Trans. Inform. Theory (1984) 30, 84-85.

2. A. Garcia and H. Stichtenoth, A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound. Invent. Math. ( I 995) 121, 2 I 1-222.

3. V. G. Goppa, Codes on algebraic curves. Soviet Math. Dokl. (1981) 24, 170-172.

4. Yu. I. Manin, What is the maximum of points on a curve over F2? J. Fae. Sci. Tokyo (1981) 28,

715-720.

5. C. Moreno, Algebraic Curves over Finite Fields. Cambridge Univ. Press, Cambridge, 1991. 6. S. A. Stepanov, On lower bounds of character sums over finite fields. Discrete Math. Appl. (1992) 2,

523-532.

7. S. A. Stepanov, Arithmetic of Algebraic Curves. Plenum, New York, 1994. 8. H. Stichtenoth, Algebraic Function Fields and Codes. Springer, Berlin, 1993.

9. M. A. Tsfasman, S. G. Vladut, and Th. Zink, Modular curves, Shimura curves, and Goppa codes, better than the Varshamov--Gilbert bound. Math. Nachr. (1982) 109, 21-28.

10. M.A. Tsfasman and S. G. Vladut,Algebraic-Geometric Codes. Kluwer Acad. Pub!., Dordrecht, 1991. 11. G. van der Geer and M. van der Vlugt, Fibre products of Artin-Schreier curves and generalized

Hamming weights of codes. J. Comb. Theory (1995) 70A, 337-348.