*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 D*0 ,
*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(D}

_{0 , }

_{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*(D}

_{0}

_{,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 C*0 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 ::;; q*112 *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 C*0 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 }* _{u}_{0 E }_{A }*I

_{either }

_{cp-}1

_{(u}_{0 ) }_{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 _{(u}_{0 ) }*

_{consists of 2·}1·-1

_{points of the form }

_{x;' }### =

_{(u}_{0 , }_{±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 thek"[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**

**L **

**L**

*(mcr-*2)

### +

### ~

**L **

**L**

**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 **

**L**

*xv<f(u))*=

**L **

x(normv(f(u)))
**L**

*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 **

**L**

*Xvlf(u))*

### =

**L **

**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 pv*12 -* 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 **

**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 **

**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

arezeros *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,*

*2'.*

**k***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 *

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

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