Fibre products of hyperelliptic curves and geometric Goppa code

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Fibre products of hyperelliptic curves and

geometric Goppa codes*

S. A. STEPANOV and F. OZBUDAK

Abstract - The purpose of this paper is to extend the results of the first author on construction of fairly long geometric Goppa codes over Fq (q = pv and v > I is even) with rather good parameters to the case of finite fields Fq consisting of q = pv elements, where v > I is an odd integer.

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

The first author was partially supported by the Russian Foundation for Basic Research, grant 94-01-01206-a.

1. INTRODUCTION

Recall the basic ideas of the Goppa construction (see [2, 31) of the linear [n, k, d]q-codes associated with a smooth projective curve X of genus g = g(X) defined over a finite field Fq. Let P =

{P

1, ••• , Pn} be a set of Fq-rational points of X and

Do = P,

+ ... +

Pn.

Let

D

be a Fq-rational divisor on

X

whose support is disjoint with D0 • We consider the

vector Fq-space of rational functions on X

L(D)

=

{f

E Fq(X)*

I

(f)

+

D ~

O}

u

{O},

and denote its dimension over Fq by l(D). · The linear [n, k, d]-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 embedding, hence k = dim C = l(D) and by the Riemann-Roch theorem

k ~ deg D - g + 1 ; in particular, if 2g - 2 < deg D < n, then

k

=

deg D - g

+

1. Moreover, we have

d ~ n - degD.

• UDC 519.72. Originally published in Di.vkretnaya Matematika ( 1997) 9, No. 3 (in Russian). Received August 13, 1996. Translated by the authors.

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Theorem 1. Let v

>

1 be an odd number, Fq be a finite field of characteristic p

>

2

consisting of q = pv elements, and lets be an integer such that

2pv +4

I < s

<

-- p<v-l)l2(p +I)_ 2

Moreover, let r be an integer such that

2·'-2((p<v-l)t2(p

+

1) - 2)s - 4)

<

r

<

2·'pv.

Then there exists a linear [n, k, d]q-code with parameters

r

<

n s; 2·1·pv,

k = r - 2"- 2((p<v-l)t2(p

+

I) - 2)s - 4), d ~ n - r.

Corollary 1. Under the conditions of Theorem 1, there exists a linear [n, k, d]q-code with the relative parameters R

=

kin and

o

=

din such that

2·'-2((p<v-l)t2(p

+

I) - 2)s - 4)

R~l-8-

.

n

In particular, for n

=

2"pv we have

(p<v-l)t2(p

+

1) - 2)s - 4

R~l-8-

.

4pV

2. NOTATION AND LEMMAS

Let Fq be an algebraic closure of the field Fq and A'+1 _{be the (s + })-dimensional affine}

space over Fq.

Lemma 1. Let /1,

Ji, ... ,

f,

e Fq[x] be pairwise coprime square-free manic

poly-nomials of the same degree m ~ 3 and Y be the fibre product in A'+1 _{given over Fq[x]}

defined by the equations

Zi

=f,(x),

z~ = !2(x),

z;

=f,(x).

Then the genus g

=

g(X) of the smooth projective model X of the curve Y is

_ {(ms - 3)2''-2 ₊_I

g - (ms - 4)2"-2 ₊_I

ifm is odd, ifm is even.

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Proof Let/ be the ideal of the curve Yin Fq[x,z1 ... ,zJ and

f'

be the projective

closure of Y in ps+I. The homogeneous ideal of

f'

in Fq[x0,x, z1, ... , zJ has the form

I,.=

{.t,,

If

E

I},

where.fi, is the homogenization off, i.e.,

f1,(xo,X,Z1, ... ,z.,) =f(xlxo,z1lxo, ... ,z)x0)xiegf.

Thus,

Y

= Yu { (0, 0, ±1, ±1, ... , ±1)} as a set, and the curve

f'

is singular at the 2-1 - 1

points P; E {(0,0, 1,±1, ... ,±1)} in general.

Let X be a normalization of

Y

which in the same time is a non-singular model of

Y

(see, for example [3], Chapter 2, 5.3). There exists a finite morphism (regular map) cp1 : X ~

Y

and a composition of cp1 with <f>'l, where <f>'l:

Y

~ P1 via (x0 , x, z1, ••• , z.,) 1--t (x0 , x) gives

a morphism cp: X ~ P1 _{of degree}_2"'_{(see, for example}_[3],_{Chapter 2,}_3.1)._Since

Y

_has

2·1 - 1 _points_P;, _{1 ::;}_i::;_2"-1, at the hypersurfacex0 = 0, the set cp-1(0, 1) consists of2·' or

2s-l points

{Q;}

~ X.

Let Q[Y] be the space of regular differential forms on Y. The space Q[Y], considered as a Fq[x, z1, ••• , z.J-module, is generated by dx and dz;, 1 ::; i ::; s. Since zf = f;(x), the

space Q[Y], considered as a Fq[x]-module, is generated by dx and dx!(z;, .. . z;J, where

1 ::; i1 < ... < ia ::; s. Next, since cp1 is a morphism, the space Q[X] is a submodule of Q[f], hence any differential form w E Q[X] has one of the form

w = F(x)dx, w = F;, .... ;Jx)dx

Z;, ...

z;.

with F, F;,, .. ,ia E Fq[x]. Thus, any regular differential form in Q[f] is regular at any

point of X, possibly except Q; E cp-1

co,

1).

Let x be the coordinate on P1, then u = x-1 _{is a local parameter at the point}_{(0, 1)}

at infinity. Since xis a rational function on P1, it defines the divisor (x) E Div(P1). Denoting cp-1 _(x)E Fq(X) by x and its divisor by (x) again, we get the pull-back divisor

(x) E Div(X).

Since cp-1 _{(0, 1)}_{consists of 2·' or 2·}1_- 1 points Q;, we have vQ;(u) = 1 or vQ;(u) = 2, therefore vQ;(x)

=

-1 or vQ,(x)

=

-2. If F(x) is a regular function on X, we have

vQ,(F(x)dx) = -(deg F(x)

+

2) or -(2 deg F(x)

+

3) respectively. Thus, F(x)dx Ee Q[X]

for any F(x) E Fq[X].

If

mis

even, then there are two cases:

(1) vQ,(x) = -1 and vQ,(Zj) = -m/2 for any j = 1, ... , s,

(2) vQ,(x) = -2 and vQ,(Zj) = -m for any j = 1, ... , s.

Since

for any j

=

1, ... , s, we have

(

F

,, .... ,,.

·

(x)dx) - m<J d F ( ) 2 V _Q; - - - e g · ₂ _{,,, ... ,,.}· X - ,

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or 11 ···'• - 2 d F ( ) 3 (p. · (x)dx) vQ; - mcr - eg ;,, ... ,;. x -Z;1 •••

z;.

respectively. Thus, P. _{,,, ... ,,.}· (x)dx _E _Q[X] Z;1 •••

z;.

if and only if mcr deg P. l(, ... ,ln · (x) < -- 2 - 2 ' or mcr 3 degP. fl, ... ,,. · (x)<- - -- 2 2

respectively. Since m is even the second inequality is equivalent to the first one. If m is odd and vQ;(x)

=

-1, then vQ;(zJ) = 2vQ;(Zj)

=

-m and we arrive at a contradiction. Thus, only one case is possible, where vQ;(x)

=

-2. In this case,

if only P. _{,,, ... ,,.}· (x)dx _E _Q[X] Z;1 •••

z;.

{(mcr - 4)/2 if cr is even, degP. · (x) < ,, ... ,. - (mcr - 3)/2 if cr is odd. Since X is non-singular, we have

g = dim.,...,, Q[X]. Thus, if m is even, then

I .,·

g

= -

L L

(mcr - 2)

=

(ms - 4)2·1

· -2

+

I,

2 _a=I I . . _s,,_{<12, ... <1":s~}.

and if mis odd, then

I ·' I "

g = -

L L

(mcr - 2)

+ -

L L

(mcr - I)

2 a= I I · Sr1 <r2, ... <r(JS,\· · . 2 a= I I<' -ll <12, ... <ln-'' . . <

=

(ms - 3)2"-2

₊

_I,

where the summation is taken over odd cr only. This completes the proof.

Lemma 2.

Let v

>

1 be an odd number, Fq be a finite field of characteristic p

>

2

with q = pv elements, and let f E Fq[x] be the polynomial

(v-1)12 (v+l)/2 f(x)

=

(x+x' )(x+x' ).

If

c is a non-zero element of Fq, then the polynomials f(x) and f(x

+

c) are relatively prime.

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Proof Letµ= (v-l)/2andf'(x) ::,/' +x,J"(x) =~r' +xsothatf'(x)J"(x) =J(x). We shall prove that (f'(x),J'(x + c)) = (1) and (f'(x),J"(x + c)) = (1) for any c E F_{1: ••}

This will imply that (f(x),f(x+ c)) = (1) for any c E F_{1: ••}

Observe that the principal ideal/' generated by J'(x) andf'(x + c) is equal to I'=

(x'

+x,x'" +x+d' +c).

The equation

aif +a=O (1)

has no solution in F;.. Otherwise al'" -1 ₌_-1._Then_afl2

" - 1 = 1, since p is odd and

hence 2

I

(p1 + I). Thus, a E Ffls'"''"+1.i,,i = Fl'. This implies that al'" -1 = I -:t. -1, and we obtain a contradiction.

Observe (using the Euclidean algorithm) that if k, l, k ~ l, are positive integers and

c E Fl'.' then the principal ideal (x" + x + c,x + x) in Ffl.[x] satisfies the relation

(x"

+x

+

c,l +x) =

(:i

+x, -/-I+I +x

+

c).

Similarly,

(-x"

+x+c,l +x) =

(:i

+x,/-1₊1 _+x+c).

Combining these relations, we find that if k ~ 21 - I and k, l are positive integers, then

(x"

+x+ c,l +x)

=

(:i

+x,x"-21₊2 _+x+_c).

By induction, if l

I

k and c E F_1:.,then

(x" +

x + c,l +x) =

(:i

+x,

(-It'x"''

+ x + c).

Applying this relation fork= pi'+1 _and_l₌_pµ,_{we find for the ideal}_I"₌_(f"(x_{+ c),f'(x))}

that

I"= (x'"

+

X,

-x'

+

X +er'+ c).

Now we observe that (g' (x), g" (x)) ::J (g'(x), (g"(x))f') for any h', h" E F₁,.[x]. There-fore

111 _::J

1

₌

cx

_1'₊_{x, -} xr> ₊

x'

₊y''"•> ₊_r)

where y = d'. We can simplify the generators of J as

/' ~,+I f+I

1 =

ex

+ x,

-x' -

x +

r

+ r)

,>' ~1+1 f+I

=

(x +x,x' +x - y' - y). Let us show that

,,..,

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Since y = d1 _and_d''_{= c E}_F_1:,,_{we can rewrite the inequality (2) in the form}

(3) The equation

(4)

has no solution in

F;,.

Indeed, raising both sides of (4) to the pl'th power, we obtain

(5) Since (I) has no non-zero solution, the last equation also has no solution in

F;,.

µ+I _,,µ+I II ,11+1 µ+l

Nowsincef"(x+c)=x' +x+c +cE I ,x +x-yP -yE ]~/",and

µ+I /'+I

d'

+

C ;it -yl - Y,

we conclude that

I"

= (1).

By symmetry (f"(x),f"(x+c)) = (1), and(f"(x),f'(x+c)) =(]).Using the uniqueness of factorization in FP, [x], we find that

(f'(x),f'(x+c)f"(x+c)) = (]),

and hence (f(x),f(x + c)) = (1 ).

(f11_(x),f1_(x+c)f11_(x+c))₌_(1),

Let

e:

Fq ~ Fq be the Frobenius automorphism of Fq over F1,, namely, B(x) = x''.

Let

x

be a multiplicative character of Fl'. We denote by Xv the character of Fq induced by

x:

xvCx)

=

x(normv(x)), XE Fq,

where

normv(x)

=

xB(x) ... ev-'(x)

=

xJ!' .. . J!,,_,.

It is easy to see that if p and v are odd numbers, Xv is induced by a non-trivial quadratic

(v-1)12 (v+l)t2

character of Fp, andf(x) = (x + x'' )(x + xP ), then (see [1], Lemma 2)

Let xv<J(x)) = {

~

ifx E

F:,

if X = 0. - f (X) .J,i,-1)12-1 .JJ(HIV2- I f(x)=-_{2 =(l+x} )(l+x ). X

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Lemma 3. Let v

>

1

be an odd integer; Fq be a finite field of characteristic p

>

2

with q

=

pv elements, c1, ... ,c.,. be distinct elements of Fq, and let Nq be the number of Fq-rational points of the affine curve Y defined by the equations

2 -Z1 =fi(x) =f(x+ c1), 2 -z2

=

fz(x)

=

f(x

+

c2),

z;

=

fix)

=

l(x

+ c.,). Then

Proof Since xif;(x))

=

xv<J(x

+

c;))

=

1 for all x E Fp•, i

=

1, ... , s, we have Nq

=

L

(1

+

xv<fi(x))) ... (1

+

Xvif.,(x)))

xeFP"

=

L

2"

=

2·''pV.

xeFJJ"

3. PROOF OF THE THEOREM

We consider the affine curve

Y: Z; 2 = f;(x) = f(x -

+

c;), 1 ~ i ~

s,

where c1, ••• ,

c.,

are distinct elements of Fq. The number of Fq-rational points of Y is Nq = 2·'·q by Lemma 3. The curve Y satisfies the conditions of Lemma 1, so the genus

g

=

g(X) of its smooth projective model X is

g

=

2.,·-2((p<v-l)t2(p

+

1) _ 2)s _ 4)

+

1.

Let S be the set of rational points on Y and S1 c S be a subset of S. Applying Goppa's

construction to

and

D = rP_,

where r

<

deg D0

=

IS

1

I

and P _ is the point of X corresponding to the point at infinity of

the projectivization

Y

of the affine curve Y, we get r

<

n ~ 2"pv, k;;:: r+ 1 - g, d;;:: n - r. Since in our case 2g - 2

<

r

=

deg D

<

n, we obtain k

=

r + 1 - g.

REFERENCES

I. S. A. Stepanov, Codes on fibre products of hyperelliptic curves. Discrete Math. Appl. ( 1997) 7, 77-88. 2. V. G. Goppa, Codes on algebraic curves, Soviet Math. Dok/. ( 1981) 24, 170-172.