Fibre products, character sums and geometric Goppa codes

Fibre Products, Character Sums,

and Geometric Goppa Codes

Serguei A. Stepanov

Abstract

The purpose of this paper is to construct some new families of smooth projective curves over a finite field F_q with extremely many F_q-rational points. The genus in every such family is considerably less than the number of rational points, and these two facts allow to produce sufficiently long geometric Gappa codes with very good parameters.

§1. Introduction

Let

F_q

be a finite field with

q

elements. The linear space

F;

can be provided with the structure of a metric space if we introduce

the

Hamming distance d( x, y)

between

x

= (

x1, ... Xn) E

F;

and

y

=

(Y1, ... , Y_n) E

F;

as the number of coordinates in which

x

and

y

differ:

d(x,y)

=

#{

i

11::;

i::;

n, x;-/- y;}.

Each linear subspace

C

s;;;

F;

is called a

linear q-ary code of length n.

The non-negative integers M =

ICI

and k = log

q

ICI

are called the

cardinality

and

log-cardinality

of the code

C,

respectively (in fact,

227

228

Stepanov

k

=

dimF

q

C). The minimum distance of the code

C

is defined as

d

=

min{

d(x,y)

I

x,y

E

C,x

-=f. y}.

A q-ary linear code

C

with

parameters n, k

and

d

is called an

[n,

k, d]q-code.

Elements of care called

code-words

(or

code-vectors),

and their components are called

positions

( or

coordinates).

Define

the

information rate

R and

relative minimum distance

8 of the code

C

as

R

=

k/n

and

8

=

d/n.

It is clear that O :S

R

'.S 1, 0 :S

8

'.S 1.

Let us briefly explain why the codes are called

error-correcting.

The following question is essentially one of the central problems of

information theory. We consider

information

presented as a very

long sequence of symbols from the

alphabet Fq,

In this sequence

each symbol occurs with equal probability. This information is sent

to a receiver over a so-called

noisy channel.

In the model that we

consider there is a small fixed probability

p that a symbol, which

is sent over the channel, is changed to one of the other symbols.

Such an event is called a

symbol-error

and

p

is the

symbol-error probability.

As a result a fraction

p

of the transmitted symbols arrives

incorrectly at the receiver at the end of the communication channel.

The aim of coding theory is to lower the probability of error at the

expense of spending some of the transmission time or energy on

redundant symbols.

The basic idea of the theory can be explained in

the sentence that follows. When we read printed text we recognize a

printing error in a word because in our vocabulary there is only one

word that resembles (is sufficiently close to) the printed word.

In

block coding

the message is split into parts of

k

symbols. The

encoding

is an injective map from

F_qk

to

F;

(where

n � k).

In other

words we take some [n,

k,

d]

q

-code

C

and fix an embedding

, : F;-.'.::+ C �Ft.

The transmission is now

R

=

k/n

times slower, which justifies the

term "information rate" for

R.

Instead of the part

z

E

F,7

of the

message we transmit the corresponding word

x = 1(z) of length n.

On the end of the channel we obtain a distorted word

x'

E

F; and

we transform it into the nearest code-word

x"

E

C

(i.e. we decode

the message on the maximum likelihood basis). This transformation

Gappa Codes.

229

can be defined by some

decoding map

( :

F;

-+

C.

If the number of

distorted symbols is at most

l d;l

J,

then

x"

=

x,

i.e. the decoding is

correct. The

maximum likelihood decoding

is an ideal that is almost

unattainable. Usually we just give a map ( :

U

-+

C,

where

U

is the

union of all

balls

B

t

(x)

= {

y E F;

I

d(x, y) St}

of radius t S

l d;l

J

centered in the elements

x

E

C.

In this case

we speak about

decoding up to t

(

or correcting

t

errors). Usually

t

=

l d;l

J,

but sometimes it is less.

Example 1

_{(binary Hamming codes).}

Let

r

be a positive integer,

n

=

2

r

- 1, and H

=

(aij) the unique

r

x

n

binary matrix whose

columns are pairwise linearly independent over the field F2 (hence

the columns of Hare distinct and differ from the zero column). The

set of all solutions x

= (

x

1, .

.. ,

x

n

) E

Ff of the system of linear

equations

H ·

XT

=

0,

forms an (

n

-

r

)-dimensional subspace C

of

F;,

The linear code

C

is

called a

binary Hamming code.

The Hamming code

C

corrects single

errors. Indeed, let x be a code-vector and x'

=

x+e a received vector.

Assume that the

error-vector e

= (

e

1 , ... , en)

has a unique non-zero

component, say

Xi

=

l. Multiplying the

parity-check

matrix

H

by

the vector

_x'

=

_x

+

e,

we obtain

H · (x

+

ef

=

H ·e

t

= s

T

Then the binary r-dimensional vector

s

= (

s

1 , ... ,

S

r

) ( called a

syn­

drome

of

_x')

is equal to the i-th column of

H.

Since we know the

matrix

H

we can compare the syndrome

s

with columns of

H

and

find the unique position where the single error occurs. This means

that the minimum distance

d

of

C

is at least 3. It is easy to see that

in fact

d

=

3, so

C is a linear [n,

n

-

r,

3]z-code.

Now we note that all the unit balls B

1

(

x)

centered in code-vectors

x E

C are disjoint, and also IB

1

(

x)

I

=

1

+

n

=

2

r

. In that case

L

IB

1

(x)I

=

2

n

-

r

lB

1

(x)I

=

2

n

-

r

2

r

=

IF:I ,

xEC

230

Stepanov so the code

C

provides a densest packing of the space

F:f.

The codes with such a property are called perfect.

Example 2 ( Reed-Solomon codes). Let P

= {

011, ... , an} � Fq

be a subset of cardinality n :::;: q. Consider a linear space

L(

m) of all polynomials

f

E Fq[u] in one variable u of degree at most m.

Its dimension over Fq is dim

L(

m)

=

m

+

1. For n > m a non-zero

polynomial

f

(

u)

E

L(

m) cannot vanish at all points of

P.

Moreover, it has at least ( n - m) non-zero values at points of the set P. Hence if n > m, the evaluation map

is injective and its image

C

is a linear [n, m

+

1, n - m]q-code called

a Reed-Solomon code of degree m

(

or RS-code). The parameters of such a code satisfy the condition k

+

d

=

n

+

1 and k

=

m

+

l can be freely chosen between 1 and n.

The construction of Reed-Solomon codes can be extended as fol­ lows. Let X be a smooth projective curve of genus g

=

g(X) defined over a finite field Fq. We describe briefly the Goppa [7] construction

of linear [n,

k,

d]q-codes associated to the curve

X.

Let { x1, ... , xn}

be a set of Fq-rational points on

X

and set Do

=

X1

+ · · · +

Xn

Let

D

be a Fq-rational divisor on

X.

We assume that

D

has support disjoint from Do, i.e. the points Xi occur with multiplicity zero in D.

Denote by Fq(X) the field of rational functions on X and consider

the following vector space over Fq:

L(D)

= {

f

E Fq(X)*

I

(!)

+

D � O} U {O} .

The linear [n,

k,

d]q-code

C

=

C(D

0,

D)

associated to the pair

(Do, D)

is the image of the linear evaluation map

Ev: L(D)-+

Ft

,

f � (f(x1), .. . , f(xn)) .

Such a q-ary linear code is called a geometric Gappa code. If deg D <

Coppa Codes 231 Dually, denote by D(X) the Fq(X)-vector space of rational dif­ ferential forms on

X

and consider the following linear space over

F_q:

D(Do

-

D)

= {

w

E D(X)*

I

(w)

+

Do

-

D �

0} u {O} . The linear map

defines a linear [n,

k,

djg-code

C*

=

C*(D

0,

D)

associated to the pair

(Do, D) .

If deg

D

>

2g -2, the map Res is injective, so that

C*

�

D(Do

-

D)

�

L(I{

+

Do

-

D) ,

where J{ 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

=

d/n is the relative minimum distance and R is the information rate of

C.

The points (

8, R)

form the set of code points i�lin � [O, 1]2. Let

u:n

denote the subset of limit points of �tin. In other terms, (

8, R)

E

U�in

if and only if there exists an infinite sequence of different linear codes

Ci

with distinct relative parameters

8i

=

8(Ci) and

Ri

=

R(Ci)

such that

Em (8i,Ri)

=

(8, R) .

i-HX>

If 8 > 0 and R > 0 such a family of codes

Ci

is called asymptotically good. The structure of U!in can be described as follows (Aaltonen [1], Manin [9] ): there exists a continuous function a�n( 8) such that

moreover, a�n(O)

=

1, a�n(8)

=

0 for

(q

_-

l)/q �

_{8 � 1, and}

a�n(8) decreases on the interval [O, (q-1)/q]. The function a�n(8) is unknown in general, but we are able to find a number of upper bounds and the Gilbert-Varshamov lower bound

where

232 Stepanov The Gilbert-Varshamov bound has a series of remarkable statistical properties. For example, it can be easily shown that the parameters of almost all linear codes lie on the curve R

=

1 - Hq( 8). For this

reason it was thought for a long time that a�n( 8)

=

Hg( 8). Only

recently this assumption was disproved (for

q

2: 49) with the help of very deep methods of algebraic geometry.

It follows from the Riemann-Roch theorem that the relative pa­ rameters

R

=

k/n

and

8

=

d/n

both for

L-

and D-constructions satisfy ( see [24], [25] ).

g-1

R>l-8---.

_-

_n

(1)

In order to produce a family of asymptotically good geometric Goppa codes for which R

+

8 comes above the Gilbert-Varshamov bound

a�n( 8) 2: 1 - Hq( 8) ,

one needs a family of smooth projective curves with a lot of Fq­

rational points compared to the genus. Examples of such families are provided by classical modular curves

X

0

(N)

and

X(N)

(Ihara [8], Tsfasman-Vladut-Zink [26], Moreno [10]), or by Drinfeld modular curves (Tsfasman, Vladut [25, Chapters 4.1 and 4.2]). So, if q

=

pv

is an even power of a prime

p,

there exists an infinite sequence of geometric Goppa codes Ci which gives the lower bound

O'.�n

( 8) 2: 1 - b - (

ylq

- 1

t

1

The line

R

=

1- 8-(Jq-lt1 _{intersects the curve}

R

=

_1-H

q(b) for

q

2: 49. A much easier proof of this result based on consideration of a sequence of (modified) Artin-Schreier coverings of the projective line IP'1 _{( Fq) was recently proposed by Garcia and Stichtenoth [3]. Later}

they [4] discovered a still easier sequence of smooth projective curves with the same property.

In terms of algebraic geometry the problem on construction of asymptotically good codes can be reformulated as follows. Let Nq

(g)

denote the maximal number of Fq-ra.tional points on a smooth pro­ jective curve X over Fq of genus g

=

g(X), and

. Nq(g)

A(q)

=

hm sup --g-+oo g

Gappa Codes

233

It follows from the Hasse-Weil bound [27] that

A(q)

s;

_2ytq.

The Serre bound [16]

IN

q

- (q

+

1)1

:s; _q[2Jq]

yields

A( q)

:s;

[2ytq]

A much stronger upper bound

A(q)

s;

_{ytq -1}

was obtained by Drinfeld and Vladut [2] . This is the best possible

upper bound, and the construction of asymptotically good geometric

Goppa codes is reduced to the construction of a family of smooth

projective curves over F

q

, for which

A

(

q)

is close to the Drinfeld­

Vladut bound. So if

q = p

11

is an even power of a prime number

p,

the result of lhara [8], Tsfasman-Vladut-Zink [26] (see also Moreno

[10] and Garcia-Stichtenoth [3], [4]) implies

A(q)2:ytq-1.

If

q is an odd power of p, the result of Serre [16] provides the

existence of an absolute constant c > 0 such that

A(q)

_2:

clogq.

In some cases the Serre lower bound was improved by Perret [15]

and Zink [28]. In particular, the Zink result yields

A(

_{q - q}

3

) >

2(

q

2

₊

-

₂

1)

In this paper we construct rather long geometric Goppa codes

coming from fibre products of superelliptic curves.

234 Stepanov

At first we consider a family of smooth projective curves )C given

over Fq by equations

(2)

where fi(u) are relatively prime square-free polynomials in Fq[u] of a special form. Every such curve is actually a fibre product of hy­ perelliptic curves. The main point of the paper is to calculate the genus g(Xs) (Lemma 1) and determine the number Nqv(Xs) of F_q­

rational points (Lemma 4) on the curve X8• We show that the ratio

g(Xs)/Nq(Xs) is small enough, and deduce from (1) that the cor­

responding geometric Goppa codes

C(D, Do)

and

C*(D,

Do) 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 [3], [4]. In particular, ifs

=

1 then the codes

C(D, Do)

and

C*(D, Do)

have the same parameters as the codes on Hermitian curves (see [24, Sec. VII.3] ). Unfortu­ nately, the parameter .s in our construction is bounded by q1_{12, and} as a result the genus g(Xs) is bounded by

(q

- 3)2,.fa-2

+

1

However, since the above upper bound is large enough for q 2 q0, the curves Xs provide sufficiently long geometric Goppa codes ( with

n

'.S q2y'q-I). Moreover, some modification of the polynomials

f,

(

x)

(Ozbudak [11], [12] ) allows to construct linear [n,

k,

d]q-cocles over

Fq for any

n'.Sq(q -l)2

q

_.

It is necessary also to note that construction of linear codes on fibre products of some Kummer coverings (2) are closely related to the well-known combinatorial problem on configurations of lines in the finite plane F_q2 _{(Ozbudak [12], Ozbuclak and Thomas [14] ).}

The genus g(Xs) can be easily c:alculated 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

D(Do

-

D).

This provides an easy way to write out generator ma­

Gappa Codes 235

Applying to curves Xs the Goppa constructions, we obtain the

following results (Stepanov [20], [21]).

Theorem 1. Let p

>

2 be a prime} v

>

1 an even integer} and Fq a finite field consisting of q

=

pv _{elements. For any positive integers} s

:S

q112 _{and r}

> (

_sq1_!2 - ₃₎₂s_{-2 there exists a geometric Coppa}

[n, k, d]q-code C

=

C(D0, D) with r

<

n :::; _(2ql_{/2 _ 8 )}

q1;22s-t , k

2'.

r -(sql/2 -3)2s-2 ' d2'.n-r .

Theorem 2. For p} q and s as before} and for any positive integer r > (sq112 - ₃₎₂s-t there exists a geometric Coppa

_[n,

_k.d]

q-code C*

=

C*(Do, D) with

r _ (sqt/2 _ 3)2s-2 < n:::; (2ql/2 _ s)q1/22s-l , k

2'.

n -r

+

(sq112 - 3)2s-2 ,

d

2'.

r -(sq112 - ₃₎₂s_-l

Corollary 1. The relative parameters R

=

k/n and 8

=

d/n of the above codes satisfy

( sql/2 _ 3)2s-2 R > l-8-

_-

_n

In particular} for n

=

(2q1l2 -s )q1!22s-l we have sq1/2 -3

R > l-8---_{- 2(2} ql/2 -s )ql/2

By a suitable concatenation one gets reasonably good codes over

FP. Indeed, let k0 > 1 be an even number. Applying a linear

[n0, k0 , do]p-code C0 to an [n, k, d]q- code C

=

C(Do, D) over Fq,

where q

=

pk0_{, we obtain an [n', k', d']P-code C' with parameters} n'

=

n0n, k'

=

k0k, d'

=

d0d .

Let us denote by Ro

=

k0/n0 and 80

=

d0/n0 the relative parameters of the code Co.

236 Stepanov

Corollary 2.

_{For any positive integers n0}

>

_{l , s} :=::; q112 and

r

>

(sq112 _{- 3)2}s-2 there exists a linear

[n',

_{k', d']}

p-code C' with nor

<

n'

=

n0n :=::; n0(2q112 - s )q1122s-l,

k' 2'. ko(r

-

(sq112 _{- 3)2}s_-2), d' 2'. d0

(

n

-

r)

Relative parameters R'

=

k' /n' and 8'

=

d' /n' of the code C' satisfy R'

+

8' 2'. Ro (;�-

-

(sql/2 _� 3)2s-2)

+

80

( 1 - �)

Applying [no, ko, do]p-code C0 to an [n, k, d)q-code C*

=

C* (Do, D) we obtain the following result.

Corollary

3. For any positive integers n0

>

l , s :=::; q112 and

r

>

(sq1_l2 _{- 3)2}s_{-2 there exists a linear}

_[n",

_{k", d"J}

p-code C" with no(r

-

(sq1_{l2 - 3)}

<

_{n" = non :=::; no(2q}112 _{- s)q}1122s-1, k" 2'. ko(n

-

r

+

(sq112 _{- 3)2}s_-2),

d" :=::; do(r

-

(sq1/2 - 3)2s-1 )

Relative parameters R"

=

k" /n" and 8"

=

d" /n" of the code C" satisfy

(

r (sq1/2 _ 3)2s_-2 ) R"

+

8"

>

Ro l - -_n

+ ---

_n

(

r

(

sql/2 -3)2s_-1 ) +80

- -

---'----n n

The above results can be easily extended to the case of codes coming from fibre products of hyperelliptic curves over a finite field Fq with q

=

p11 elements, where v

>

l is an odd number (Stepa.nov and Ozbudak [22] ).

Theorem 3.

Let v

>

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

>

2 consisting of q

=

p11 _{elements, and s an integer} such that

1

< <

_{- -}

s ---2p11

+

4 p(v-1)/2(p

+

1) _ 2

Gappa Codes

Moreover_} let r be an integer satisfying

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

4)

< r < 2spv

Then there exists a linear

[n,

k , d]_q-code C = C(D0, D) with

parameters

r < n '.S 28pV_,

k = r - 2s-2 ( (p(v-1 )/2 (p + l ) - 2)s - 4) , d ?_ n - r .

237

A similar result holds for codes coming from fibre products of superelliptic curves

zt

=

fi (u) ,

(3)

where µ ?. 2 is a positive divisor of q - 1. The following theorems

(Ozbudak [11] ) are based on the use of lower bounds for character sums obtained by Gluhov [6] and on the construction of geometric Goppa codes coming from superelliptic curves (Ozbudak and Gluhov

[13]).

Theorem 4.

Let T be a positive integer_} Fq,. a finite field of characteristic p > 2 with qr _elements

} and µ ?. 2 a positive divisor

of q - 1 . Then:

(i) If T > 1 is an odd number} s an integer such that

1

<

S

<

2µ( q

r

+

1)

- _{- (}

µ - l ) (q(r-1 )/2(q + l ) - 2) J

and r is an integer satisfying s-1

y

((µ - l ) (q(v-1)f2(q + 1 ) - 2)s - 2

µ) < r < µsqr J

there exists a linear [n, k , d]_qr -code C

=

C(D0, D) with

parameters r < n '.S µs_q7_, s-1 k ?_ r

- Y

((µ - l ) (q(r-l)f2(q + 1 ) - 2)s - 2 µ) J d ?_ n - r ;

238

Stepanov moreover) if r

>

_µs

-1 ((µ - l)(

q(T

-1)/

2

(q + 1) -2)s -2µ)

i then s-1

k

=

r

- T

((µ - l)(

q(T

-l)f2(

q

+ 1) -2) -2µ)

(ii) If T

=

2 ( mod 4)

₁ s an integer such that

2

µ

(

qT

+

1)

1

<

_s

<

_J

- - (µ -l)(q

T

_/

2

_-l(q

2

_{+ 1) -2)}

and r an integer satisfying

s- 1

T

((µ - l)(q

T

_/

2

_-1(q

2

_{+ 1) -2) -2µ) <}

_r

_<

µsqT !

there exists a linear

[n,

k, d]qr -code C

=

C(Do, D) with parameters s-1

k ;:::

r- T

_{((µ - l)(q}

T

/

2

-1(q

2

_{+ 1)-2)s -2µ) ,}

d ;::: n-r ; moreover1 if r

>

_µs

-1 ((µ - l)(q

T

/

2

-l(q

2

+

1)-2)s -2µ)

i then s- 1

k

₌

r

- T

_{((µ - l)(}

qT

f2-1(

q 2

+

1)-2)s -2µ)

(iii) If T

=

0 ( mod 4)

₁ s an integer such that

2

µ

(

qT

+ l)

1

<

s

<

₁

Coppa Codes

and r an integer satisfying s_-1

T

((µ -l )(qTf-1(q2 + l)s -2q) -2µ)

<

r

<

µsqT }

there exists a linear

[n,

k, d]qr -code C

=

C(D0, D) with parameters r < n :S µs_qT_, s_-1 k 2 r

-T

((µ -l )(q7 !2-1(q2 + 1) -2q)s -2µ) , d 2 n-r ; moreover} if r

>

µs-1 ((µ -l )(qT/2-l(q2 + 1) -2q)s

-

2µ) J then s_-1 k

=

r - T

((µ-l )(q7/2-1(_q2 + 1)-2_q)s-2_µ)

239

Theorem 5 . Let T be a positive integer} Fq,. a finite field of

characteristic p

=

2 with q7 _elements

} and µ 2 2 a positive divisor

of q - l . _Then:

(i) If T

>

l is odd} s an integer such that

l < s < _{- - (} 2µ(qT + l )

µ-l )(q( T-1)/2(q + l )-2q) J and r an integer satisfying

s_-1

T

((µ-l )(q( T-1)/2(q + 1) -2q)s -2µ)

<

r

<

µsqT }

there exists a linear

[n,

k, d]qr -code C

=

C(Do, D) with parameters

r < n :S µs_qT_, s-1

k 2 r

-T

((µ -l )(q( T-l)/2(q + 1) -2q)s

-

2µ) , d 2 n-r ;

240

Stepanov moreover, if

r >

_µs

_{-1 ((}

µ

- l)(qCr-1)/

2

(q + 1) -2

q

)s -2µ)

then s

_-1

k

=

r

- Y

((µ - l)(q

(r

-l)/

2

(q + 1)-2q)s -2µ)

(ii} If T

=

2 (mod 4),

s an integer such that

2

µ

(

qT

+

1)

1 <

- - (µ - l)(q

S

<

_r !

/

2

_{-l (q}

2

_+1)-2q

2

₎

and r an integer satisfying

s

_-1

y

((

µ

- l)(q

T

/

2

-1 (q

2

+ 1)-2q

2

)s -2µ) <

r

<

µSqT I

there exists a linear

[n,

k, d]qr -code C

=

C(Do, D) with parameters s

_-1

k �

r

- Y

((µ - l)(q

7/2

-1 (q

2

+ 1) -2q

2

)s -2µ) ,

d � n-r ; moreover, if r

>

_µs

-1 ((

_µ

- l)(q

r

/

2

-1(q

2

+ 1) -2

q2

)s -2µ)

then s

_-1

k

=

r - Y

((

µ

- l)(

qr f2

-1 (

q2

+

1) -2

q2

)s -2µ)

(iii} If T

=

0 ( mod 4),

s an integer such that

2

µ

(

qT

+

1)

1 <

S

<

/

Gappa Codes

and r an integer satisfying

s-1

T

((µ - l ) (qT/2-1 (q2

+

1)

-2q)s - 2µ)

<

r

<

µsqT J

there exists a linear

[n,

k,

d]_qr -code

C

=

C(D

0,

D)

with

parameters s_-1 k � r

- T

((µ - l ) (q7!2-1 ( q2

+

1) -

2q)s - 2µ) , d � n - r ; moreover_} if r

>

µs-1 ((µ - l ) (qT /2-l (q2

+

1) -

2q)s - 2µ) J then s-1 k

=

r

- T

((µ - l ) ( q7/2-1 (q2

+

l ) - 2q) s - 2_µ)

24 1

The advantage of Theorems 4 and 5 is that the fibre products of

superelliptic curves provide a class of much longer linear codes than

the class of codes coming from fibre products of hyperelliptic curves.

Moreover, for odd

µ ,

_{the fibre products of superelliptic curves over}

a finite field of characteristic 2 gives a possibility to construct rather

long binary linear codes with very good parameters.

A similar construction of non-singular projective curves with a.

lot of F

q

-rational points based on the use of fibre products of some

special Artin-Schreier curves was independently considered by van

der Geer and van der Vlugt [5]

§2. Notation and Lemmas

Let F'

q

be an algebraic closure of F

q

and A/

+1

be the (

s

+

1 )­

242 Stepanov

Lemma 1 .

Let f1, . . . , fs be pairwise cop rime square-free manic polynomials in Fq[u] of the same odd degree m 2:

1 ,

and Y the fibre product in _As+l _{given over Fq by equations}

z;

=

fi ( u) , 1 :S i :S s

Then the genus g

=

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

g

= (

ms - 3)2s-2

+

1

Proof.

Let

X

be a smooth projective model of the curve

Y.

Denote

by Vx the canonical valuation of the function field F'_q(X), and by

D[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 7/J :

X

-t

Y,

which

is an isomorphism between

Y

and 7/J-1_{(Y). Hence it follows that}

g = g(X) = g(Y).

The rational map ( u, z1, . . . , z .. ) 1-t u of the curve

Y

in A 1 deter­

mines a morphism 'P : X -t JfD1 of degree 2s, so that for u₀ E A 1 ei­

ther t.p-1_(u

0 ) consists of 2s points of the form x'

=

(u0 , ±z1 , . . . , ±zs)

in each of which vx,(t)

=

1 for a local parameter t at u0, or else

t.p-1 ( _{u0 ) consists of 2}s-l _{points of the form x?}

= (

_u

0 , ±z1 , . . .

±

Z i-1, 0, ±zi+l, . . . , ±zs), and Vx11_I (i)

=

2.

Let us consider the point at infinity u₀₀ E JfD1 . If the coordinate on A.1 _{is denoted by u, then t}

=

_u-1 _{is a local parameter at u}

00•

If t.p-1 ( _u00

)

_{were to consist of 2}s _points

xt;,J,

_{then at each x}

00

=

xtl the function t would be a local parameter. Hence it would follow that Vx₌(t)

=

1 and Vx₌(fi(t))

=

-m. But since m is odd,

this contradicts the condition that Vx₌ (fi ( u))

=

2vx₌ ( Z i). Thus

t.p-1_(u00

)

_{consists of r}

=

₂s-l _{points xtl, 1 :S T :S r, with projective}

coordinates xtl

=

(0, 1, ±1, . . . , ±1, 0). It follows that X

=

Y U { x�,l } U · · · U

{xt,l } .

At any such point

x

₀₀

=

xtl we have Vx= (u)

=

-2 and Vx₌ (z i)

=

-m.

Let us now find a basis of the space D[X] over the field Fq. Any

Gappa Codes

differential forms w0

=

P0

(

u) du and

Pi1 , ... ,i" ( u) du

W i1 , ... ,ic,-

=

'

Z i1 . . . Zi"

243

where i1, . . . , i17 are integers satisfying the condition 1 :::; i1 < · · · <

i17 :::; s and Pi₁ , ... ,i" are polynomials in F'q [ u]. Indeed, the differential

form

du

w'

1.. 1 ' • • • ,i

.

o-

=

�---is regular at any point u0 E A 1 with the condition Z i ( u0)

#

0 for i E

{i1, . . . , i17 }. Now if z i(u0 )

=

0 for a unique i E {i1, . . . , i17 }, then Zi

is a local parameter at

xi'

=

(uo, ±z1, . . . , ±z i-1, 0, ±zi+I , . . . , ±zs) ,

so that Vx11(z i)

=

1 and Vx11(u - uo)

=

2. Therefore, Vx11(du)

=

1 and

again wt ,.' .. ,i" is regular at uo. The form wb

=

du is �lso regular at any point u0 E A 1. Thus, the differential forms wb

=

du and wi1, ... ,i,,.

form a basis of the .F'q[u]-module D[Y] .

It remains to clarify which of the forms w0 and Wi1 , ... ,i" are regular

at points x�,l,

. . . ,

xt,l. Let x00 be one of these points. If t is a local parameter at x00, then u

=

t-2u', Z i

=

t-m zL where u' and

<

are

units in the local ring Ox₌· Therefore wt , ... ,i" = tm µ-3B i₁, ... ,i"dt, with

ei1 l ... i a unit in O' CJ' x 00 ' hence (w� ; ) "1 l " " J " O"

=

(mO" - 3) . Xoo. Thus, the

differential form

Pi1 , ... ,i" ( u) du

W i1 , ... ,ic,-

= ----­

Z i1 . . . Zi"

is regular at x00 if and only if

v (P · (u)) > -(mO" - 3) Xoo i1 , . . . ,i cr

-This means that deg Pi1, ... , i"(u) :::; (m0" - 3)/2 and hence

deg P,, , . ,,. (u) :S {

ffi(J" - 4 2 ffi(J" - 3 2 if O"

=

0 (mod 2) if O"

=

1 (mod 2)

The differential form w0

=

Po du is not regular at x00 for any non-zero polynomial Po E k"[u], so the regular differential forms

I I

w i1 , ... ,i" ' uw i1 , ... ,i" '

n I

244

where 1 :S i

1

< · · · < i

r;

:S

s

and

{

mcr -

4

n

-

-

_{mcr - 3}

2

2

if

er

_ 0 ( mod 2)

if

er

=

1 (

mod

2)

form a basis of the space D[X] over

F

_q

. Therefore

dim - D[X]

_Fq

=

!

2

o- = 1 _{l <i1 < . . ·<ia- < s}

(mcr - 2)

o-:O ( mod 2) -

-1

+-

₂

o-=1 _{l < i1 < . . ·<ia- < s} o- : l ( mo d 2) -

-(mcr - 1)

Stepanov cr=l o-:O ( mo d 2)

(;) -t � (;)

o- :: 1 ( mo d 2)

1

=

- (ms2

s

-l - 2

8 -

2

s

-l

+

2)

2

and hence

9

=

g(X)

=

dimpq D[X]

=

(ms

- 3)2

s

-

2

+

1

This completes the proof.

Let p be a prime number,

11

a positive integer and Fq a finite field

with

q

=

p11

_{elements. The field Fq is a Galois extension of the prime}

finite field F

p

of degree

II

with the cyclic Galois group of order

11 .

The action of a generator 8 of this group on an element x E Fq is

given by the rule

_O(x)

=

xP_.

_{The map}

1 v - 1

norm

v

(x)

=

X . O(x) . . . ev- (x)

=

X . xP . . . xP

of F

q

onto F

p

is the

norm

of the element

x.

Let x be a multiplicative character of the field Fp and x an element

of F

q

. Set

Gappa Codes

245

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

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

x

a non-trivial quadratic character of Fp. Consider the character sum

Sv(f)

=

L

Xv(J(u))

=

L

x(normv (f(u)))

and recall the well-known Weil bound [27] (see also Stepanov [19, Chapters 1 and 5] ):

I

Sv(f)

l:S

2

[

m

;

l]

q1 12

The following result (Stepanov [18, Theorem 3] ) shows us that Weil's 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 char­}

acteristic p > 2 and Xv the character of F_q induced by a non-trivial quadratic character X of the field Fp. If v

>

1 then for the square-free polynomial f E Fq[u] given by

{

U

+

uP"/2 if

I/

=

Q ( mod 2) f(u)

=

_(u

₊

_uP( v-1)/2 )(u

+

Up( v + 1 )/2 ) if

I/ =

1 (mod 2) J we have

L

Xv(f(u))

=

{ q - q ( 1 / 2

1)

1 / 2

q

- 1

if v _if v

=

=

1 ( mod 2) 0 ( mod 2) u E Fq

Proof. Let v > 1 be an even number. As far as uP"

=

u in Fq, then for any u E F_q we have

V V

normv (f(u))

=

II (u

+

upv/2 t-1

=

_{II (u}Pi-1

+

upv/2+i-1 )

i=l i=l

v/2 v/2

II (

uPi-1

+

uPv/2+i-1 )

II (

uPv/2+j-1

+

uPj-1 )

i=l j=l

246

Stepanov

Therefore,

L

Xv (J(u))

=

L

x(norm

v

(J(u)))

=

q-

N ,

where N is the number of elements of the set A

= {

u E F

q

I

f ( u)

=

0}.

Since f(u) = u(l

+

u

P

"12 -

1 )

we have A = {O} U B, where

v/2 1 }

B

= {

u E Fq I l

+

u

P

-

=

0

is the set of roots of the polynomial 1

+

u

P

"12 -1 in F

q

. Taking into

account the equality

gcd(p

v

/2 - l, p

v

- 1)

=

p

v

/2 - 1 ,

we obtain from the Euler criterion that the number of roots of the

polynomial 1

+

u

P

"12 -1 is equal to p

v

/2 - l. In that case

N

=

IAI

=

₁

+

IBI

=

₁

+

_(p

v

/Z - 1)

=

q

1

/2 ,

and hence

L

X

v

(J(u)) = (q

l

/2 - l)q

l

/2 .

uEFq

Gappa Codes

Let now v > l be an odd number. In this case for any u E Fq

norm

v

(f (

u))

=

II (

uPi_{- 1}

+

uP("- 1 ) /2+ i -1 ) ( uPi-1

+

_uP("+ 1 J /2+ i -1 ) i=l ( v- 1 ) / 2 V

=

II (

uPi_-1

+

_{uP( v-1 ) /Z+ i- 1} )

II (

uP i-1

+

uP( ..,+ 1 ) /Z+ i -l ) i=l _{i= (v+l )/2} ( v- 1 ) / 2 V x

II (

uPi_{- 1}

+

uPc"+ 1 ) /2+ i-1 )

II (

uPi-1

+

uP(..,+ 1 ) /2+ i -1 ) i=l _{i= (v+l )/2} (v+l ) / 2 (v- 1 ) /2

=

II (

uPi-1

+

_uPc "- 1 ) /2+ i -1 )

II (

uPc ..,+ 1 ) /2+; -1

+

uPj-1 ) i=l j=l ( v - 1 ) / 2 (v+l ) / 2 x

II (

uPi_-1

+

_uPc "+ 1 )/2+ i - 1 )

II (

uP"- 1 ) /2+; -1

+

uP; -1 ) i=l j=l (v+l ) / 2 ( v - 1 ) / 2

II (

pi -1 p( l-'-l )/2+i-l

)2

II (

pi-1 pC "+1 )/2+i-l

)2

=

u _{+ u} u _{+ u}

i=l i=l

and hence

L

Xv(f(u))

=

L

x(norm

v

(f(u)))

=

q -

N'

,

247

where N' is the cardinality of the set A =

{ u E

F

q

I

f ( u₎

O}.

Clearly

_N'

=

l and therefore

L

Xv (f(u))

=

q - l

uEFq

This completes the proof.

Lemma 3.

Let

F

p be a a prime finite field of characteristic p

> 2,

F

q

=

F

p

"

be an extension of

F

p of even degree v

> l

and A the set

of roots in

Fq

of the polynomial

248

Then:

Stepanov

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

{

A1

=

A,

A

2, . . . , Ar} is the set of all cosets in F// A and

{ a1 , a2 , . . . , _{ar} are distinct representatives of the cosets,}

(

v_/2

Ji u)

= (

u

+

ai)

+ (

u

+

ai )P

(1 ::;

_{i ::;}

_{r )} are pairwise coprime polynomials in Fq[u] ;

(iii} r

=/

F//A

/=

p11/2.

(4)

Proof. The main point is (i). First of all we note that f (O)

=

0. Now, if a and /3 are zeros of f (u), then

f (a

+

/3)

= (a +

/3)

+ (a +

f3)P"12

=

a + aPv/2

+

(/3

+

/3Pv/2 )

=

f ( a) + f (/3)

=

0 ,

so that a

+

/3 is also a root of the polynomial f ( u

).

Thus A is a subgroup of F/ .

To prove (ii) let us suppose that fi (u) and fi (u) for i -=/- j have a common root in Fq, say u

=

e.

In that case

and therefore This yields

(

v/2

ai

-

aj

+

ai

-

aj )P

=

0 ,

and we find that ai

-

aj is a root of f (u), hence ai

-

aj E A. But ai

-

aj (/. A according to the choice of a1 , . . . , ar, and we arrive at a contradiction.

Finally, since / A /

=

p11_l2 _{we find that}

Coppa Codes

249

This completes the proof.

Lemma 4.

_{Let Fp be a prime finite field of characteristic p}

>

_2,

Fq an extension of Fp of even degree v

>

1 and s

:S

q112 a positive integer. Let Nq be the number of Fq-rational points of the affine curve Y given by equations

(2)

with polynomials

v/2

fi(x)

=

(u

+

ai )

+

_(u

+

ai)

P

defined by

(

4) .

Then

Proof. We have

Nq

=

L

( 1

+

Xv (fi (u)))

· · ·

( 1

+

Xv (fs(u)))

and hence

It follows from Lemma 2 and Lemma 3 that

Xv (fi(u))

= { �

if u E Ai _{if u E F} q

\

A

and since any two distinct sets

A

and Aj have no common element

we obtain

Nq pv

+

E

(;)

(pv

-

(Jpv/2) pv

+

_{(2s - l}

)pv

_

_s2s-lpv/2

(2pv/2 _ S _{)pv/2 2s-l}

=

_{(2ql/2 _ S )ql/22s-l} This proves the lemma.

250

Stepanov

Now let F

q

be a finite field of characteristic

p

>

2 with

q

=

p

v

elements, where v

>

I is an odd number. Consider again the smooth

projective curve X given over Fq by equations (2). Using the same

arguments as in the proof of Lemma 1 we obtain the following result

(Stepanov and Ozbudak [22]).

Lemma 5 .

Let

Ji, . . . ,

fs

E Fq [u]

be pairwise coprime square-free manic polynomials of the same even degree

m 2 4

and X the smooth projective curve in

JP>

s + l _{defined over}

_Fq

_via

Then the genus g

=

g(X) of the curve X is

g

= (ms - 4)2

s

-Z

+

1

Next, one can show that if v

=

l(mod 2) and

!(

_u

_{) (}

₌

_{u + u}

pC v-1 ) /2 ) (

_{u + u}

p( v+l ) /2 )

_,

then the polynomials

f(u

+

a) and f(u

+

/3)

are relatively prime for

any distinct a, /3

E Fq (Stepanov and Ozbudak [22]).

Lemma 6.

Let v

>

I

be an odd integer,

F

q a finite field of

characteristic p

>

2

and

!(

_{) (}

p( v-1 ) /2 )

(

p(v+l )/2 )

F [ ]

u

=

u + u

u + u

E

q U

For any distinct

a, /3 E F

q the polynomials f (

u

+

a)

and f(

u

+

/3)

are coprzme.

Using the result of Lemma 2 we can easily calculate the number

of Fq-rational points on the affine curve Y given by equations

Gappa Codes 251

Lemma 7.

Let v

>

l be an odd integer, Fq a finite field of char­ acteristic p

>

2 with q

=

pv _elements

! a1, . . . , a5 distinct elements of FqJ and Nq the number of Fq-rational points on the affine curve Y defined over Fq by

Then

Proof. Since _XvUi( u))

=

Xv(]( u + ai))

=

1 for all u E Fq, l

S

i

S

s,

we have

Nq

=

L

(l

+

Xv(f1 (u))) . .

·

(1

+

XvUs (u)))

uEFq

Now we turn into the consideration of fibre products of superellip­ tic curves (3) with polynomials f i(u), 1

S i S

s of a special form. At first we calculate the genus of the smooth projective model of such a product. The following result is a generalization of Lemma

1 .

Lemma 8.

Let f1,j, . . . , fs,j E Fq [u] be pairwise coprime square­ free manic polynomials of the same degree mj_J for j

=

l, 2. Let

µ1, µ₂ be positive integers! µ 2'.

2

a positive divisor of q

-

l! and

m

=

m1µ1

+

m2µ2 2: µ

+

l . Let Y be the fibre product in As+l given over Fq via

Assume that

(m,

µ)

=

l or

(m,

µ)

=

µ. Then the genus g

=

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

µs-1

9

= -

₂- ((µ - l)s(m1

+

m2) - (µ -

1))

+

1

if

(m,

µ)

=

l! and µs-1

252

if (m,µ) = µ.

Stepanov

Let

_{µ1, µ}2

be positive integers. Consider the polynomials

f (u)

=

(1

+

uq( r-l)/2 _1)µ1 (1

+

uq( r+ 1 )/2 _l t2 '

for

T

=

1 ( mod 2), and

g(u)

=

(1

+

uqrf2-1 _1)µ1 (u

+

uqr/2+1 _1) mu2 '

(5)

(6)

for

T

O (mod2). Then we have the following result ( Ozbudak

[1 1] ) .

Lemma 9. Let T

>

1 be a positive integer, and Fqr a finite field of characteristic p

>

₂

with q7 _{elements. Then:}

(i) If T

=

1

(mod 2L

the polynomial f ( u

+

a) and f ( u

+

/3) are co prime in Fqr [u] for any distinct a, /3 E Fqr ;

(ii) If T

=

2 (mod

4 ) and T

>

2

} the polynomials g ( u

+

a) and

g( u

+

/3) are coprime in Fqr

[u]

for any distinct a, /3 E Fqr ; (iii) If T _

0 (

mod

4),

there exists an additive subgroup A C _{F/;. of}

cardinality q2 _{such that the polynomials g( u + a) and g( u}

+

_/3)

are coprzme in Fqr [u] for any a, /3 E Fqr with the condition a

-

/3 (j. A.

Now we calculate the number of Fqr-rational points on the corre­

sponding fibre products. The following is a modification of Gluhov's

result [6] .

Lemma 10. Let T

>

1 be a positive integer, Fqr a finite field of characteristic p

>

2

with q7 _{elements, and µ a positivt: divisor} of q - 1 . Let µ1, µ2 positive integers such that µ1

+

µ2

=

µ} and a1 , . . . , a5 distinct elements of Fqr . Then the following holds:

(i) If T

=

l

( mod 2)

and

Gappa Codes

the number Nq,,. (Y) of Fq,,. -rational points on the curve Y defined over Fq,,. by

(ii) If T

2 (mod 4),

T

>

2,

and

g(u)

=

(1

+

uqr

/2-1 _1 )µ1(1

₊

uqr/2+1 _1 )µ2

J

the number Nq,,.(Z) of Fq,,.-rational points on the curve Z defined over Fq,,. by

zf

=

g(u

+

ai),

253

(iii) If T

O (mod 4),

elements a1 , . . . a8 lie

m

distinct cosets of F/;./A, and -u

=

( 1

+ uq r/2-1 _1 ) µ1

( 1

+ uq r/2+1 _1 ) µ2 g(

)

₁

₊

_{uq-I 1}

₊

_uq-I

'

the number Nq,,.

(

Z)

of Fq,,. -rational points on the curve

Z

defined over Fq,,. by

zf

=

g ( U

+

CYi),

is Nqr(Z)

=

µ8q7•

Similar results for fibre products of superelliptic curves over finite

fields

Fq,,.

hold in characteristic

p

=

2 (Ozbudak [1 1]).

Lemma 11. Let T

>

1 be a positive integer, Fq,,. a finite field of

characteristic p

=

2 with q7 _{elements, and f(u), g(u) polynomials} defined by

(

5)

and

(

6) .

Then:

(i) If T

=

1 (mod 2),

the polynomials f(u

+

a) and f(u

+

/3) are

254

Stepanov {ii) If T

=

2 (mod 4)

_{and T >}

2,

the polynomials g(u

+

a) and g( u

+

/3) are coprime in Fq,. [u] for any a, /3 E Fq,. such that a -/3 (/. Fq2

;

{iii) If T

=

0 ( mod 4),

the polynomials g( u

+

a) and g( u

+

/3) are coprime in Fq,. [u] for any a, /3 E Fq,. such that a -/3 (/. Fq2 .

The number of Fqr-rational points on the corresponding fibre prod­

ucts is determined as follows.

Lemma 12. Let T >

l

be a positive integer, Fq,. a finite field of

characteristic p

=

2 with q7 _{elements, and µ a positive divisor of}

q -

1 .

_{Let µ1, µ2 be positive integers such that µ1}

+

_µ2

=

_{µ, and}

a1 , . . . , as distinct elements of Fq,. . Then the following holds:

{i) If T

=

l (mod 2),

elements a1 , . . . , a8 lie in distinct cosets of

Fqr

I

Fq , and

J'(u)

=

(

1

+

₁

u _{+ u}q(r-1)/2_1_q-t ) µ1

(

_l_+_u __ _ _uq(r+l)/2 _1_q-t ) µ2

'

the number Nqr (Y') of Fqr -rational points on the curve Y' defined over Fq,. by

zf

=

_J'

(

u

+

a;), is Nqr (Y')

=

µ8 q7;

(ii) If T

=

2 ( mod 4),

T

>

2,

elements a1 , . . . , as lie in distinct cosets of Fq,.

/

Fq2 , and

the number Nqr (Z') of Fqr -rational points on the curve Z' defined over Fq,. by

Gappa Codes

255

(iii) If T

O ( mod 4)

J elements

a

1,

. . . ,

CY8 lie in distinct cosets of

Fqr

I

Fq2 J and II U

=

( l

+ u qr/2-1 _1 ) µ1

( 1

+ uq r/2+1 _1 ) µ2

g ( )

₁

₊

_uq-l

₁

₊

_uq-l

_'

the number Nqr (Z") of Fqr-rational points on the curve Z"

defined over Fq,. by

zf

=

g

11

(

U

+

_{CYi ) ,}

§3. Proofs of the Theorems

1 :S; i :S;

s

,

Let p > 2 be a prime number,

Fq

an extension of a prime finite

field F

p

of an even degree v > l, and s ::S; q

1

1

2

a positive integer.

Let

f1, . . . , fs

be pairwise coprime polynomials in

F_q[u]

of the same

degree q

1

1

2

_{defined by ( 4), and Y C _A,8+}

1

_{the affine curve defined}

over k' by equations (2). Let Y C r

s + l

_{be the projective closure}

of

Y ,

and

X

a non-singular projective model of

Y over an algebraic

closure

.F'q

of the field

Fq.

Since curves

Y

and

X

are birationally isomorphic, we have g =

g(Y)

=

g(X) ,

and by Lemma 1

g

=

(sq

l

/2 - 3)2

s

-2

+

1 .

Next, let N

q

be the number of F

q

-rational points of Y and M

q

be

the number of F

q

-rational points of

X.

We have Mq �

Nq

+

l, and

by Lemma 4

Let

n

::S; N

q

be a positive integer, let

x1, . . . , Xn

be Fq-rational

points of the curve

X

at the finite part of

X,

and x

00

be a point of

X at infinity. Set

256

Stepanov

Applying to X the £-construction for r > ( sq

1

1

2 -

3)2

s

-2 and n > r

we obtain a geometric Gappa [n,

k ,

d]

q

-code

C

=

C(Do, D)

with

r < n ::; (2ql/2 _

5

)q1/22

s

-1 ,

k 2:

r -

g

+

1

=

l - (sq

1

₁

_{2 -}

₃₎₂

s

-

2 ,

d 2: n - r .

This proves Theorem 1.

Now, applying to X the !1-construction for

r > (sq

1

₁

_{2 -}

_s)q

1

₁

2

₂

s

-

1

and n > r - (sq

1

1

2 -

3)2

s

-2 we obtain a geometric Goppa [n, k, d]

q

­

code

C*

=

C*(D

0 ,

D)

with

r _ ( sql/2 _ 3)2

s

-2 < n ::; (2ql/2 _

5

)q1/22

s

-l ,

k 2: n - r

+

(sq

1

1

2 -

3)2

s

-2 ,

d 2:

r - ( sq

1

12 - 3)2

s

-l

This gives the result of Theorem 2.

It follows from ( 1) that the relative parameters

R

=

1.� / n and

8

=

d/n

of the codes

C

=

C(Do, D)

and

C*

=

C*(Do, D)

satisfy

( sql/2 _ 3)2

s

-2

R 2: 1 - 8 -

---This proves Corollary 1.

The results of Theorems 3, 4 and 5 follow in the same way from

Lemmas 5, 6, 7, Lemmas 8, 9, 10 and 8, 11, 12, if we consider fibre

products defined by equations

z;

=

J

(

u

+

ai),

1 :S i :S s ,

and also by equations

and

respectively.

zf

=

f(u

+

ai),

zf

=

g(u

+

ai),

zf

=

g(u

+

ai),

zf

=

J'(u

+

ai),

zf

=

g' ( u

+

ai),

zf = g"( u

+

ai),

1 :S i :S s ,

1 :S i :S s ,

1 :S i :S s ,

1 :S i :S

s ,

1 :S i :S s ,

l :S i :S s ,