IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 6, NOVEMBER 1992 1821

9. The exponent of G is 3. Hence, the extension field is GF(22). The mixed-radix number system considered has mixed-radixes m, = 3 and m, = 3. Let (Y be an element of order 3 in GF(2*) Then, for a nine-tuple,

the DFT is given by

The conjugacy classes are {00), {Ol, 02), {10,20), {11,22), and {12,21). There are 5 conjugacy classes and every conjugacy class can take either zero or nonzero elements. Hence, there are total 32 codes including the trival codes. The transform vectors of the idempotent generators of all the nontrivial codes are listed in Table I. In Table I, the codes are labeled Ql, i = 1 to 15 and dual code of Q, is denoted by @*. The entries 0 and 1 represent additive and multiplicative identities of the extension field GF(22).

Explicitly, the codewords of the abelian code Q13 are

0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 1 1 0 1 1 0 ’ 0 0 0 0 0 0 0 0 0

It is seen that the codewords are closed under cyclic shifts also. It can be checked that codes Q2,Q,3,Q,5,@2,@13,@,5 are cyclic codes also. These are the only cyclic codes of length 9. If one were to study these codes without considering them as abelian codes the extension field required is GF(2‘).

Example 2: Consider binary, length 25, cyclic codes and abelian codes where the abelian group is the direct product of two cyclic groups of order 5 each. The exponent of this abelian group is 5. The least integer m such that 25 divides 2’” - 1 is 20. Hence, the cyclic codes require the extension field GF(220). Whereas in the case of abelian codes since the least integer m such that 5 divides 2” - 1 is 4, the extension field required is GF(24). It can be shown [12] that, also in this case, all the cyclic codes are included in the set of abelian codes.

VI. CONCLUSION Abelian codes over finte fields are characterized in the trans-

form domain using DFT defined over an appropriate extension field. A simple transform domain description of the dual code of a given abelian code is obtained. Also, it is shown that the idempotent generator of the dual of a given abelian code can be easily obtained from the idempotent generator of the given abelian code. It will be of considerable interest to extend the transform decoding that has been obtained for cyclic codes 191 to abelian codes.

REFERENCES 111 W. W. Peterson and E. J. Weldon, Error Correcting Codes, 2nd ed.

Cambridge, MA: MIT Press, 1972. [2] F. J. MacWilliams and N. J. A. Sloane, The 7heory of Error

Correcting Codes. Amsterdam: North Holland, 1977. [31 S. D. Berman, “On the theory of group codes,” Kibemetzka, vol. 3,

no. 1, 1967. 141 -, “Semisimple cyclic and abelian codes,” Kibemetika, vol. 3,

[5] P. Camion, “Abelian codes,” Tech. Rep. No. 1059, Mathematical Res. Center, 1971.

pp. 21-30, 1967.

[61 P. Delsarte, “Automorphisms of abelian codes,” Phillips Res. Rep.,

[7] F. J. MacWilliams, “Binary codes which are ideals in the group algebra of an abelian group,” BSTJ, vol. 49, pp. 987-1011, 1970.

[SI R. E. Blahut, “Algebraic codes in the frequency domain,’’ CISM Courses and Lectures, no. 258, pp. 447-494, 1979.

[9] -, Theory and Practice of Error Control Codes. New York: Addison-Wesley, 1983.

[lo] M. U. Siddiqi, “A study of permutation-invariant linear systems,” Ph.D. thesis, Indian Inst. of Technol., Kanpur, 1976.

[ll] M. Hall, Jr., The Theory of Groups. New York: MacMillan, 1964. [12] H. S. Madhusudhana, “On abelian codes which are closed under

cyclic shifts,” M. Tech thesis, Indian Inst. of Technol., Kanpur, 1987.

vol. 25, pp. 389-402, 1970.

Weighted Reed-Muller Codes and Algebraic-Geometric Codes

Anders Bjzrt Sprensen

Abstract-A generalization of the Reed-Muller codes, the weighted Reed-Muller codes, is presented. The code parameters are estimated and the duals are shown to also be weighted Reed-Muller codes. It is shown how the minimum distance of certain algebraic-geometric codes in many cases can be determined exactly or an upper bound can be found, using subcodes which are weighted Reed-Muller codes.

Index Tems-Error-correcting codes, Reed-Muller codes, finite pro- jective geometry, algebraic-geometric codes.

I. INTRODUCTION

The so called generalized Reed-Muller codes were intro- duced by Kasami, Lin, and Peterson [l], [3] and Weldon [2] about 1970. The multivariable approach [4] and the introduction of algebraic-geometric codes in the eighties inspired Lachaud [5] to introduce the projective Reed-Muller codes. Their param- eters and cyclic structure were determined in [6] by the author. In this correspondence, we generalize some of the results of [6] by the presentation of so called weighted Reed-Muller codes and study their connection to algebraic-geometric codes.

In Section 11, we define the affine and projective version of the weighted Reed-Muller codes, the WARM-codes and WPRM-codes, and we determine their code parameters (for the minimum distance in the projective case we give only an esti- mate). In Section 111, we show that the dual of a WARM-code is again a WARM-code and that the dual of a WPRM-code is almost a WPRM-code. In Section IV, we show that a certain class of algebraic-geometric codes are in fact WARM-codes, and as an example we show that the curve used by Pellikaan et al. in [7], as a basis to prove that all linear codes are weakly algebraic-geometric, gives codes which are WARM-codes. Hence, the minimum distance of these codes are determined. Furthermore, we study the Hansen-Stichtenoth codes [lo], [ l l ] and we give an upper bound of the minimum distance using maximal WARM-subcodes.

Manuscript received July 23, 1991. This work was supported by the Danish Natural Science Research Council and by the Danish Research Academy.

The author is with the Department of Mathematics, University of Aarhus, 8000 Aarhus C, DK-Denmark and the Institute of Experimental Mathematics, University of Essen, Ellernstrasse 29, 4300 Essen 12, Germany.

IEEE Log Number 9201545.

0018-9448/92$03.00 0 1992 IEEE

1822 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 6, NOVEMBER 1992

11. WEIGHTED AFFINE AND PROJECTIVE REED-MULLER CODES

A. The Definition

Let Fq be the finite field with q elements, q a prime power. By A"(Fq) resp. P m ( F q ) we mean the m-dimensional affine resp. projective space defined over Fq. We will often just write Am and P"'. Fq[X1;.., X,] is the ring of polynomials in m variables with coefficients in Fq. If we attach to each variable X, a natural number w,, called weight of X,, we speak about the ring of weighted polynomials, W q [ X 1 ; . . , X,]. If all weights are 1 (trivial weights), we have just the usual polynomial ring. The weighted degree of F E WFq[X1;.., X,], is defined as deg, ( F ) = deg,,,(F(X,;.., X,)) = deg(F(X;I;.., X?), where deg is the usual degree. We will, without loss of generality, always assume that the weights are ordered w1 I w2 I ... I w,.

The general set-up in the affine case is the following: Con- sider the evaluation map

(6: m q [ X I , . . . , X,l --f (F,)'"

( 6 ( F ) = ( ... , F ( P , ) , .'. ), given by

where P, runs through Am(Fq). The W is actually not needed in this definition, since we do not use the weights. We will in the following therefore sometimes omit it.

The definition of the affine Reed-Muller code, (also called the generalized Reed-Muller code in the literature) ARM(u, m, q) , of order U and length 4"' is now

where A R M ( u , m , q ) = 4(V(u)) ,

V ( U ) = { F E F ~ [ X ~ , . . . , X ~ ] I ~ ~ ~ ( F ) I U},

with 1 I U I m(q - 1). Definition I : Let there be given a natural number w and

weights (w,}: corresponding to the ring of weighted polynomi- als Wq[X,;.., X,].

The weighted afine Reed-Muller code, WARM (0, m, q ) of weighted order w and length qm, corresponding to the weights {w,}: is defined by

WARM(w,m,q) = 4V, , , (w) ) ,

V,(w) = { F E WF,[X,;..,X,]Ideg,(F) I 0).

where

The set-up in the projective case is the following: Consider the evaluation map

*:wF,tXo,...,XmI + (Fq)"

* ( F ) = ( ... , F ( P , ) , ... ), given by

where (P,] is an arbitrary, but fixed, set of representatives of the points in Pm(Fq) and n = (qm+' - 1)/(q - 1). Like before we will omit the W, when suitable.

The definition of the projective Reed-Muller code, PRM(u, m, q), of order U and length n is

where

U ( u ) = { F E F ~ [ X ~ ; . . , X , ] I ~ ~ ~ ( F ) = U ,

with 1 I U I m(q - 1).

PRM(V,m,q) = * ( U ( V ) ) ,

F homogeneous} U {0},

Definition 2: Let there be given weights (w,}: corresponding to the ring of weighted polynomials Wq[X0;.., X,]. Let fur- thermore w and s be natural numbers such that 0 I w < ( q - l)C;=ow, and 0 < s < q.

The weighted projective Reed-Muller code, WPRM (0, s, m, q ) of weighted order w , order equivalent to smod(q - 1) and length ( q m - l)/(q - l), corresponding to the weights {w,}Eo, is defined by

where U , ( w ) = ( F E Wq[Xo,...,X,]Ideg,(F) I w ,

B. The Code Parameters First we introduce some convenient notation and'some useful

lemmas. For a polynomial F in Fq[XI;.., X,], we will by denote the

reduced form of F . That is the polynomial of lowest degree equivalent to F modulo the ideal (Xp - X,, i = l;.., m). For any subset M of F q [ X 1 , . . . , X m ] , the set II? denotes the set of reduced elements of M.

Lemma I: For F E F,[X,;..,X,] and G E Fq[X0;..,Xm],G homogeneous, we have

a) for every P E A,: F ( P ) = 3~); b) for every P E P": G ( P ) = G(P);- c) if F ( P ) = 0 every P E A", then F = 0; d) if G ( P ) = 0 for every P E P", then G = 0.

WPRM(w, v , m , q ) = $ ( U w ( w ) ) ,

F homogeneous, deg ( F ) = s mod ( q - 1)) U {O}.

Proof Statements a) and b) are obvious. Statement c) is 0

Lemma 2: Consider the evaluation maps (6 resp. 9 defined above and their restriction to some F,-linear subspace V of polynomials resp. U of homogeneous polynomials. We have

proved in [8] and statement d) follows from c).

4: WF,[X,,.-, X,] 2 V + (FJ?

9 : Wq[X0;.*, X,] 2 U + (F, )" ,

where n = ( q m c l - l ) / ( q - 1). Then,

a) ker(4) = ( F E VIE= 01, b) ker(q9) = {G E UlG = 01, - c) 4 , ~ is injective, m( V) = (6(_V) and V/ker (4) = v, d) I++ is injective, $ ( U ) = # ( U ) and U/ker(+) = U.

Proof Easy consequences of Lemma 1. 0

It will be interesting for a given weighted degree w to find the monomials of weighted degree less than or equal to w, which have the maximal, resp. minimal (ordinary) degree, and for a given degree U to find the monomials of degree less than or equal to U, which have the maximal, resp. minimal weighted degree.

Definition 3: Given natural numbers w, U, and a set of weights {wL}EF (ordered), such that 1 I U I m(q - 1) and 1 I w I ( q - l)C~!ew,w,. Here, E = 1 in the affine case and E = 0 in the projective case.

Let

where ~, , ,dw) = Qe(q - 1) + R E ,

Q - l + e

Q , = m a x Q l w > { 1 = e ( q - I ) r , )

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 6, NOVEMBER 1992 1823

and Q , - ~ + E

R 1 0 2 ( q - l)wi + RwQ<+, i = E

Furthermore, let

i = m - Q + l

and

I m

R' = max Rlw 2 ( q - l )w , + R w , , _ ~ , . i r = m - Q ' + l

Next, we define for a fixed degree v the interesting interval of weighted degrees. Let

m

wmax(v) = C ( q - l ) w , + Rwm-Q r = m - Q + l

and a - ] + €

amin(') = C (9 - 1)wi + ( R + 1>wp+6 - 1, I = <

where v = Q(q - 1) + R , 0 I R < q. Lemma 3: We observe that

a) v,,(w) is the maximal degree of a reduced polynomial of weighted degree less than or equal to w ,

b) vml,(w) is the maximal degree such that all reduced poly- nomials of degree less than or equal to this degree have weighted degree less than or equal to w ,

c) w,,(v) is the maximal weighted degree of a reduced polynomial of degree less than or equal to v,

d) w,,,(v) is the maximal weighted degree such that all reduced polynomials of weighted degree less than or equal to this degree have degree less than or equal to U.

The defined numbers have the following characteristic proper- ties:

e) v,,(w) = min {v lw I wm1,,(u)}, f) U,,,( w ) = max (VI w 2 om,( VI}, g) w,,(v> = min{wlv I v,,,,,(w)}, h) w,,,(v) = maxiwlv 2 vm,(wH, i) w 2 w,,(v> * v I vmln(w>, j) w I wmln(u) * v 2 v,,(w>.

Proof: For a given weighted degree w , the corresponding degrees v,,,(w) resp. vmaoax(w) are found simply by constructing a monomial of lowest, resp. highest possible degree that has weighted degree less than or equal to w.

Since the weights are ordered by I , these monomials are of the form

m

Q , - l + <

where Q', Q,, R' , R , are defined in Definition 3. For a given degree v , the weighted degrees wmin(u) resp.

U,,,,( v ) are found by building monomials from top (i.e., mono-

mials of high indexes), resp. from bottom (i.e., low indexes) of degree v. Note that wmln(v) is not just the weighted degree of the polynomial of degree v constructed from the bottom, but this weighted degree plus we+ - 1.

Now statements a)-d) are clear. Using the construction just mentioned one easily establishes the following:

w < wm,(v> * v > vmin(w>,

> wmin(v> * v < vnmx(w)

This implies e)-j). Lemma 4: With the notation as before, the following holds:

a) WARM(wmln(v>, m, q ) 5 ARM(v, m, q ) G WARM(w,,(v), m, q),

b) ARM ( vmln( w ) , m, q ) c WARM (0, m, q ) WARM(v,,(w), m, q) ,

c) WPRM(o,,,(v), vmod(q - l ) , m , q > G PRM(v,m,q) C WPRM(w,,(v), v mod(q - 11, m, q),

d) PRM (vmln(w) - T ' , m , q ) c WPRM ( w , s, m , q ) C PRM ( U,,,,( w ) - T , m , q ) where T = ( R , - s) mod(q - 1) and T' = ( R ' - s)mod(q - 1).

Proof: Use the interpretation given in Lemma 3 (statements a)-d)) and Lemma 4 (statements a)-c>> follows immediately. In statement d), one has to be careful that the degree is equivalent to s modulo ( q - 1).

It is well known (see [l], [4]) that the ARM(v, m,q)-code is a [q", k , d]-code with

and d = (4 - R ) q m - ' - l ,

where v = Q(q - 1) + R , 0 I R < q - 1 , 0 I Q I m - 1. A similar result holds for the WARM(w, m , q)-code.

Theorem 1: Given a natural number w and a corresponding set of (ordered) weights (wi}E such that w I ( q - 1)Z: Iwi. The WARM(w, m, 9)-code is an F,-linear [qm, k , d ] code with

and d = q m - Q - l ( q - R ) ,

where Q and R are given by v,,(o) = Q(q - 1) + R , 0 I R <

Proof From Lemma 2, it is clear that we only have to study

( q - 1).

__ V( w ) . The following set of monomials

- is a basis of V ( w ) and the expression for the dimension then follows immediately.

Consider the polynomial

1824 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO, 6, NOVEMBER 1992

- G is a polynomial in Y ( w ) of highest possible degree. Fur-

thermore, G is zero at all points of Am(Fq) except at points of the form Consider the polynomial

is a basis for U , ( W ) . The expression for the dimension then follows immediately.

(hi,..', AQ, /*, Z Q + ~ , " ' , Z m ) ,

where A, E F , p E A, and z, is chosen freely in Fq. There are exactly qm-"' ' ( 4 - R ) such nonzeros and therefore, we get an upper bound on the minimum distance:

d I qm-Q- ' (q - R ) .

From the multivariable description of affine Reed-Muller codes (see [4], [6]), it is known that a polynomial of degree ( q - 1)Q + R can have at most q m - qm-Q- ' (q - R ) zeros and equality follows. 0

Remark 1: Notice that the proof together with Lemma 4 (statement b)) immediately gives that the parameters of the WARM (0, m, q)-code are worse than those of the ARM(v,,(w), m, q)-code, since the minimum distances are equal, but the dimension of a WARM(u, m, q)-code is less than that of a ARM( vmW( w) ,m, q)-code. In other words, the optimal weights are the trivial ones.

In [6], it is shown that the PRM(v, m, q)-code is a [ n , k , d]- code with

and

where v - 1 = Q'(q - 1) + R ' , 0 I R' < q - 1, 0 I Q' 5 m - 1. For the WPRM(w, s, m, q)-code, we have the following theo- rem.

Theorem 2: Given natural numbers w , s and a set of (ordered) weights {w,} such that 1 I w I ( q - l ) X ~ l w r , 0 < s < q, define

e- 1

r = O I Q* = max QIw 2 (9 - 1) W , + WQ + (S - l)wQ+l ,

i i , = o

e- 1

i Q** = max Qlw 2 ( q - 1)

The WPRM ( w , s , m, q)-code is an Fq-linear [n, k , d ] code with

w, + swo .

m

(e,;..,e,)I w,e, I w , i = O

m

ei = v mod(q - I), 0 I e j < i = O

and

Proof: From Lemma 2, it is clear that we only have to study U,( w ) . The following set of monomials

{ fi X:l E w,e, I w , r = O 1 - 0 r = O

Q*-1 s - 1

G = X p r = O n (xp-'-xg;:1)n(A,x~*-x~.+l), J = 1

where A, E Fq, are distinct. Since Q * - 1

deg,(G) = (4 - l ) ~ , + wQ. + ( S - I ) W Q * + ~ r = O

by definition of Q*, we have of Pm(Fq) except at points of the form

E U,( w ) . G is zero at all points

(O:..-:O: 1: A: z ~ . + ~ : " ' : z,,,),

A # A,, j = l;..,s - 1, and z, chosen freely in Fq. There are exactly qm-Q*'- ' (q - s + 1) such nonzeros and we, therefore, get an upper bound on the minimum distance:

d I q m - Q * - l ( q - s + 1).

From the projective Reed-Muller codes (see [61), it is known that a reduced polynomial of degree ( q - 1)r + s + 1 can Lave at most (qm+ l - l)/(q - 1) - qm- ' - ' (q - s) zeros. Now G is not necessarily a polynomial in U,( w ) of highest possible nor- mal degree equivalent to s modulo ( q - l), so we are at the moment only able to give a lower bound by replacing Q* with

Remark 2: As in the affine case we immediately from Lemma 4 conclude that the optimal weights for WPRM-codes are the trivial ones. That means that also here no good new codes are produced.

e**.

111. THE DUAL OF WEIGHTED REED-MULLER CODES

In this section, we will show that the dual of a WARM-code is again a WARM-code and that this almost hold for WPRM-codes, too. We recall from [6] that the same situation happened for PRM-codes. For ARM-codes this has been known for a long time (see [4]).

Definition 4: Let m

(e,;..,e,)IO I e, < q , C e, = I = 1

m

i = 1

I , m

m

. C e , = s m o d ( q - r = O

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~

1825

ProoF Properties a)-c) follow using the 1-1 correspon- dence between n , X : and FIrXP-I-e'. Properties d)-f) can easily be proved using the same technics as in [6, Lemma 61.

Theorem 3: The dual of a weighted Reed-Muller code is given by

a) WARM(w,m,q ) l= WARM(w,,, - w - l ,m,q) . b) w P R M ( ~ , s , m , q ) ~ = WPRM(w,,, - w - l , q - 1 -

s, m, q), s f 0. c) WPRM(w,O, m, q I L = WPRM(w,,, - U - 1,0, m, q)*,

where C* is the shortened code of C, defined by C* =

span{u2;.., uk} where C = span(1, uZ;.., uk} and 1 means the vector (l;.., 1).

Proot Using [6, Lemmas 7 and 81, it is clear that it is enough to prove that the dimensions add to q" resp. (q"+l - l ) / ( q - 1). This follows, however, immediately from Lemma 5 (d)-f)). Like in the projective Reed-Muller case one has to be careful in the case when the degree is 0 modulo ( q - 1). The constants then have to be counted only once.

IV. WEIGHTED REED-MULLER CODES AND ALGEBRAIC-GEOMETRIC CODES

In this section, we study the WARM-codes from an algebraic-geometric point of view. Goppa introduced the alge- braic geometry approach to coding theory about 1980 by the following beautiful construction (see [9]).

Let C be a projective, nonsingular, absolutely irreducible curve defined over F, with functionfield F,(C) and of genus g. Let D = C:=,P,, and G be two divisors defined over Fq such that G and D have disjoint support, where the Pz"s are distinct F,-rational points on C. We define an F,-linear evaluation map

cy,: L ( G ) + F:.

by c y I . < f ) = ( ... , f(p,, ... 1.

sponding algebraic-geometric code C,(C, D, G) is now As usual L(G) = {f E F,(C) 1 div(f) 2 - G} and the corre-

C,(C, D , G ) = a,(L(G)) . Let us now specialize to the following situation, see [9]. Definition 5: We will call a curve with the following proper-

a) C c P"(F,), b) G = wQ, Q a F,-rational point on C, w E N ,

ties:

C) D = Ey:lP,, P, E P" \ H ,

where H is a hyperplane and H n C = {Q}, an afjne covering cun'e.

Consider an affine covering curve C. Denote the coordinates of P" by x , , and assume w.l.0.g. that H = Z ( x , ) . Let us now consider linear systems of monomials in the affine coordinate- functions y, = x,/x,,, i = l;.., m. That means we consider vec- torspaces generated by monomials of the form

m

ny:, e, E N .

For a given finite set of such monomials, the vectorspace gener- ated by these monomials is a subspace of an L(nQ) for n large enough (larger than the maximal pole order in Q of the basis monomials). When the linear system generates an L(nQ> for some n, the linear system is said to be complete.

I = 1

We have the following theorem.

Theorem 4: Let C c P"(F,) be an affine covering curve with notation as in definition 5. Let y , denote the corresponding

affine coordinates and let vQ(y,) be the discrete evaluation of y , at Q. Assume w.l.0.g. that vQ(y,) I vQ(y,) for i I j .

If C,(C, D, wQ) is the corresponding algebraic-geometric code and WARM ( w , m, q ) the weighted affine Reed-Muller code corresponding to weights w, = I vQ(y,)I, then

a) WARM(w, m, q ) G C,(C, D, wQ), b) WARM( w,m, q ) = C,(C, D , w e ) , if an only if L ( w Q ) is

generated by monomials of the form rIE ly:, e, E N .

Proof It is clear that monomials n ~ = , y f ~ of weighted de- gree less than or equal w are contained in L( wQ). This gives a). In b), the "if' part is obvious, and the "only if ' part follows from the fact that ker:(a,) in the case of affine covering curves coincide with ker ($1 from Lemma 2.

From Theorem 4, we immediately get an upper bound on the minimum distance for codes from affine covering curves.

Corollaiy I : With notation as in Theorem 4, we have

min.dist.(WARM(w,m, 4 ) ) 2 min.dst. (C,(C, D, w e ) ) .

Example I (The Pellikaan-Shen-van Wee Curve): The curve C used in [7] is given by the complete intersection of hypersurfaces

xp" - . , "x i - ' + x , + l x i - xpt1x0 = 0 ,

where i = l,...,m - 1. C is almost a curve on the form previ- ously given. It goes through all F,-rational points in P m \ H, where H = Z ( x , ) . It intersects the hyperplane H in exactly one point Q, but is highly singular in that point (and that is the only singularity). It is shown in [7] that there lies only one smooth point above Q on the normalization of C. If we identify Q and the q2 points in P" \ H with their preimage on the normaliza- tion (see [7], for a precise description) and likewise with the coordinate functions y, = x,/x,,, we are now in a situation like before: we can think of C as an affine covering curve.

Let uQ denote the discrete evaluation at Q. In [7], it is shown that

U Q ( Y , ) = - q m - ' ( q + 1)l- l .

It is, furthermore, shown that

If we define weights {q"-'(q + l)L-l)zl, we have that, from the theorem, the corresponding code a,(L(wQ)) is equal to

WARM (0, m, 4 ) .

Thereby, the minimum distance is also found but, as noted in Remark 1, the code itself is actually not interesting, since there exists a Reed-Muller code with the same minimum distance and length but with a higher dimension, namely ARM (vmm( 01, m, 4).

Example 2 (The Hansen-Stichtenoth Curve): We consider the curve, C, given by the affine equation

xg - x.2 - x p ( x y - X I ) = 0 ,

where q =p2"+ ' , qo =p" , n E No, i E N , It has q2 + lF,-rational points, where exactly one point, Q,

lies in the hyperplane (line) at infinity, and the rest, P,, covers the affine plane A2(Fq). It is known (see [lo], [ll]) that there is only one smooth point above Q; so, as in the preceding example, we can think of the curve as an affine covering curve.

Using Goppa's construction we get codes C,(C, wQ, D) of length n = q2, dimension k = dim (a,(L(wQ)>), and minimum distance d 2 q2 - w.

1826 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 38, NO. 6, NOVEMBER 1992

In [Ill, it is shown that the functions (considered as elements [71 R. Pellikaan, B. Z . Shen, and G. J. M. van Wee, “Which linear codes are algebraic-geometric?” IEEE Trans. Inform. Theory, vol. 37. ut. 1. DD. 503-602, Mav 1991.

in the functionfield Fq(xl , x , ) )

g1 = f q =x1,

g, = f q + q / q o

g2 =fq+qo =x2, x q / 4 0 - 4 / 4 u + l x1 >

[8] S. iang,’&bra. New Ybrk: Addison Wesley, 1965. [9] V. D. Goppa, Geometry and Codes, Mathematics and Its Applica-

tion, Soviet series 24. Dordrecht, The Netherlands: Kluwer Ac. Publ., 1989.

[lo] J. P. Hansen and H. Stichtenoth, “Group codes on certain alge- braic curves with many rational points,” AAECC, vol. 1, no. I, pp.

[ l l ] J. P. Pedersen and A. B. Sgrensen, “Codes from certain algebraic function fields with many rational places,” MAT-Rep. no. 1990-11, Matematisk Inst., Danmarks Tekniske Hgjskole, June 1990. A. Holm, Masters thesis, Matematisk Inst., Aarhus Univ., 1990.

= x j / q o x q / q o - l + f q / q o q + q / q o ’ g4 = f q 2 / q ; - q + q / q o + 1

67-77, 1990. “generate” the vector space L( w Q ) in the following sense:

L( w e ) = span ng,’c eiwi I w ,

where the weights wi are the subindexes of the f-function corresponding to gi.

( i r , I i I , ] [12]

Permutation Decoding of Abelian Codes

Herv6 Chabanne and, therefore, the minimum distance of WARM (U--”( w),2,q) is an umer bound for the minimum distance of C,‘ic. WO. D).

I I I - - - ’ I I

The minimum distance for the WARM ( vmax( w ) , 2, q)-code Abstract-A permutation decoding procedure for abelian codes is introduced hy using the Grcebner bases theory. This method is valid for decoding all the binary abelian codes. Some examples are given to show with the weights q and q + qo is

where (1)

how powerful this method can be. d = ( 4 - b)q’-‘, Index Terms-Abelian code, Grcebner basis, permutation decoding.

a = 0, for w I q ( q - 1)

for q(q - 1) < w I ( q - 1)(2q + qo - 1)

I. INTRODUCTION Abelian codes are defined as ideals in the group algebra of an

abelian group over a finite field. They have been studied by many authors (our principal reference is Camion [l]). Recently, Jensen [4] showed how to construct good ones. The purpose of this correspondence is to show how to decode those which are

proper definition of the information symbols in the case of the group algebra’s representation of these codes (see also Sakata [5], [6]). Following him, we show how to calculate syndromes via the Grcebner bases theory and then generalize a permutation decoding procedure due to Mac Williams [3].

and

a = 1,

and

b = 1 - ‘(’ - ’) for a = 1 and b = , for a = 0.

This is in fact the exact minimum distance for w = q(q - c), c = I,..., q, since here the minimum distance coincides with the lower bound

ideals in a binary semisimple group algebra. Imai [21 gives 4 + 40

d > q 2 - w

deduced from the Riemann-Roch theorem. By complete com- putersearch it is shown ([121) that for other values of w , (1) is not the exact minimum distance of the Hansen-Stichtenoth codes.

11. PRELIMINARIES AND DEFINITIONS

basic definitions about binary abelian codes. In fact, we confine ourselves in the coding theory context (2-D cyclic codes) introduced by Sakata in [7]. The generaliza- tion to binary abelian codes (multidimensional cyclic codes) follows naturally. Details and proofs can be found in [l] for abelian codes and in [10]-[12] for Grebner bases.

First, we recall

ACKNOWLEDGMENT

This work was done during a visit at RISC-LINZ, Austria. The author would like to thank the faculty at RISC for their hospital-

REFERENCES T. Kasami, S. Lin, and W. W. Peterson, “New generalizations of the Reed-Muller codes, Part I: Primitive codes,” IEEE Trans. Znform. Theory, vol. IT-14, Mar. 1968. E. J. Weldon, “New generalizations of the Reed-Muller codes, Part 11: Nonprimitive codes,” IEEE Trans. Znfom. Theory, vol. IT-14, Mar. 1968. T. Kasami, S. Lin, and W. W. Peterson, “Polynomial codes,” IEEE Trans. Inform. Theory, vol. IT-14, Nov. 1968. P. Delsarte, J. M. Goethals, and F. J. MacWilliams, “On general- ized Reed-Muller codes and their relatives,” Znform. Control, vol.

G. Lachaud, “The parameters of projective Reed-Muller codes,” Discrete Math., vol. 81, pp. 217-221, 1990. A. B. Sgrensen, “Projective Reed-Muller codes,” IEEE Trans. Inform. Theory, vol. 37, pp. 1567-1577, Nov. 1991.

16, pp. 403-442, 1970.

A. Abelian Codes Let K be the finite field GF(2), we denote by R N , , N , the

quotient algebra RNl, N, = K[x, y]/(xN1 + 1, y N 2 + 1>, where N , and N2 are odd integers. A binary 2-D cyclic code of area N1N2 is an ideal C of the semisimple algebra RNIrN2. Each codeword ii = ( a - . ) is represented as a bivariate polynomial (modulo ( x N 1 + l ,y22 + 1))

N, -1 N,-1

i = o j - 0 a ( x , y ) = c a;,x‘y’.

Manuscript received December 12, 1991; revised February 25, 1992. The author is with INRIA, Project CODES, Batiment 10, Domaine de

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