Codes on fibre products of Artin-Schreier curves

(1)

Discrete Math. Appl., Vol. 11, No. 2, pp. 133-143 (2001) © VSP 2001.

Codes on fibre products of Artin-Schreier curves*

S. A. STEPANOV and M. H. SHALALFEH

Abstract - The purpose of this paper is to construct a new family of smooth projective curves over a finite field Fq with many Fq-rational points using fibre products of Artin-Schreier curves. We show that for any curve X in this family the ratiog(X)/Nq(X), where g(X) is the genus and Nq(X) is the number of Fq-rational points, is small enough to get geometric Goppa codes with good parameters. This paper extends the results of Stepanov and Ozbudak concerning the construction of long codes.

1. INTRODUCTION

Let p be a prime number, and let Fq be a finite field with q

=

pv elements. A linear

[n, k, d]q-code C is a subspace of F;, where n is the length of the code C, k

=

dimFq C

is the dimension of C, and d is the minimum Hamming distance between the non-zero elements of C. Each linear [n, k, d]-code C defines a pair of its relative parameters (8, R),

where 8 = d / n is the relative minimum distance and R = k / n is the transmission rate of

the code C.

Using the Gappa construction for linear codes associated with smooth projective curves over finite fields, one can prove the existence of long linear codes. Although the geometric Goppa codes associated with maximal curves are interesting codes since they have quite good relative parameters, the length of these codes is bounded by 1

+

q3. This is a direct consequence of Hasse-Weil bound and the fact that the genus of a maximal curve X satisfies the inequality g(X) ::: q(q - 1)/2 (see [10]).

In [7], Stepanov constructed long geometric Gappa codes using fibre products of hy-perelleptic curves over a finite field Fq, where q

=

pv and v is even. In [8], Stepanov and Ozbudak extended this result to the case where v is odd. In [ 4], Ozbudak constructed longer codes than those in [7] and [8] using fibre product of Kummer coverings. The aim of this paper is to construct geometric Gappa codes which are much longer than those in [6], [7], [8], and [4] using fibre products of Artin-Schreier curves.

Throughout this paper, Fq (x) denotes the space of rational functions of the projective line P 1. An element f (x) E Fq (x) is said to be degenerated if there exists g (x) E Fq (x) such that

f

=

gP - g

+

a, where a E Fq, otherwise

f

is called non-degenerated.

If C is a code over Fq and Tr v : Fq ---+ Fp denotes the trace map from Fq to Fp, then

Tr(C)

=

{Trv(c)

I

c E C} is a code over Fp, which is called the trace code of C.

'UDC 519.4. Originally published in Diskretnaya Matematika (2001) 13, No. 2 (in Russian). Received February 2, 2001. Translated by the authors.

(2)

134 S. A. Stepanov and M. H. Shalalfeh

The relation Trv(a)

=

0 if and only if there exists f3 E Fq such that f3P - f3

=

a

connects trace codes with Artin-Schreier curves XI with defining equation

yP - y

=

f(x) E Fq(x). (1) The curve (1) is absolutely irreducible if and only if f(x) is non-degenerated, (cf. [3]). It is

known that if f (x) E Fq [x], deg f (x)

=

m, and (m, p)

=

1, then f (x) is non-degenerated,

(cf. [5], p. 55).

The number Nq (Xf) of Fq-rational points of the smooth projective model of (1) is equal to

Nq(Xf)

=

pN

+

1,

where N

=

l{x E Fq: Trv f (x)

=

O}.

Let A be Fq-linear subspace of Fq[x]; A is called non-degenerated if each f(x) E A is non-degenerated. If we regard A as a vector space over Fp, then we see that there is a basis

{!1 (x), ... , Is (x)} of A over Fp and can write

A

=

Fpfi (x)

+ ... +

Fpfs(x).

We call such a basis a canonical basis if any subset of this basis consisting of equal degree polyn~mials has Fp-linearly independent leading coefficients. Then the system of equations over Fq

Yi - YI= fi(x),

yf -

Y2

=

h(x),

yf -

Ys

=

fs(x)

defines an irreducible curve in the (s

+

1)-dimensional projective space ps+l (cf. [3]).

In [9], G. van der Geer and M. van der Vlugt considered the fibre products

yf -

Yi

=

lliX.Jq+l,

where ai E F; satisfies the equality a(i

+

ai

=

0 if q

=

pv and v even, and p p<v+l)/2+1 (v-1)/2 p<v-1)/2+1

Yi - Yi

=

aix - ai x ,

where ai E F; if q

=

pv and v odd.

As a result, they got smooth projective curves Xr such that g(Xr)

=

I

(pr - 1).,/q /2,

(pr - 1),Jiiq/2,

1 ~ r ~ v /2, v is even, 1 ~ r ~ v, v is odd,

and the number of Fq-rational points of the curve Xr is N (X)

=

lprq

+

1,

q r pr q

+

1,

1 ~ r ~ v /2, v is even, 1 ~ r ~ v, v is odd.

(3)

Codes on fibre products of Artin-Schreier curves 135

Remark 1. For even v, the curve X r is a maximal curve for any r, 1 ::: r ::: v /2. In this paper, we construct a family f;(x} E Fq[x] of non-degenerated polynomials, which form a canonical basis for a linear subspace of Fq[x], and we apply the fibre product (2) to this family of polynomials to obtain smooth projective curves defined over finite fields with many rational points and considerably small ratio g(X)/ Nq(X). We obtain the following results.

Theorem 1. Let q

=

pv and Fq2 be a finite field with q2 _{elements. Then for any integer}

s such that 1 ::: s < q, (s, p)

=

1, there exists a smooth projective curve X s with the genus g(Xs) and the number of Fq2-rational points Nq(Xs), which satisfy the relations

where N1 (s)

=

s - [s/p]

+

2[logp((s(q

+

1) - 1)/q)].

Theorem 2. Let q

=

p2_{v+I and Fq be a finite field with q elements. Then for any} integers such that 1 ::: s < pv-l, (s, p)

=

1, there exists a smooth projective curve Xs with the genus g(Xs) and the number of Fq-rational points, which satisfy the relations

g(Xs) < !(PN2(s) - l)(s(pv+I

+

1) - 1),

where N2(s)

=

(2v

+

l)(s - [s/p]).

We apply Goppa's construction to the family of curves of these theorems to obtain the following result.

Corollary 1. Let Fq2, s, N1 (s) be such as in Theorem 1. Then for any l > !(qN1(s) - l)(s(q+ 1) - 1)

there exists a geometric Goppa [n, k, d]q2-code C(Do, D) with parameters

z < n ::: qN1 (s)+2,

k ::::_ l - !(qNi(s) - l)(s(q

+

1) - 1)

+

1, d ::::_ n -1.

Corollary 2. Let Fq, s, N2(s) be such as in Theorem 2. Then for any

l > !(PN2(s) - l)(s(pv+I

+

1) - 1)

there exists a geometric Goppa [n, k, d]q-code C(Do, D) with parameters

z <

n:::

qpN2(s),

k ::::_ l - !(pN2(s) - l)(s(pv+I

+

1) - 1)

+

1,

(4)

136 S. A. Stepanov and M. H. Shalalfeh

Remark 2. The longest code in Corollary 1 has the length p 2v pv((p-l)p"-1_+v),_{and the}

longest code in Corollary 2 has the length p<2v+l)(p-I)p"-2_,_{these codes are much longer} than those in [ 4], which had already extended the results in [7] and [8]. The longest code obtained in [4] has the length (p - l)P" pv.

Remark 3. The relative parameters R

=

k / n and 8

=

d / n of the codes in Corollaries

1 and 2 satisfy, respectively, the inequalities

(qNi(s) - l)(s(q

+

1) - 1) - 1

R > l - 8 - - - -

_- _qNi(s)+2

₊

₁ _'

(pNi(s) - l)(s(pv+I

+

1) - 1) - 1

R>l-8-

.

- qpNi(s)

+

l

The proof of the main results in this paper are arranged as follows. In Section 2, we give an explicit formula for the genus of the curve (2). In Section 3, we construct a canonical basis for a non-degenerate subspace from Fq [x] of special form and then we calculate the number of Fq-rational points of the curve defined by (2) for this canonical basis. In the last section, we prove Theorem 1, Theorem 2, and their corollaries.

2. THE GENUS CALCULATION

Let Fq be a finite field of characteristic p with q

=

pv elements. Let f(x) E Fq[x] be a polynomial of degree m with (m, p)

=

1. Then the curve X defined over

Fq

by the equation

yP-y=f(x) (3)

is irreducible. The following lemma is well known, but we include the proof because a part of it will be used in the proof of Theorem 3.

Lemma 1. The genus of the curve X is

g(X)

=

!<P - l)(m - 1).

Proof The curve X provides a covering ¢ : X -+ PI _{of degree}_p_{of the projective line}

P1• The only ramification point of¢ is the point at infinity. Let p00

=

[O : l] be the point at infinity and x be the coordinate in A 1. If we take t

=

l / x as a local parameter at

p00 , then the inverse image of p00 is the point x00 in the curve X which corresponds to the

discrete valuation Vx00 with Vx00 (x)

= -

p and Vx00 (y)

=

-m. A local parameter at x00 is

s

=

1/(xayb), where ap

+

bm

=

I. Since Vx00 (1/(xayb))

=

1,

Using d(yP - y) = df(x), or equivalently, dy = - J'(x)dx, we get

((bxf'(x)

+

(5)

Since

Codes on fibre products of Artin-Schreier curves

Vx00(bxf'(x)

+

ay)

=

min{vx00(bxJ'(x)), Vx00(ay)}

=

Vx00(bxf'(x))

=

-mp,

we deduce that the degree of the canonical divisor is

Vx00 (dx)

=

mp - m - p - l.

Thus, 2g(X) - 2

=

mp - m - p - 1, which gives g(X)

=

1(p - l)(m - 1).

Since <p is a covering of degree p of the projective line P 1, we have 2g(X) - 2

=

p(2g(P 1) - 2)

+

deg Rep, where Rep is the ramification divisor of <p.

137

(4)

Since the only ramification point of <p is x00 , Rep

=

ax00x00 • Combining equation (4)

and the value of the genus, we get ax00

=

(p - l)(m

+

1).

Now let A be a linear subspace of Fq [x ]. Let {!1, ... , fs} be a canonical basis of A such that deg .fi (x)

=

mi and (mi, p)

_=

1. Assume that mi are ordered such thatm1 :'.S .•. :'.S ms, Let Xs be the curve given over Fq by the fibre product

yf -

YI= fi(x),

yf -

Y2

=

h(x),

yf -

Ys

=

fs(x). (5)

Theorem 3. The genus of the curve Xs given by (5) is

s-l (p - l)(ms - 1) (p - l)(m2 - 1) (p - l)(m1 - 1)

g(Xs)=p + .. ·+p

+

.

2 2 2

Proof For 1 :'.Sr :'.S s, let Xr be the curve in ps-r+I defined by removing the first r - 1

equations in (5). Then for any 2 :'.Sr :'.S s, Xr-1 is a covering of Xr and Xs is a covering of the projective line P 1 , each of the covering is of degree p. Thus, we have the coverings

X1 -+ X2-+ ... -+ Xs-+ P1•

To prove the formula of the genus, we use induction on s. Fors

=

1, the result is contained in Lemma 1. Suppose that the result holds true for s - 1, then

s-2 (p - l)(ms-1 - 1) (p - l)(m2 - 1) (p - l)(m1 - 1)

g(X2)

=

p 2

+ ... +

p 2

+

2 .

Since X 1 is a covering of degree p of X2, we have

(6)

where R<h. is the ramification divisor of the covering ¢2: X 1 -+ X2. If we bear in mind that the only ramification point of ¢2 is the point at infinity and the (smooth projective) curve X2

near the point at infinity looks like the projective line P 1, then by the arguments of Lemma 1 we see that

deg R<h.

=

(p - l)(m1

+

1). Thus (6) implies that

g(X1)

=

pg(X2)

+

!(P - l)(m1 - 1),

which is the required result.

We emphasise that the ordering m 1 :::: . • . :::: ms of the degrees of the polynomials

Ji

(x), ... , fs(x) is crucial. Indeed, if we regard each equation

yf -

Yi

=

f;(x) in (5) separately as a covering of the projective line and Xi00 as the inverse image of the infinity

point [O : 1] E P1, then Xi00 corresponds to the discrete valuation Vx;00 such that Vx;00 (x)

=

-mi. Thus, we first have to consider the covering with the minimum valuation of x and this is clearly the covering with maximum mi, say ms. Repeating the same thing with the projective line replaced by the curve yf - Ys

=

fs(x) and bearing in mind that the smooth projective model of this curve near the point at infinity looks like the projective line, we see the necessity of the ordering.

It is worth pointing out that one could use the genus formula for elementary Abelian p-extensions given in [1] or Theorem 2.2 in [3] to prove Theorem 3, but the advantage of our proof lies in its simplicity and closeness, in spirit, to the geometry of the curve Xs.

Remark 4. It is important to note that the value of the genus given in Theorem 3 is bounded by (ps - l)(ms - 1)/2.

3. THE NUMBER OF Fq-RATIONAL POINTS

Let p be a prime number and Fq be a finite field with q

=

pv elements. We denote by Tr v : Fq -+ Fp the trace map. The number N of solutions of the equation

yP - y

=

f(x) E Fq(x)

in Fq x Fq is given by

N = pl{x E Fq: Trv f(x) = 0}!.

Using this well-known fact, we obtain the following assertion.

Lemma 2. Let p be a prime number, q

=

pv, where vis a positive integer, and let Fq2 be a finite field with q 2 elements. Then for any a E F*z such that a _q

+

aq

=

0 and any integer j ~ 1, the number of Fq2-rational points of the affine curve

yP - y

=

axj(l+q) is equal to q 2 p.

(7)

Proof It is clear that

2v-1

Tr2v(axi(l+q))

=

L

(axi(l+q))Pk k=O

v-1 2v-1

=

L

aPk xipk(l+q)

+

L

aPk xipk(l+q)

k=O k=v

v-1 v-1

=

L

aPk xipk(l+q)

+.

L

aPk+v xiPk+"(l+q)

k=O k=O

v-1 v-1

=

L

aPk xipk(l+q)

+

L

-aPk xiPk(q+q2).

k=O k=O

139

Since xq2

=

x for x e Fq2, we have Tr2v(axi(l+q))

=

0 for any x e Fq2. This gives the result.

Lemma 3. Let Fq2 be as in Lemma

2.

Then for any a e

F;

2 and any integer j, linebreak/3] 0 < j ::: v - I, the number of Fq2-rational points of the affine curve yP - y

=

ax 1+p•+i - aPv-i x 1+p•-i is equal to q2 p.

Proof It is clear that

Tr2v(axl+Pv+j - aPv-j xl+Pv-j) 2v-1 2v-1

=

L

(axl+pv+j)Pk -

L

(aPv-j xl+p•-j)Pk k=O k=O v-j-I 2v-1

=

L

(axI+p•+i)Pk

+

L

(axI+p•+i)Pk k=O k=v-j v+j-I 2v-l

- L

(aPv-j xl+p"-i)l -

L

(aPv-j xl+p•-i)l

k=O k=v+j v-j-1 v-j-1

=

L

aPkxpk(l+p•+i) _

L

aP2v+\pk(p•+i+q2) k=O k=O v+j-1 v+j-1

+

'°"'

_{~ a}pv-j+k pk(p•-i+q2) _x _{- ~ a}

'°"'

pv-j+k pk(l+p•-i) _x _{_}

(8)

Using the fact that aq2

=

a and xq2

=

x for x E Fq2, we obtain

for any x E Fq2 . This gives the result.

Let S1, S2 C Fq [x] be the sets of polynomials having, respectively, the forms

/J

(X)

=

ajXj (l+q),

where ai E Fq2 are such that a;

+

ai

=

0, ai are Fp-linearly independent, and 1 :::: j < q,

(j, p)

=

1; and

I+ v+j pv-j I+ v-j

fz(x)=aix P -ai x P ,

where ai E Fq2 are Fp-linearly independent and 1 :::: j .:::: v - 1.

Using these sets of polynomials and Lemmas 2, 3, we arrive at the following result.

Lemma 4. Let Fq2 be as in Lemmas 2, 3. Let 1 :::: s < q be an integer such that

(s, p)

=

1, and let Xs be the affine curve defined over Fq2 by the fibre product

Yi -

YI= f1(x), yf - Y2

=

fz(x),

yf - Ys

=

fs(X),

where .fi (x) E S1 U S2 are such that deg fi(x) .:::: s(q

+

1), i

=

1, ... , s. Then the number Nq2 (Xs) of Fq2-rational points of Xs is

Nq2(Xs)

=

qN1(s)+2, where N1 (s)

=

s - [s/p]

+

2[logp((s(q

+

1) - 1)/q)].

Proof The family of polynomials S1 U S2 forms a canonical basis for a non-degenerate linear subspace of Fq[x], hence the given system of equations defines a curve. By virtue of Lemmas 2 and 3, for any x E Fq2, the fibre over x on the curve Xs contains pN(s)

Fq2-rational points, where N (s) is the number of the defining equations of the curve. Writing

s in the forms

=

np

+

r, 1 :::: r .:::: p - 1, we see that the number of integers j such that

1 :::: j .:::: s and (j, p)

=

1 is equal to

n(p - 1)

+

r

=

np

+

r - n

=

s - n

=

s - [s/p].

Thus, the number of the defining equations with polynomials coming from S1 is equal to

v(s - [s / p ]). The number of the defining equations with polynomials coming from S2 is equal to 2vm, where mis the largest integer such that

(9)

Thus,

m = [logP((s(q + 1) - 1)/q)),

and hence

N(s) = v((s - [s/p]) +2[logp((s(q + 1) -1)/q))) = vN1(s),

which completes the proof.

The following two lemmas concern the case of Fq, q

=

p 2v+l.

141

Lemma 5. Let p be a prime number, q = p2_v+l,_{where v is a positive integer, and} let Fq be a finite field with q elements. Then for any a; E F; and any integer j ~ l, the number of Fq -rational points of the affine curve

yP - y

=

ajXj(pv+t+l) -

a(

Xj(p"+l)

is equal to qp.

Proof. It is clear that

Tr2v+l (axi (p"+1_{+l) -} _{aP" xi}_(p"+l))

2v 2v

=

L(axi<P"+1+1))Pk - L(aP" xi<P"+l))Pk

k=O k=O

v-1 2v

=

L

aPk xiPk(p"+l+l)

+

L

aPk xiPk(pv+l+l)

k=O k=v V 2v '"" pv+k jpk(pv+l) ' " " pv+k jpk(pv+l) - ~ a x - ~ a x k=O k=v+l V-1 V

=

La

Pk xiPk(pv+l+l)

+

L

aPv+k xipv+k(pv+l+l)

k=O k=O

V V-1

- Z:aPv+k xiPk(pv+l) - LaPlv+k+l xiPv+k+l(pv+l)

k=O k=O

V-1 V

=

La

Pk xiPk(pv+l+l)

+

L

aPv+k xiPk(plv+l+pv)

k=O k=O

V V-1

"'""' pv+k jpk(pv+l) "'""' p2v+k+l jpk(p2v+l+pv+l)

- ~ a

x

- ~ a

x

.

k=O k=O

Using the fact that aq

=

a

and

xq

=

x

for

x E

Fq,

we obtain

• ( V+\ 1) V • ( V 1)

Tr2v+l

(ax1 P

+

-

aP x1 _P

_{+ )}

=

₀

(10)

142 S. A. Stepanov and M. H. Shalalfeh

If we take ai in the previous lemma to be a basis of Fq over Fp and argue as in the proof

of Lemma 4, we get the following assertion.

Lemma 6. Let Fq be as in Lemma 5. Then for any integers, 1 ::: s < pv-1, (s, p)

=

1,

the fibre product

yfj - Yij

=

aiXj(pv+l+l) - a( xj(p"+l) where 1::: j:::: s, (j, p)

= 1,

has

qp(2v+l)(s-[s/ p]) Fq-rational points.

4. PROOF OF THE THEOREMS

In this section, we prove Theorem 1 and Corollary 1. The proofs of Theorem 2 and Corol-lary 2 are completely similar.

Proof of Theorem I. Let Fq2 be a finite field consisting of q2

=

_{p2v elements, and let}

1 ::: s < q, (s, p)

=

I. Let Xs be the affine curve defined in Lemma 4. By that lemma Xs

has qNi(s)+2 Fq2-rational points, where

N1(s);, s - [s/p] +2[logp((s(q

+

1)-1)/q)].

By normalization of the curve Xs, we obtain a non-singular model Xs without losing

Fq2-rationality of these points. Taking into account the point at infinity p00 of the curve Xs, we

obtain

Nq2 (Xs)

=

qN1 (s)+2

+

1. By Theorem 3 and Remark 4 the genus of Xs is less than

!(qNi(s) - l)(s(q

+

1) - 1). The proof is complete.

Before giving the proof for Corollary 1, we recall the well-known Gappa construction of the linear [n, k, d]q-codes associated to a smooth projective curve X over a finite field Fq, where q

=

pv (see [2]).

Let X be an absolutely irreducible smooth projective curve of genus g over Fq. Let {p1, ... , Pn} be a set of distinct Fq-rational points on X and Do

₌

Pl+ ... + Pn· Let D be a Fq-rational divisor on X such that the supports of D and Do are disjoint. The linear space

L(D)

=

{f E Fq(X)*

I

(!)

+

D ~ O} U {O} yields the linear evaluation map

(11)

Codes on fibre products of Artin-Schreier curves 143

The image of this map is the linear [n, k, d]q-code C

=

C(Do, D), which is called a geo-metric Goppa code associated to the pair (Do, D).

If deg D < n, then Ev is an embedding, hence k = dim C = l ( D). By the Riemann-Roch theorem,

k

=

l(D)::: deg D - g

+

1. In particular, if 2g - 2 < deg D < n, then k

=

deg D - g

+

1.

The minimum distanced of the code C

=

C(D, Do) satisfies the inequality

d ::: n - deg D.

Hence it follows that the relative parameters R

=

k/n and 8

=

d/n of the linear [n, k,

d]q-code C

=

C(Do, D) satisfy the relation

R>l-8-g-l.

- n

Now, applying Goppa's construction to the curves of Theorem 1, we prove Corollary 1.

Proof of Corollary 1. Let Xs be the (smooth projective) curve defined in Theorem 1. Let S be the set of Fq2-rational points of Xs at the finite part of it, and let p00 denote the

point of Xs at infinity. By Lemma 4

ISi

=

qN1(s>+2. Let n ~ qN1(s)+2 and Do

=

Pl

+ · · · +

Pn, D

=

lpoo.

Applying Goppa's construction to these Fq2-divisors on Xs for any integer l such that !(qN1(s) - l)(s(q

+

1) - 1) < l < n,

we get the required code.

REFERENCES

I. A. Garcia and H. Stichtenoth, Elementary Abelian p-extensions of algebraic function fields.

Manuscr. Math. (1991) 72, 67-79.

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

3. G. Lachaud, Artin-Schreier curves, exponential sums, and the Carlitz-Uchiyama bound for geo-metric codes. J. Number Theory (1991) 39, 18-40.

4. F. Ozbudak, Codes on fibre products of some Kummer coverings. Finite Fields and their Appl. (1999) 5, 188-205.

5. S. A. Stepanov, Arithmetic of Algebraic Curves. Plenum, New York, 1994. 6. S. A. Stepanov, Codes on Algebraic Curves. Plenum, New York, 1999.

7. S. A. Stepanov, Codes on fibre products of hyperelliptic curves. Discrete Math. Appl. (1997) 7,

77-88.

8. S. A. Stepanov and F. Ozbudak, Fibre products of hyperelliptic curves and geometric Goppa codes. Discrete Math. Appl (1997) 7, 223-229.

9. G. van der Geer and M. van der Vlugt, How to construct curves over finite fields with many points. In: Algebraic Geometry E-prints, alg-georn/ 9511005, 1995.

10. C. Xing and H. Stichtenoth, The genus of maximal function fields. Manuscr. Math. (1955) 86,