A simple approach for construction of algebraic-geometric codes from affine plane curves

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 4, JULY 1994 1003

A Simple Approach for Construction of Algebraic-Geometric Codes from Affine

Plane Curves Gui-Liang Feng and T. R. N. Rao, Fellow, ZEEE

Amstract-The current algebraic-geometric (AG) codes are based on the theory of algebraic-geometric curves. In this paper we present a simple approach for the construction of AG codes, which does not require an extensive bnckground m algebraic geometry. Gwen an &ne plane irreduciBk curve and its set of aM rational points, we can find a sequence of monomials x'yj b a d on the equation of the curve. Usiag the first r monomials as a basis for the dual code of a linear code, the designed minimum distance d of the linear code, called the AG code, can be easily detemdned. For these codes, we show a fast decoding procedure with a Complexity O(n7'3), wbkh can correct errors up to [ ( d - 1) / 2J. For this approach it is neither necessary to know the genus of curve nor the basis ef a differential form. This approach can be easily understood by most engineers.

Zmhx Tsnns-Algebraic geometric codes, fast decoding, minimum distance, error-amecting codes.

I. INTRODUCITON HE most important development in the theory of T error-correcting codes in recent years is the introduc-

tion of methods from algebraic-geometric curves for the construction of linear codes. These so-called algebraic- geomehic codes (AG codes) were introduced by Goppa [1]-[3]. In 1982, Tsfasman, Vladut, and Zink [4] obtained an extremely exciting result: they showed the existence of a sequence of AG codes that exceeds the Gilbert- Varshamov bound [SI. For that work, they received the IEEE Information Theory Group Paper Award in 1983. Since then, many papers dealing with AG codes have been published [6]-[14]. However, these AG codes are based on the theory of algebraic-geometric curves [15]-[17]. In [6], Justesen er al. first gave a description of the algebraic-geometric codes defined only by monomials, and further gave a decoding procedure. For that paper, they received the IEEE Information Group Paper Award in 1991. Following that description of AG codes, in this paper we present a simple approach for the construction of AG codes from algebraic geometric curves in an affine plane. We intend to construct a class of linear codes that

Manuscript received August 6, 1992; revised November 23, 1993. This work was supported in part by the Office of Naval Research under Grant NOOO14-93-5-1117 and by the National Science Foundation under Grant

' The authors are with the Center for Advanced Computer Studies, University of Southwestem Louisiana, Lafayette, LA 70504.

IEEE Log Number 9403847.

NCR-9305038.

are very similar to the current AG codes, but the construction is not directly based on the theory of algebraic- geometric curves. These aspects form the essence of this paper.

Initially, a technique to construct AG codes using monomials of the form x'y' as a basis of a linear space was presented in [6]. In this paper we will present a simple approach, which can easily construct AG codes from a very large class of affine plane irreducible curves without directly using the theory of algebraiogeometric curves. Furthermore, we will show that the designed minimum distance of the AG codes constructed by this approach can be easily determined without directly using the Rie- mann-Roch theorem. For some cases, the designed minimum distance obtained by this approach is more precise than that obtained by using the Riemann-Roch theorem. We will also prove that for these AG codes, a fast decoding procedure that can decode such AG codes up to the designed minimum distance with a complexity O(n7I3) [18], can be easily realized. This approach can also be developed and applied to some AG codes from curves in high-dimensional affine spaces [191.

This paper is organized as follows. In Section 11, for easy reference, we include a novel approach for the construction of linear codes, as well as a general method to determine a minimum distance bound for the linear codes, including the cyclic codes. The well-known minimum distance bounds, such as the Hartmann-Tzeng bound 1201, the Roos bound [21], and the van Lint-Wilson AB-method [22], can be also derived by this method. The designed minimum distance for the current AG codes can also be determined by this method without directly using the Riemann-Roch theorem. In Section 111, a simple approach for the construction of AG codes based on the novel approach in Section 11, is presented. For a very large class of affine plane curves, the corresponding AG codes can be easily constructed and the designed minimum distance of the constructed code can also be determined by the method in Section 11. In Section 111, we compare the designed minimum distances obtained by this approach and by the Riemann-Roch theorem for well- known AG codes. In Section IV, we show that the fast decoding procedure of [18] is very efficient for the AG codes constructed by the approach shown in Section 111.

0018-9448/94$04.00 0 1994 IEEE

1004 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 4, JULY 1994

Finally, some of the problems proposed for future research are mentioned in Section V.

and

11. A CONSTRUCTION OF LINEAR CODES AND A MINIMUM DISTANCE BOUND

In this section we first introduce a simple approach for the construction of linear codes, including some of the

(2.5)

current AG codes. Then we present an essential method, which plays a key role in determining the minimum distance bound for the linear codes constructed by this approach. For the current linear codes, the Vandermonde matrix, the Cauchy matrix, and the Riemann-Roch theorem, are very powerful tools in determining the minimum distance bounds. However, all these results are special applications of the essential method.

{hl, h2,..., h,,..., hu} be a sequence of vectors in F:, where h, (hr l , hr2; - - , h,,,), and let S(r) be the linear :pace, oyer Fq :pannTd by the first r vectors of H. Let H {hl, h2,--., hCL,-.-, hJ be a supplementaq sequence and S(r, U) be the linear space over Fq sparme: by only the first r vectors of H and all the yectors of H . In many cases, the supplementary sequence H may be empty, i.e., u = 0. When U = 0, the linear space S(r,O) = S(r). Let hi, (hi, h j l , hi, h j2 , - . - , hi, hjJ. In this p?per, we are interested only in such sequences H and H, which satisfy the conditions given in (2.1142.31.

Let H

For1 si < j I U , hj @ S ( i ) . (2.1)

For each pair of ( i , j ) with i + j I U, if hij I? then there exists an integer + ( i , j ) s U such that

where & , j ) is an increasing function of i and j . A function +(i , j ) is increasing with respect i and j , if + ( i , j ) > +(i - 1, j ) and t$(i,j) > +(i, j - 1) (refer to Examples 2.1 and 2.2). The condition (2.2) forms the basis for the definition of well-behaving matrices (see Lemma 2.2).

It will be clear that d, in practice can be easily estimated. [ tr] to be a parity check matrix of a

linear code over Fq, denoted by C,. For the special case of u = 0, H: reduces to H,. The following examples illustrate that the Reed-Solomon codes and the current AG codes are particular cases of the new linear codes.

Example 2.1: Let n = 2"' - 1 and a be a primitive element of GF(2"'). Let x i express the vecto! (1, ai, C X ~ ~ , - - - , (1, x , x 2 , - - - , x"-l) and H = 0, i.e., U = 0. Then the linear code C, defined by Hr, is an (n , n - r ) Reed-Solomon code. For Reed-Solomon codes, we obtain that &(i,j) = i + j - 1, because b, =

' 1 Y 1 9 1 W , P Erample 2.2: Let ,y be an algebraic geometric curve

with the genus g and let Pl, P2,-.., P,,, and P, be the set of rational points over a finite field Fq. Let H {fi,fi,...,f,) be a sequence of functions such that for a given r > g, the set of the first r fi;nctions is a basis of L((r + g - 1) P,) and further let H 0. Then, the linear code C, defined by HT is an (n, n - r ) AG code C J D , G), where D = Pl + P2 + ,.e., + P,, and G = ( r + g - 1) * P,. For the AG codes, if i, j > g, we have fi E

L[(i + g - 1) P,], and f i E L[( j + g - 1). P,]. Then f, - E ~ [ ( i + j + 2g - 2) - Cl = L[(+(i, j ) + g - 1) - P,1.

Therefore, we have + ( i , j ) = i t j + g - 1. Now we describe a simple method for determining the

minimum distance bound of a linear code defined by HT. Let r be a received vector, c be a codeword of C,, and e be an error vector. We have r = c + e. Define the following syndromes:

(2.6)

S,, = h,,] rT. (2.7)

We define HT

e t H

h = x i - l and h, , = x i + j - 2 = x W . j ) - l = h

si = hi - rT and 3j = ̂ hj - rT

and

From (2.0, (2.21, and the definition of C,, we have

si = hi eT and 3. I 1 = ̂ h. eT, for 1 I i I r , 1 ~j I u (2.8)

Sij = hi . eT, if hi, (2.9) E S ( r , U). and

Any d, - 1 or fewer columns of

are linearly independent (2.3) 7 1

where d, is an integer, We Ffer to these syndromes as known syndromes. When e = 0, these known syndromes are all equal to zero. Any other syndrome is termed as unknown.

S(r , U) and h' E S(r + 1,u). Then we say that s , + ~ and s:+~ h' rT are consistent, and that s:,~ is a consistent element of s,+~. Obviously, si+ can be expressed by Cy' :upi + Cy= lbj3j, where a,+ # 0. There may be several such vectors of h'.

Let h' be a vector such that h'

(2.4)

FE" AND R A 0 CONSTRUCIlON OF AG CODES FROM AFFINE PLANE CURVES 1005

We use to express all syndromes defined by all such h' vectors except h,+l. Thus, Si,,, = s,+, or s:+,, if + ( i , j ) = r + 1. We have the following lemma.

Lemma 2.1: If si = 0 for 1 I i I r + 1 and ij = 0 for 1 I j I U , then s:+~ = 0. If si = 0 for 1 I i I r, $, = 0 for 1 I j I U , and s,+, # 0, then s:+~ # 0.

For the matrix S = [Si,], i,, ,, we use the representa- tion S s , , < , for [ S i j ] l s i s u , l s j ~ u . If some elements of the matrix S are _unknown, we call it an incomplete matrix, and denote it as S . We are only interested in well-behaving matrices, in which whenever S,, is known, all the elements of S, ,, , are known.

We define

A [Si , l1si , j s r + l * (2.10)

Lemma 2.2: S( ,+') is a well-behaving matrix. Proof- If S,, is known, then +(h, k ) I r. From (2.21,

for any 1 I i I h and 1 I j I k, + ( i , j ) I r. Hence, S i j is also known. 0

Lemma 2.3: Suppose r is a codeword c 2 (c,, cZ;--, c,). If there are d nonzero s , + ~ and its consistent elements s:+, in S(,+l), then the weight of the codeword c is at least d.

Proof: The matrix S(,+') can be decomposed into XYX', where

Y =

c1 0 ... 0 i, Ct ..- 0

(j 0 ... . . . . . .

cn -+

Since HT - cT = 0, all syndromes s, = 0, for 1 I i I r, and $j = 0, for 1 I j I U . However, there are d nonzero s , + ~ and its consistent elements. From Lemma 2.1 and Lemma 2.2, S('+ ') is a well-behaving matrix and the known Si, are all zero. Thus, rank (SCr+')) 2 d. This implies that rank

0 Theorem 2.1: Suppose HT is a parity-check matrix of a

linear code C,. For any w, where r I w < U , if the number of s,+, and its consistent elements sa+, in S(,+l) is at least d:, and d: I if,, then the code C, has the minimum distance of at least d:.

h o f i For any nonzero codeword e, if the value of s,+, is not zero, from Lemma 2.3, we obtain that the weight of c is at least d:. If the value of s,+ , is zero, then from (2.3) and d: I d,, we know that there is the smallest value of p < U , such that sh = 0 for 1 s h I p , ik = 0 for 1 I k I U, and s,+~ # 0. Thus, c is also a codeword of codes Cr+,,--- ,C0. Since spfl # 0, the number of s,+, and sb+ , in S ( P + ') is at least d; . Furthermore, regarding c as a codeword of C,, from Lemma 2.3, we obtain that the weight of c is at least d:. Therefore, the minimum

0 Remark 2.1: Theorem 2.1 plays an essential role in the

(Y) 2 d . Hence, the weight of c is at least d.

distance of C, is at least d:.

linear codes, similar to the role of the Riemann-Roch theorem in the current AG codes. When U = 0, Theorem 2.1 is reduced to that derived by the Riemann-Roch theorem. In other words, for the current AG codes the designed minimum distance can be determined by Theo- rem 2.1 instead of the Riemann-Roch theorem. When r I g, for the current AG codes, the designed minimum distance can also be determined by Theorem 2.1. How- ever, for the same case, the Riemann-Roch theorem cannot be used. On the other hand, we also mention that the Hartmann-Tzeng bound, the Roos bound, and the van Lint-Wilson AB-method for cyclic linear codes are also special cases of Theorem 2.1, provided Theorem 2.1 is applied to a suitable form of S'" '). As an application of Theorem 2.1, we will show that the

designed minimum distance of the current AG code can be derived by Theorem 2.1 as the special case of U = 0, instead of directly using the Riemann-Roch theorem. If the total number of s,+ and si+, in S(,+') is increased (may not be strictly increased) with r + 1, we call this sequence H a fine sequence. If a sequence is fine for all r 2 a, we call it a quasi-fine sequence with a. Obviously, a fine sequence is a quasi-fine sequence with a = 1.

Example 2.3: Let us consider H = (1, x , x', . . . , x"-l) given in Example 2.1. It is easy to see that St,, =sI+I-~. Therefore, the number of s,+, and si+, in S(,+') is exactly r + 1. Hence, this H is a fine sequence.

Ewmple 2.4: Let us consider H = {f,, f2,..., f,} in Ex- ample 2.2. By a proof similar to the proof of Theorem 3.2 in [8], or the proof of Theorem 3.1 in the next section, it is found that the number of s,+, and s:+, in S(,+l), is at least r - g + 1 for r z 2g. Hence, this sequence H is a quasi-fine sequence with 2g. (It should be mentioned that the number of s,+ , and s:+ in S(,+') is exactly r - g + 1, for r z 2g.)

For a qwi-jine sequence H, with a, we have the following theorem to determine the minimum distance lower bound.

Theorem 2.2: If r > a and the number of s,+, and its consistent elements in S(,+') is d:, where d: I d,, then the minimum distance of C, is at least d:.

It is known that for the current AG codes, U = 0, i.e., fi = 0, d , = U - g + 1 and a = 2g. From Example 2.4 and Theorem 2.2, we have the following.

Theorem 2.3 (Equivalent to the Riemann-Roch Theorem): For an AG code C,(G,D), the designed minimum distance is r - g + 1, provided r 2 2g, where g is the genus of the algebraic geometric curve.

Remark 2.2: For the current AG codes, in order to determine the designed minimum distance for the case of r < 2g, Theorem 2.2 sometimes is better than Theorem 2.3 (see Example 3.2).

111. A SIMPLE CONSTRUCTION OF AG CODES FROM h N J ! PLANE CURVES

Given an algebraic geometric curve in a projective space, a basis of L(G) can be found, and then a series of

1006

the current AG codes can be constructed. However, this is not a unique approach; From the previous section, we know that the set { H , H ) is essential for the construction of a series of AG codes. A basis of L(G) is only a special example of such a set, i.e., it is a basis of L(G) = { H , 0 ) . In this sectiqn, we prefer a direct construction of some simple { H , H ) for finding a basis of L(G) for a given affine @ n e curve. For convenience, first we restrict H and H initially to some special vectors, that is h, =

[p , (x l , y l ) , pr(x2, y 2 ) , - ~ - , p r ( x n , Y J I , where (xi, yJ are rational points and p , (~, y ) is a polynomial of two variables x and y . For simplicity, we denote H =

Ipl(x, y ) , p2(x , y),..., p,(x, y),..., p u b , y)) . In the Same way, H = {ql(x, y ) , q Z ( x , ~ ) , - * . , q J x , y)} . At a later stage, we will restrict H and H further to special polynomials, i.e., monomials.

In order to construct H and Z-? such that the conditions (2.1) and (2.2) are satisfied, let us define the order of a polynomial f ( x , y). The order is very similar to the nota- tion K,(P) introduced in the thesis of Porter [231. Each polynomial f ( x , y ) is associated with a nonnegative integer f ( x , y ) , which satisfies the following conditions:

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 4, JULY 1994

f ( x , y ) * g ( x , y ) = f ( x , y ) +g(x,y). (3.2)

We define the order of a constant as 0. We will discuss later the procedure to @d the order for a class of curves. Thus, for such H and H, the conditions (2.1) and (2.3) are satisfied if the following conditions (3.3) and (3.4) are satisfied:

--

and if pi (x , y ) -p j (x , y) E Z-?, then

and

where p i ( x , y) + p j ( x , y ) = p + ( x , y ) , & , j ) is an increasing function o E i Z j , i m j ) > 4 ( i - 1, j ) and 4 ( i , j ) > +(i , j - l), when i, j > g, & i , j ) = i + j + g - 1, and g is a constant related to H. Henceforth we will use conditions (3.3) and (3.4) instead of (2.1) and (2.21, respectively.

Let Z be the set of orders of polynomials in H, that is, Z {p i (x , y)li = 1,2,..., U). If an integer p E Z and 0 I p I p u m p is called a gap of I. Let g* denote the n u m i i i a l l the gap: of I , and let g’ = g * + U. We call g‘ the genus of {H, H } . We will see later that the action of g’ is the same as that of g in the current AG codes. Let r’ = r + U. Then, n - r’ is the dimension of a linear code C,.

Hereinafter, we always assume that { H , Z-?) satisfies the conditions (3.3) and (3.41, and that d , of (2.3) can be estimated. From (3.4) and (3.1) and (3.21, we know that

{ H , I?> has the following property: If s, t E I and s + t I p , , (x , y) then s + t E I . (3.5)

Lemma 3.1: If 2g* I s I p u ( x , y ) , then s E I , that is, from 2g*, the orders of polynEG&k H are consecutive.

Proofi Assume that 2g* I s I p , ( x , y ) and s E I, then we have

s = 1 + s - 1 = 2 + s - 2 = ... =g* + s - g *

where 1, s - 1,2, s - 2,..., g* - 1, s - g * + 1 are distinct from each other as well as from g*, s - g*, for s > 2g*. From ( 3 3 , it is known that there exists at least one gap among each of the following: (1, s - 11, (2, s - 2},.-.,{g*, s - g*). Thus, there are at least g* gaps in [O,s - 11. On the other hand, we also know that s is a gap. Therefore, there are at least g * + 1 gaps in I . This is in contradiction with the assumption that there are only

We introduce the following important theorem regarding the designed minimum distance.

Theorem 3.1: If r > g* (i.e., r‘ > g’) and r - g * + 1 I d,, then the minimum distance of an (n, n - r ’ ) linear code C, defined by [ :], is at least r - g* + 1 (i.e., r’ - g ’ + 1).

Proofi Let us consider S(,+l) associated with C,. From definition (2.71, for i, j I r + 1, S I , = C;= lrkpl(xk, yJp ( x k , y k ) . We construct a new integer

+ p , ( x , y ) . From (3.4), and the definition of srtl and

sL+ 1, it is known that the number of s,+ and sL+ in S(,+l) is equal to the number of p,+ l(x, y ) in Pcr+l).

On the other hand, let A be this matrix

p,+ gx+ l(x, y).Obviously, Pcr+l) can be obtained by ?Eking all the rows and columns that are associated with the gaps in Z from A. Thus, P(‘+l) can be obtained by deleting g* rows and g* columns from A. Since the original number of p r + l in A is pr+l + 1, the number of P , + ~ in P ( , + ~ ) must b e t least p , z + 1 - 2g*. F r o m the definition of g*, thZZGndition of r > g*, and Lemma 3.1, we have that p,+ = r + g* and p r + l + 1 - 2g* = r - g* + 1 = r’ - g‘Tl . Obviously, thisvalue in- creases with r. From Theorem 2.2, the minimum distance

0 Remark 3.1: By the proof of Theorem 3.1, Example 2.4

can also be proved. The value of r’ -g’ + 1 = r -g* + 1 is called the

designed minimum distance. In the proof of Theorem 3.1, when r < 2g*, there are some rows and columns that are associated with the gaps, and that have common P , + ~ . Thus, deletion of some P , + ~ ’ S may be counted tFE. Therefore, when r < 2 g * 3 1 s theorem can more pre- cisely be stated as follows.

Theorem 3.2: Let d , be the minimal number of ~ ~ + ~ ( x , y ) in PCs+l) for s = r , r + 1,...,2g*, when r < 2g*; and let d, be the number of p,+ l(x, y) in P(,+l),

g* gaps in I . 0

matrix p ( r + 1 ) = f~s,](r+1)x(r+l), where pi, = p i ( x , ~ )

[ 4 , 1 ( p r + , + l ) x ( p , + , + 1 ) and 4, = i + i - 27 where

of C, is at least r’ - g ’ + 1 = r - g * + 1.

FENG AND RA0 CONSTRUCTION OF AG CODES MOM AFFINE PLANE CURVES 1007

when r 2 2g*. If d, I d,, then the minimum distance of code C,, defined by [ t,], is at least d,.

Thus, when r < 2g*, we can determine a more precise minimum distance bound (see Example 3.2) using Theo- rem 3.2.

In order to construct such H and I? easily, we are interested in simple polynomials. A monomial is a polynomial with exactly one nonzero term, i.e., it has the form xayb, where a, b 2 0 in the current context. For more convenience, in the following discussion we restrict p i ( x , y ) and q j ( x , y ) to bf monomials, that is, H p {xaayb+ = 1,2 , -*- , v) and H {xCJydJl j = 1,2,---, U). We have

From (3.3) we have x"lybl = 1. Let a (similarly p ) be a value such that there are at most a (similarly p ) rational points having the same yi (similarly xi) . We have the following useful lemma.

Lemma 3.2: If

{x'y'lo I i I a - 1,0 I j I q - 2) G H , U I? (3.6)

or

{x'yjlO I i I q - 2,O I j I /3 - 1) c H, U I? (3.6')

then all the columns of [z"] are linearly independent, that is,

d , = U + U + 1 . (3.6" )

Proofi Let us prove the result under the assumption of (3.6). Assume that (3.6) is satisfied, and that (3.6") is not satisfied. Suppose that the 1:)th columns for k =

l,..., h and p = l,**',pk are linearly dependent, where ylhk) have the same value Yk. From (3.61, we have

for i = 0, l , . . . , a - 1; j = 0, l,..., q - 2

for i = 0, l,..., a - 1; J = 0, l,..., q - 2. since the number of distinct nonzero Yk is at most q, from the above equations, we have

Pk

cq) * + ) = 0 for each k and i = 0, 1, .**, ff - 1. p= 1

On the other hand, since the number of distinct q k ) is at most a, and they are all distinct from each other (because y$k) are the same), we have

Cif) = 0 for each k

This is in contradiction with

and p = 1,*",pk.

the condition that c $ k ) # 0. 0

From the above discussion, it is clear that the smaller the value of g ' , the greater the efficiency of the linear codes C,. In the following di-ion, we will give an approach to determine H and H for a given affine plane curve such that

1*) the conditions (3.3) and (3.4) are satisfied, and d, can be estimated, and

2*) g' is as small as possible.

If there exist integers i*,w, and a monomial xu'yb* such that for i > i*,

then we call the smallest w satisfying the above equation and i* as the width and the start value of H , respectively. Later on we will see that w is the block width of Hankel- block Hankel syndrome matrix. The smaller the value of w, the faster the decoding. Thus, the width w is an important parameter in the decoding procedure. This parameter should be considered during the construction of H and H.

A. Type I of Plane Affine Curves

affine plane irreducible curves over Fq: In this section, we will consider the following type of

x a + yb + g ( x , y) = 0 (3.7)

where gcd(a, b ) = 1 and a > b > deg g(x, y). Set H = 0 for the above curves. In order to construct

H, first we give an important lemma. Lemma 3.3: Given two relatively prime integers a and

b , there exists the smallest integer g* such that

1') for m 2 2g*, m can be written as m = as + bt, where s, t 2 0; and

2') in [0,2g* - 11 there are exactly g* such m,'s where i = 0, l , . . . , g* - 1, m, = as, + bt,, and sa, t , 2 0.

Roo$ Since a and b are relatively prime, ( a - 1Xb - 1) is even. Let g* = (a - 1Xb - 1)/2.

First we prove 1'). Since a and b are relatively prime, from Euclidean algorithm, there are positive integers A, B, A', and B' such that

where A, B 2 l , A +A' = b , and B + B' = a. From the definition, 2g* = ( a - 1Xb - 1) = a(A - 1) + b(a - 1 - B') , where A - l , a - 1 - B' 2 0. Thus, 2g* can be written as m = as + bt, where s, t 2 0. Now suppose that for m 2 2g*,m can be written as m = as + bt, where s , t 2 0, we will prove that m + 1 can also be written as m + 1 = as' + bt'. Since m 2 2g* and m = as + b t , s 2 A' or t 2 B' (if it is not true, i.e., s <A' and t < B' , then, ab - 1 - a - b = ab - aA + bB' - a - b 2 aA' + bB' - a - b 2 as + bt 2 ( a - 1Xb - 1). This is impossible). If s 2 A', then using (3.8), we have m + 1 = a(s - A ' ) + b(t + B). Thus, s' = s -A' 2 0 and t ' = t + B 2 0. Sim-

1008 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 4, JULY 1994

ilarly, if t 2 B', we have m + 1 = as' + bt'. Thus, the proof of 1') is completed.

Now we prove 2'). There are (a - 1) ( b - 1) pairs of (x,y)such that 0 I X I b - 2andO I Y I a - 2. Among them, there is a l-to-1 map JI: ( x , y ) - ( x * , y * ) such that x = b - 2 - x * and y = a - 2 - y * . It can be easilyseen that if a * x * + bey* 2 ( a - 1Xb - l), then a - x + b - y = 2(a - 1Xb - 1) - 2 - a * x * - b - y * < ( a - 1Xb - 1). Thus, there are half pairs of ( x , y ) , for which a * x + b - y < ( a - 1Xb - l), that is, there are g* integers, which are in [ 0,2g*) and can be written as m = as + bt. Thus, 2') is proved.

From the above proof, it is easily seen that g* is the 1 smallest integer that satisfies the conditions 1') and 2'). 0

Now we have the following theorem. Theorem 3.3: For a plane affine curve of (3.71, fi = 0

and there are two H's such that 1*) and 2*) are satisfied.

H"): all x h y k for 0 I h I a - 1, and their orders form an ascending sequence. Its width w is a.

H"): all x h y k for 0 I k I b - 1, and their orders form an ascending sequence. Its width w is b.

For H(') and El('), 5 = b, y = a, and 2g* = (a - 1) ( b - 1) are defined in Lemma-3.3.

Proof Let us prove that H(') satisfies 1*) and 2*). Since 3 = b, y = a, and since h < a if xhyk E W'), it

can easily be checked that if (h, k) # (h', k') and 0 I h, h' I a - 1, then x h y k # x h ' y k ' . On the other hand, their orders form an a E d i n g F u e n c e and 1 E H('). There- fore, (3.3) is satisfied.

For any s and t , we have, xhsyks -xh'ykt = ~ ~ s + ~ i Y k s + k t .

Repeatedly using (3.7), we can write this as a linear combination of x h y k for h I a - 1 and bh + ak I b(h, + h,) + a(k, + kJ . Thus, these x h y k E H'". Therefore,

xh*yks .xh'ykr E S(r) , where xa,ybr= b(h, + h,) + a(k, + k,). On the other hand, it is easilyseen that xhsyks .xhrykt 4 S(r - 11, because x " r - ~ y ~ r - ~ < b(h, + h,) + a(k, + kf ) . Thus, (3.4) is satisfied.

It can be easily checked that there are at most a = a rational points having the same y,, /3 = b rational points having the same x t , and that (3.6) is satisfied. Thus, d , can be estimated.

Now we are going to prove that there are g* gaps in I. First, we prove that if s 2 2g*, then s E I. From the proof of Lemma 3.3, 2g* = a(A - 1) + b(a - 1 - B'). Furthermore, since 0 I B' < a, we have 0 I (a - 1 - B') < a. Thus, 2g* can be written in the form of as + bt, where 0 I t < a. Suppose that for any m 2 2g*, m = as + bt, where 0 I t < a, we want to prove that m + 1 =

as' + bt', where 0 I t' < a. We consider two cases. Case I: t 2 B'. In this case, using (3.81, we have m + 1

= a(A + s) + b(t - B') . Thus, 0 s t - B' a t ' < a. Case 2: t < B'. In the same manner, we have m + 1 =

a(s - A') + b(t + B). Thus, 0 I t + B a t ' < a. because

(B + B') = a. From these two cases, we know that m + 1 can be

written in the form as' + bt', where 0 I t' I a - 1. Therefore, for any s 2 2g*, s E 1.

The proof showing that there are g* gaps of I in [0,2g*] is similar to that of Lemma 3.3. On the other hand, g* is as small as possible. Thus, the proof that H(') satisfies 1*) and 2*) is completed. The above proof is also true for H"). 0

From 2g*, the orders in I are consecutive. On the other hand, (a - l)b > 2g* (similarly ( b - l)a > 2g*). Thus, we have the following theorem.

Theorem 3.4 (Regarding the Properties of H") and H(2)): For H(') , all the monomials after x a - ' can be written in the form

xhlyk/ -yP for p = 0,l.e. and 1 = 1,2 e - . a

where

h, = a - 1, k, = 0

and

h,,, = h, -A' and k,,, = k, + B for h, >A' h,,, = h , + A a n d k , + , = k , - B ' forh,<A' .

For H(*), all the monomials after yb-' can be written in .the form

x h l y k / - x P for p = 0 , l and 1 = 1,2... b

where

h, = 0 and k, = b - 1

and

h,,, = h, + A and k , + l = k, - B' h,, = h, -A ' and k,, , = k, + B

for k, 2 B' for k, < B'.

For a fixed p , the terms x h / y k / . y P for 1 = 1,2;..,b (respectively, 1 = 1,2,-.., a) are called a periodic sequence with width b (respectively, a ) for H(' ) (respectively,\H(')). For each periodic sequence of H( ' ) (respectively, H'')), if there are t such 1 with h, < A' (respectively, h, < B'), we call it a t-changed sequence. If t = 0, we call it a single ascending sequence. These properties are useful in design- ing a fast decoding procedure for these AG codes.

Corollaly 3.1: If A' = 1 (similarly B' = l), then the periodic sequence of H(' ) (similarly H'')) is a single ascending sequence.

Example 3.1: Let us consider a curve with the equation x5 + y 3 + y = 0 over GF(29. From Theorem 3.3, we have x = 3, y = 5, 2g* = 8, i.e., g* = 4, and -

H(' ) = { 1, x, y , x 2 , q, x3, y 2 , x 2 y , x 4 y p , q 2 y p , x 3 y y p , y3yp, x 2 y 2 y p , for p = 0,1,2 } . I(1) = {0 ,3 ,5 ,6 ,8 ,9 ,10 ,11 ,12 ,13 ,14 ,15 ,16 ,17~~-} , w = 5 , i* = 8.

FENG AND R A 0 CONSTRUCTION OF AG CODES FROM AFFINE PLANE CURVES 1009

{ x 4 y P , xy'yp, x3yyP, y3yP, x2y2yP) is a 2-changed sequence.

H(') = ( I , x , y , x 2 , x y , x 3 , y 2 x p , x'yxp, x4xp, for p = 0 , 1 , 2

Z ~ 2 ~ = { 0 , 3 , 5 , 6 , 8 , 9 , 1 0 , 1 1 , 1 2 ~ ~ ~ } , w = 3 , i * = 6 . }.

{xpy' , xPx2y, xPx4) is a single ascending sequence.

Comparing these two H's, we find that H(') is better than H('), because the width of H(') is less than that of H(l) .

Example 3.2: Let x 5 - y 4 - y = 0 over GFC14). This is an affine Hermitian curve over GF(24). From Theorem 3.3, g* is exactly equal to the genus g of the Hermitian curve, and

Let xhlykl and xhzyk2 belong to H"). Thus, 0 I h, +

a) If 0 I h , + h, I a - 1 , then ~ ~ 1 + ~ z y ~ 1 + ~ z E H(') c

b) If a I h, + h, I :(a - 1) and 0 I k, + k, I c - 1,

h , I 2(a - 1). We consider the following three cases.

S(r,

then ~ ~ l + ~ 2 .yk1+k2 E H c S(r, U), for some r.

for r.

which was discussed in [MI. When r > g, the minimum c) If a I h, + h, I 2(a - 1) and c I k, + k,, then distance bound obtained by Theorem 3.1 is equal to the ~ ~ l + ~ z y k l + k z = - x " ~ + ~ z - ~ y k l + k z + b - ~ ~ l + ~ z - ~ *

designed minimum distance derived by the ykl+k2-c - g ( x , y ) . Obviously, ~ ~ 1 + ~ 2 - ~ . yk l+kz+b E H('). If Riemann-Roch theorem. When r I g , the all the monomials in ~ ~ 1 + ~ z - ~ - y k l + k z - c g ( x , y ) belong to Riemann-Roch theorem cannot be applied. However, either case a) or case b), then ~ ~ 1 + ~ z . y k l + k z E H(') U fi. Theorem 3.2 can be applied. When g < r I 2 g , for some , Otherwise, repeatedly using (3.7), they can be eventually cases, the minimum distance bound obtained by using .

Theorem 3.2 is better than the designed minimh distance obtained by using the Riemann-Roch theorem. For example, for r = 11, that is, the rows of parity-check matrix H,, are formed from

from Theorem 3.2 we obtain that the minimum distance of C,, defined by H,, is at least 8. From [24], the true minimum distance of C,, is exactly 8. But the designed minimum distance obtained by using the Riemann-Roch theorem is 6.

B. l jpe IZ of Plune AfFne Curves In this section, we are going to discuss another type of

plane affine irreducible curves over Fq:

(3.9) where gcd(a, b) = 1 and a + c, b +,c > deg g(x , y ) .

Theorem 3.5: There exists-a { H , H ) satisfying (1*) and (2*), where H = H(') and H = {xsy'ls = a, a + 1,.. . ,2(a - 1) and t = 0, l , . . . , c - 1). Thus, U = ( a - l ) c , g ' = ( a - 1Xb - 1 ) / 2 + ( a - 1)c.

Proofi From Theorem 3.3, it is known that (3.3) is satisfie! and d , can be estimated. Now we prove that {H('), H ) satisfies (3.4).

xayc + yb+c + g ( x , y ) = 0

reduced to the monomials, which belong to, either case a) or case b). Thus, ~ ~ 1 + ~ z -yk l+kz E H(') U H. According to the definition of order, ~ ~ 1 + ~ z -ykl+kz E S(r, U). From (3.1) and (3.2), it can be easily seen that ~ ~ l + ~ z y ~ 1 + ~ z I S(r - 1, U). Thus, (3.4) is satisfied.

Therefore, {H(') , fi) satisfies ( I*) . It-can be easily checked that the number of monoFials

in H cannot be decreased. Since the monomials in H are all linearly independent and each one is a product cf some two monomials in H('), we know that { H ( ' ) , H } satisfies 2*). CI

Example 3.3: Let us consider an affine Klein quartic curve with the equation x 3 y + y 3 + x = 0 over GF(2'). We have 5 = 2, - y = 3, respectively.

H(') = ( 1 , X , Y , X 2 , xy,Y2,X2Y, xy2,Y3,X2Y2, xy3,Y4 .-}, w = 3 and i* = 3 .

I ( " = {0 ,2 ,3 ,4 ,5 ,6 ,7 ,8 ,9 ,10 ,11 ,12**~} , g* = 1. --- For Theorem 3.5, we obtain that fi = { x 3 , x4). Thus, U = 2, g ' = 3. In this example, g' = g , where g is the genus of Klein quartic curve.

Example 3.4: Let us consider an example in [61 with the equation x 5 y 2 + y 9 + x 2 = 0 over GlQ5). We have x =

7 , y = 5. -

H(') = ( 1 , Y , X , Y * , ~ , X 2 , Y 3 , x y 2 , x2y,y4,x3,xy3,x2y2,y5,~3y,xy4,

x 4 , x 2 y 3 , y 6 , x 3 y 2 , x y 5 , x 4 y , x 2 y 4 , y 7 , x 3 y 3 , xy6 . - a }, w = 5 , i* = 16.

Z(l) = {0,5,7,10,12,14,15,17,19,20,21,22,24,25,26,27, 28,29,30,31,32,33,34,35,36,37 * * e } , g * = 1 2 .

fi = { x 5 , x6, x 7 , x 8 , x5y, x6y, x7y, x8y) .

Thus, we obtain that U = 8 and g' = 12 + 8 = 20.

1010 IEEE TRANSACl7ONS ON INFORMATION THEORY, VOL. 40, NO. 4, JULY 1994

C. Other Plane Afine Curves An affine plane irreducible curve that cannot be in-

cluded in any of the above types may be transformed to one of the above two types. We show two examples that illustrate this transformation procedure.

For example, let us consider the following type of affine plane irreducible curves over Fq:

xac + yb" + g ( x " , y ) = 0 (3.10)

where gcd(a, b) = 1, gcd(c, a) = 1, q - 1 is not divisible by c, and the degree of g(xc, y ) < ac and the degree of g(xc, y ) < bc. Let z = x", then the equation is reduced to za + ybC + g(z, y ) = 0. This is Type I and we can follow Theorem 3.3 to construct H('). Similar to the proof of Theorem 3.3,1*) and 2*) are satisfied:Since q - 1 cannot be divided by c, xi # xi implies zi # zj . This implies that these two curves have the same set of rational points. Thus, we have the following theorem;

Theorem 3.6: There exists a { H , HI satisfying 1*) and 2*), whe? H = H(') = {x'"yj10 I i I a - 1, 0 I j I q -

We illustrate the above theorem with the help of an example.

Example 3.5: Let x 6 + y 4 + y = 0 over GF(28). Let z = x2, then the equation is reduced to z3 + y 4 + y = 0. We have a = 3 , b c = 4 , 2 = 4 , i.e., x = 2 , and - y = 3 , respectively.

2) and H = 0, d, = U (9 - 1).

g* = 3 and w = 3.

= {0,3,4,6,7,8,9,10,11,12,13 - - - }. -- A = 0 , u = o .

In this example, g' = 3 + 0 = 3 and w = 3. Now we show the second example. Example 3.6: Let C be a projective Hermitian curve

over GF(r2) with the equation

u r + l + u r + l + ,,,'+I = 0.

The affine Hermitian curve can be transformed into the Type I (see [ill).

Remark 3.2: Following the ideas in Sections 111-B and 111-C and based on the result in 111-A, some new types of plane curves can be derived. These results are however, not included here.

IV. A FAST DECODING PROCEDURE In this section, we prove that the AG codes derived by

the approach in the previous sections can be fast decoded up to the designed minimum distance using the decoding procedure in [18].

Let P = (Pl, P2,-.-, P,) be a set of n rational points on the curve X over F,. We consider the AG codes Cr

defined by a parity check matrix it,], obtained from X over Fq using the construction in

Here we introduce some important concepts. Let A =

[A,,l]os,,,<,, be a Hankel matrix, i.e., A,,, = a,,,. A (p, m)-block Hankel matrix is a pm X pm matrix of the form A = [A,,l]oir,l< , where each A,,, = A,+, is an m x m matrix. A ( p , m5-Hankel-block Hankel matrix is a

pm x p m matrix of the form A = [A,,l]Oi,,l<p, where each A,,, = A('+]) is an m X m Hankel matrix, that is, At,:,) = at:{). In [MI, we presented a modified Gaussian elimination (algorithm BH) on A, which can be accom- plished in 0(m3p2) accumulations. Let A' =

[A, , ,I, E I , , , E 1'. We say that A = [A,, ,I, E 1, , E J can be embedded in A', if ZcZ' and JcJ'. If A' is a Hankel (respectively block Hankel, Hankel-block Hankel) matrix, then A is called an embedded Hankel (respectively, block Hankel, Hankel-block Hankel) matrix. If each A,, , = A('+') is an m X m matrix formed by some Hankel matrices, then A = [A,, ,lo ,, , < is called a (p, m) general HankeZ- block Hankel matrix.

If a matrix A can be embedded in a Hankel, block Hankel, Hankel-block Hankel, or general Hankel-block Hankel matrix, then many matrix problems for A can be solved by applying efficient algorithms on the embedding host. The complexity is mostly no more than that for the embedding host. This is a popular technique in numerical linear algebra. Algorithm BH can be easily adopted for embedded matrices. We call this the modified Algorithm BH*. Algorithm BH* was discussed in detail in [181.

Let U, e, and c be a received vector, an error vector, and a code vector, respectively. Then we define the syndromes as in (2.6) and (2.71, and the syndrome matrix S(r+l) as in (2.10). We must mention that for the AG codes constructed in the previous sections, H's are all either H(')'s or H(2)'s. From Theorem 3.4, we have the following Lem- mas and an important theorem.

Lemma 4.1: Let us consider such a submatrix

e previous sections.

(Si,)* s 1 < k , p s 1 i 47 where X"'+lyb,+l = X',+A Y h-8'

x a j + ~ y b j + ~ = X a l + A Y b1-B'

for h 5 i < k forp i j < q .

Then it is a Hankel matrix.

conditions, we have h f i From definitions (2.3)-(2.6) and the above

St+,., = ht+l.] . rT n

p= 1 n

= .y:t+1 ' X a J . Y b J . r P P P

= c ,>+A .y; , -B' . x a , . y ; .

= x:' .y;' . q + A .y ; -B'

=, X ; . y ; z . x a , + l . y ; + l .

- St,,+l*

P rP

- rP

p= 1 n

p= 1 n

P 'P P= 1

-

FENG AND RA0 CONSTRUCTION OF AG CODES FROM AFFINE PLANE CURVES 1011

Fig. 1.

Thus, we see that the submatrix ( S I J ) h s r j k , p s J j q is a Hankel matrix. 0

From Theorem 3.4 and Lemma 4.1, we can derive the following lemma.

Lemma 4.2: If {xalyblyplZ = 1,2,...,u} is a t-changed sequence, then a submatrix (S1J)t41s t + a - l , A s l s A + a - l is a matrix formed by ( t + 1)' sub-Hankel matrices, where xaeybt and xa*yb* are the initial monomials of two periodic sequences, with width U for H(').

has the form, shown m Fig. 1, where A, are the Hankel matrices for s = 1,2,--- , 9, and the elements at the same line are equal to each other.

From these lemmas we have the following main theorem.

Theorem 4.1: S@+') is an embedded general Hankel- block Hankel matrix.

h f i We only prove this theorem for the case of H = H"). The proof for the case of H = H(*) can be similarly derived. For the case of H = H('), from Theo- rem 3.3 and Theorem 3.4, we have

and

For when = 2, ( S i j ) f s r _ < ~ + a - l , A \ r ] i A + a - l

w = a

X a l + w y b , + w = x " , y b t y .

Now we consider a submatrix ~ S l , l ~ t ~ l ~ f + a ~ l , h l , s h + a ~ l of Scr+'), where x a ~ y b t and x"*yb~ are the initial monomials of two periodic sequences, with width a for H('). We first prove that it is a block Hankel matrix.

From definitions (2.3)-(2.6), we have

Thus, S(,+ ') is a block Hankel matrix.

On the other hand, from Lemma 4.2, it is known that each submatrix is a matrix formed by ( t + 112 sub-Hankel matrices. Since the result is valid for i, j 2 i * , we claim that S@+ ') is an embedded general Hankel-block Hankel matrix. 0

From the decoding procedure based on a majority voting scheme in [8] and [18], any [(d - 1)/21 or fewer errors can be corrected by performing a Gaussian elimination on the syndrome matrix S(,+ '), where d is the number of s,+ and si+ From Theorem 2.1, d is the designed minimum distance. On the other hand, following the same method as that used to derive Algorithm BH* in [MI, a general- ized algorithm BH*, which has the same complexity O(n7/3), can be derived. Thus, using the decoding procedure based on a majority voting scheme in 181 and [MI and this GBH* algorithm, the decoding of C, up to the designed minimum distance can be realized.

V. CONCLUSIONS In this paper, we introduced a new construction of

linear codes and presented a very general method to determine the minimum distance lower bounds. Some current AG codes can be obtained by the new construction. Moreover the well-known BCH bound, HT bound, Roos bound, and van Lint-Wilson AB method for cyclic codes, and the designed minimum distance for the current AG codes can be derived by a simple method. Based on the new construction, we have shown a simple construction (the basis is formed by monomials only) for plane AG codes, without directly using the theory of algebraic geometry, and further indicated that their designed minimum distance can be easily determined. For some cases, the results obtained by the new approach are better than that obtained by the existing approach for the construction of AG codes. For the codes obtained by the new approach, a fast decoding procedure that can decode these codes up to [(d - 11/21 with a complexity O(n7/3), was also presented and proved. This approach can easily be understood by most motivated readers without any deep knowl- edge of algebraic geometry.

This is a primary work on deriving general AG codes not requiring the theory of algebraic geometric curves. However, for more general or all plane curves, the procedure to construct AG codes, using only a part of monomials as the basis, still remains unsolved.

ACKNOWLEDGMENTS The authors would like to thank M. S . Kolluru and P.

Xiao for useful discussions and valuable comments during the preparation of this paper. The authors are very grate- ful to the referees, the Associate Editor, Prof. A. Tietavainen, Dr. I. Honkala, Prof. R. Pellikaan, Prof. T. Haholdt, and Prof. H. E. Jensen for their many valuable suggestions and comments during the revision of this paper.

REFERENCES [l] V. D. Goppa, "Codes associated with divisors," problemy Peredachi

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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 4, JULY 1994

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A simple approach for construction of algebraic-geometric codes from affine plane curves