538 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 2, MARCH 1994

The Automorphism Group of Double-Error-Correcting BCH Codes

Thierry P. Berger

Abstract-Using the description of primitive cyclic codes in a modular algebra, we characterize the permutations of the support of a cyclic d e which leaves the code globally invariant. Applying thin result to the binary double-error-correcting BCH d e s , we prove that the automorphism group of such a code (of length 2 m - 1, m > 4) is the semi-linear group of GF( 2") over GF( 2m) , and, in the special case m = 4, the semi-linear group of GF(16) over GF(4).

Index Terms-Error-correcting codes, cyclic codes, BCH codes, auto- morphism group, permutation group.

I. PERMUTATION GROUPS OF PRIMITIVE CYCLIC CODES

A. Primitive Cyclic Codes

In this section, we recall some well-known results on primitive cyclic codes. For the proofs see [9].

A cyclic code of length n over a finite field I< of characteristic p, K = GF(q) . is an ideal of the algebra R = K [ A - ] / ( X " - 1 ) . In this paper we only consider the codes of length n = qm - 1. These codes are called primitive cyclic codes. Let cy be a primitive root of the finite field G = GF(q"), and i E J , J = (0 ,..., TL - 1). The minimal polynomial of cy* over A- is the polynomial

nc,(X) = n(x - a J )

JET

where T = { i , q i , q ' i , . . .} is the q-cyclotomic coset of i modulo n. The quotient ring I<, = A - [ X ] / m , ( X ) is isomorphic to the finite field GF(q'), where I is the number of elements of the cyclotomic coset 1.

Let { i l , . . . , z'k} be a set of coset representatives mod n. It is well- known that R is isomorphic to the Cartesian product A-zl x ... x IC,, . More precisely,

R = A - [ X ] / ( X " - 1 ) --+ x . . . x li,,

h ( X ) I--..+ ( / ! ( a y , . . . . h ( a ' * ) ) (1)

and, for j E J , h(cyJq) = ~ ( C I ' ) ~ .

In the algebra R, each ideal is principal, and is generated by a well-defined polynomial g ( X ) . If {cy",. . . ,cys*} is the set of roots of g ( X ) , and T = {SI,. . . , s t } , then T is called the defining-set of the ideal C generated by g ( X ) .

Manuscript received July 27, 1992; revised May 6, 1993. This paper was presented at the 1993 IEEE International Symposium on Information Theory, San Antonio, TX, January 17-22, 1993.

The author is with the Npartement de Mathtmatiques, Universite de Limoges, 123 AV. A.Thomas, 87060 Limogek Cedex, France.

IEEE Log Number 9215544.

The defining-set is a union of q-cyclotomic cosets. Conversely, if a subset T c 4 J is invariant under multiplication by q mod n, the correspondin 4 ideal is completely determined by the formula (2).

B. Primitive Cyclic Codes in a Multiplicative Group Algebra The representation of primitive cyclic codes introduced here fol-

lows from a paper of F. Laubie [6]. Let A4 = li[G*] be the group algebra of the multiplicative group

G* of G = GF(qm), over the field K = GF(q). An element of A4 is a formal sum T = E,,,* x g ( g ) , x9 E I<. The addition in M is defined by the following formula:

and the multiplication is the extension of the multiplication in G':

The algebra hf is isomorphic to R as follows: if cy is a prinitive root of G, then

4 : R - M n - 1 n - 1

* = O * = O

For s E J =: { O , . . . , n - l}, we define the map p. of M into G in the following way:

gEG' g € G *

the sum If I = 4 ( h ( X ) ) , then p S ( z ) = h ( c y s ) .

xggy not being a formal sum, but calculated in G.

The following diagram is commutative:

A cyclic code C in M, of defining-set T is then

C = {z E M/p , (z ) = 0,Vs E T }

C. Permutation Gmups of Cyclic Codes

R in the following way: A permutiition U E S ( n ) of the set J = (0,. . . , n - 1) acts on

- - - l n - 1

0018-9448/94$04.00 0 1994 IEhE

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 2, MARCH 1994 539

The permutation group of a cyclic code C is the group Per(C) of permutations of S(n) , which leaves C globally invariant: a ( C ) = C. The representation of cyclic codes in the algebra M is very interesting, because the permutations are permutations of the elements of the finite field G (more precisely G*), and we can use the geometric properties of G. It is known (see, e.g., [7] Ch.7) that the symmetric permutation group S( G) of the finite field G is isomorphic to the group (for composition of polynomials) of invertible elements of the ring G[X]/ (Xqm - X ) .

Each permutation U E S ( G ) admits a unique polynomial represen- tation of degree less than qm: a(g) = E::;' X,g', A, E G. More precisely, the associated polynomial is

We obtain the equality 1 ... 1 1

/? ( p . . . pd-1 pz / j 2 Q . .. p 2 q t -

p'-1 p(t-119 . . . /3(t-1)qt-l

1 . . .

The matrix

f ( X ) = a ( g ) ( l - (X - g)ql'l-l) S E C

Unfortunately, the converse problem is very difficult: how to know if a polynomial f ( X ) of G[X] / (Xqm - X) is associated to a permutation, i.e., f ( X) is invertible. From the "Hermite's criterion" (cf. [7, p.349, Theorem 7.4.]), we know however that the degree of such a polynomial is at most qm - 2 = n - 1. We can remark that a permutation polynomial f ( X ) is associated to a permutation of G* if and only if f (0) = 0, i.e., XO = 0. Such a polynomial is of the

Theorem I : Let C be a primitive cyclic code, and T its defining- set. A permutation a E S(G*), with associated polynomialf(.Y) = C:=;'X,X' is a permutation of C if and only if, for all s E T, the polynomial f ( X ) ' mod(Xqm - X ) has all its exponents in T, i.e.,f(X)' = E,,, p 3 X 3 , for some p 3 E G.

ProoJ Let s E T; let U be a permutation, and f ( r U ) its associated polynomial,

form f ( X ) = ~ ~ ~ ; x , X z .

f ( X ) " = C;:;pJXJ. For I E M , I = CgEG* I9(9), 4.1 = E S E G * .,(47))>

n - I

If I is an element of C, then pj(x) = 0, for all j E T. Hence

p s ( 4 . ) ) = P 3 P 3 ( " )

34T

Now, suppose that for all s E T, the polynomial f(X)' has all its exponents in T. Then for any x E C and any s E T, we have p s ( u ( r ) ) = 0. That means that a(z) is an element of C . Hence U

is a permutation of C, i.e., a E Per( C). Conversely, suppose that ~7 E Per(C). Let s E T, and r $! T.

Let t be the size of the q-cyclotomic coset of r , t = IFl. If h-, = A-[X]/m,(X) then Ii, = GF(qt). Let p be a primitive root of KF=,. For each b'', i = 0, . . . , t - 1, there exists an element zz E M suchtha tp , (z , )=Ofor j $ ! f , p , ( e , ) = P ' , a n d p , , ~ ( r , ) =,?~'~l.In accordance with the isomorphism (1) in Section I-A, such an element I, exists. Moreover, I, is an element of C , since for all j E T, j $! 7 ( T $! T) , and pJ(xZ) = 0.

By hypothesis, U E Per(C), and for i E ( 0 , . . . , t - l},

fS('(2z)) = C p J P I ( z z ) = C p J p 3 ( z z ) 34r 3 E+= t - 1

= c p r q , . l p ' = 0 1=0

is a Vandermoiide matrix, and is invertible, since the pql are nonzero distinct elements of Ii, (p is a primitive root of I<, = GF(qt); see, e.g., [9] p. 116)

Then it follows that pr = 0, prq = 0, ..., prq t - l = 0. The same result holds for any r $! T. Finally:

f ( ~ ) " = p3xJ, for all s E T. J E T

0

The permutation group of a cyclic code C overIi- = GF(q), of length n = qm - 1 contains the permutations of the form a(g) = ngq',c, E G*, i E { O , l , . ..,m - 1).

7 he defining-set T is invariant under multiplication by q mod n, the mociated polynomial of U is f (X) = axq', and for s E T, f ( X ) ' = aXSq' . The hypothesis of Theorem (1) holds.

Remark: A,, a Referee pointed out, this Theorem can be proved in another way by use of the Mattson-Solomon transform (see, for example, 18, ch. 81, 5 6): A primitive cyclic code, of length n = qm - 1 over K = GF(q) can be described by means of an index set i which is a union of cyclotomic cosets of the set J = {0,1,. . . , n - 1) . The code C associated with I contains the vectors c = ( ( (1) . ~ ( c u ) , . . . , c(a"- ' ) ) where c ( X ) is a polynomial of the form

For example, we obtain a well-known result: Corollary 1

ProoJ

c X) = c c , X * , with cz E G,rtq = cy. (a) 1 E I

Note that the condition rtq = c: means exactly that all the compo- nents ~ ( 0 ' ) of c belong to the subfield Ii of G. Th: connection with the defining-set is T = { s / ( n - s) E I}, or I = T. The polynomial c ( X ) is in fact the Mattson-Solomon polynomial of c. Clearly, a permutation of C can be considered as a permutation on the a ' , i.e., as a permutation of G*. If f ( X ) is a polynomial associated to a permutation of C, then, for all c ( X ) satisfying (a),

f ( c ) = (c(f( l ) ) ,c(f(m)) , . . . ? C ( f ( o " - ' ) ) )

Then c(f(X)) must be a polynomial satisfying (a). In fact, it i:, sufficient to consider the monomials c ( X ) = X ' ,

z E I , i.e., i E 9. The polynomial f(X)* must be a polynomial satisfying (a), this is the condition of the Theorem 1 for the dual of the code C.

D. Extended Primitive Cyclic Codes T, it

is possible to extend this code by adding an overall parity check (see, e.g., [91 p. 27). For example, if I E M, then let IO = - CgEG, zg = -po (z ) , the extension of I is then s* = ( 2 0 , ~ ) .

Let C be a primitive cyclic code, and T its defining-set. If 0

540 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 2. MARCH 1994

Let A = h-[G] be the group algebra over K of the additive group of the finite field G. An element of A is the formal sum CBEC; X ~ A - ~ , with the usual operations:

An extended cyclic code can be naturally embedded into the algebra A (cf. [3]).

Theorem 2: The permutation group of the primitive cyclic code C of M is the subgroup of the permutation group of the extended code C* of A which leaves zero invariant.

Proo) Straightforward. An ideal of the algebra A is a IC-vector space which is invariant

under multiplication by X 6 for all b E G, i.e., which is invariant under the translations of G: Ub (9) = g + b. A very important class of codes is the class of extended primitive cyclic codes that are ideals of the algebra A. Let C* be the extension of a cyclic code C. If C' is an ideal of A, then the permutation group of C* contains the affine group:

GA = {ua,6 E S(G)/ca,b(g) = ng + b,a E G', b E G }

More precision on this class of codes may be found in [S, 4, 31. This class of codes contains the generalized Reed-Muller codes, the extended BCH codes, and the extended Reed-Solomon codes.

If C* is such a code, and C the corresponding cyclic code, then the permutation group of C* is the group generated by the permutation group of C and the translations of G (cf. 111).

11. AUTOMORPHISM GROUPS OF DOUBLE-ERROR-CORRECTING BCH CODES

A. Double-Error-Correcting BCH Codes

The narrow sense BCH code (cf. [9]) over I< = GF(p) of designed distance tl and length n = p" - 1 is the primitive cyclic code, the defining-set of which is the smallest T containing {1,2, . . . , d - 1 ) such that pT = T modulo n.

A double-error-correcting code is a code with minimum distance at least 5.

Definition I : The binary double-error-correcting BCH code is the BCH code over G F ( 2 ) of designed distance 5 and length 2" - 1 (m > 2). Let Dm denote the double-error-correcting BCH code of length 2" - 1. The defining-set of B, is

} T = T u 3 = 2 z , 2 z + 2 2 " + L / / 1 ~ { ~ , . . . . ~ ~ - i } { In the binary case, the automorphism group of a code C is its

permutation group (for the nonbinary case see [9] p. 238 or [l]). If m = 3, then T = {1 ,2 ,3 ,4 ,5 ,6 } . Then the code L33 is the

repetition code of length 7; its automorphism group is obviously S(7). We now suppose m > 3 .

Let U be a permutation of B,, and f ( X ) = Cy=;' X , X L be the associated permutation polynomial. Since the defining-set T is the union of T and 5, it is sufficient to apply the criterion of Theorem 1 for s = 1 and s = 3. For s = 1, we deduce

, -I m-I

f ( X ) = n,X2' + b 2- Y2'+2L+1 r = D * = O

Since f(X)' = ET:;' a:- lX2L + E:;' b ~ - l X 2 ' + 2 ' + 1 where, by conventicm b-1 = b m - l , we obtain for s = 3

The permutation (T preserves B, if and only if f ( X ) is a permutation polynomial and f (X)3 has exponents in T. Recall that d = (0.1. . . , n - 1). Let the binary expansion of I E J be - - j = E:;' j i 2 * , j, E ( 0 , l ) and let us define the 2-weight of j as J z ( j ) = cziljz.

Lemma1 ~ t ~ E [ O , m - I ] a n d j E [ O , m - l ] I ) If i and j satisfy one of the following equalities:

i = j o r i = j+l o r j = i + l , then ~~2(2'+2"'+2J+2J+' = 2 (the sums are calculated modulo 7 1 ~ ) ;

2) If not. then we have w2(2' + 2'+' + 2J + 2.'+l) = 4. Moreover, if k is an exponent in (I), and L J Z ( ~ ) = 4, then k = 2 + 2"' + 2J + 2.'" for some i and j.

Proo) Let s = 2' + 2''' + 2J + ZJ+'. For i = j , we obtain s = 2'+' i- 2'+', and W Z ( S ) = 2. For i = j + 1, we obtain s = 2J + 2j+3 j # j + 3, since ni > 3, and W ' Z ( S ) = 2. For the same reason. if j = i + 1, then w 2 ( s ) = 2.

Obviously, if i # j and i # j + 1 and j # i + 1, then w2(s) = 4. In (A), ~ ~ 2 ( 2 ' + 2 3 ) = 2 if i # j and 1 otherwise. In (B),

q ( 2 * + 23 -- 2 j ' l ) = 1 if i = j, 2 if i = j + 1 and 3 otherwise. cl Lemma 2 Let (T be a permutation of B, and f ( X ) =

its associated polynomial. For m > 4, if two distinct b; are nonzero, then all the h; are nonzero. Moreover, if b = blh;', then b, = bb:-,.

Pro08 Let s E J. If L J ~ ( S ) = 4, then s $! T, and the coefficient in (I) associated to s is zero. Using Lemma 1, we obtain:

"1 a,X"+ Cm-l b ; X 2 ' + 2 z + 1 L O t=O

b,b;-,+bjb?-, = O . f o r j $ ! { i - l , i , i + l } (3)

Without loss of generality, suppose that bz # 0.

.

Let us assume that there exists a k > 2 (i.e., k $! { 0 , 1 , 2 } ) such that b k # 0. Since rn > 4, then j = k + 1 $! { 1 , 2 , 3 } . According to ( 3 ) we have bzb;-l + hJb;-, = 0, which means b& + bk+lh: = 0. Since b:! and b k are not zero, we then deduce that bk+L # 0 and bl # 0. By induction, for all IC', k 5 k' 5 m, b p # 0 For j = 3 and i = 1 in (3), we obtain b l b i = bsbi # 0, which implies b3 # 0 and bo # 0. We can choose k = 3 and for all k', 3 5 k' 5 m , b k , is not zero. Then all the h , are nonzero. Now suppose that b k = 0 for all k , 2 < k < m ,

a. If b l # 0, let i = 1 and j = 3 in (3). We have blbz = babi with b l b i # 0. Then bo # 0 and bs # 0, whch is a contradiction. If bo # 0, let i = 0 and j = 3 in (3), 3 $! { m - 1.0, l } since nr > 4. We have bob: = h3b;,-, with bob; # 0. Then b3 # 0 and b,n-l # 0, which is a contradiction.

b.

Only b: is nonzero. Let us study the case where all the b; are nonzero. From (3), for j =. i + 2 we obtain

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40. NO. 2, MARCH 1994 541

Likewise, for j = i + 3, j f { i - l , i , i + 2}, since m > 4, b, - bt+3 - - -.

b?-l b?+2 Then, for all i ,

0

Theorem 3: For m > 4, the automorphism group of the binary double-error-correcting BCH code B, is the semi-linear group of GF(2"') over GF(2"):

H = { U E S ( G ) / a ( g ) = ag2 ' ,a E G',

i~ { O , ..., m - I}} Proo$ It is clear that Per(B,,) contains H (cf. Corollary

1). Conversely, let o E Per (Bm) , and f ( X ) = E:;' a,X2' be its associated polynomial. Using lemma [2],

only three cases may occur: either all the b, are nonzero or only one b, is nonzero or all the b, are zero.

1 ) Let us assume that all the b , are nonzero, and b, = bbZ-_,.

7n-1 bIX2'+2'+' + E,=o

Let 9 6 E S(G*) be the multiplication by b in G*: a ( g ) = bg. The permutation U is an element of Per(&) if and only if

o U is an element of Per(&). The associated polynomial of db o U is

m-1 TT-1

b f ( X ) = a:X2' + b:X2'+2'+', z=O n=n

a', = ba,, b: = bb,. b' Then & = % =

Withou: loss of generality, we can suppose now that b = 1; in particular b, = bg'. In the equation (I), the only exponents of 2-weight 3 are obtainedin (B), and, for j f {i-2,i- l , i, i+l}, the coefficient

is a, b:- + a?- b, . Then:

x % = 1 bo

of x2 '+2J+2J+' .

a,b:-l + a?-lb, = 0,

for j

Inparticular,forj = i+2 since m > 4. We obtain the relation:

{ i - 2 , i - l , i , i + 1) (mod m ) , j f {i-2, i - - l , i , i+l}

a,b:+l = a?-lb,+a for all i E (0 , . . . ,m - 1 ) (4)

Suppose that a , = 0 for some i . The relation (4) implies that all the a, are zero. The polynomial f (X) is of the form:

f ( X ) = m-1

OX^)^', since 2' + 2'+' = 3 x 2z *=n

Then u ( g ) = T r ( b o g 3 ) , where T r ( g ) = ET:&' g2' is the trace function, and U is not a permutation of G. Suppose that all the a, are nonzero. From the relation (4), we obtain

a , - h + 2 = 1 a:-1 bT+l - - -

By immediate induction, a, = a;*, hence

U is not a permutation. Then it is impossible that all the b, are nonzero.

Now suppose that only one b, is nonzero. Without loss of generality,we can suppose that bo = 1. Then

m - 1

f ( X ) = a,X2' + x3 Z=O

The eqhation (I) becomes

f (XI3 = c 2 X2t+23

where, by convention a1 = am-l. For m 5, the exponents 9, 11 and 14 are not in the defining- set T (T = T U 3). For all i E (0 , . . . ,m - l} it is clear that 2' -k 3 # 9 (mod n) and 2' + 6 # 9 (mod n). The monomial X9 does not occur in (B') and (C'), and occurs in (A') for i = 0 and j = 3, or i = 3 and j = 0. Its coefficient is 1 + c'oag + at-la3, this implies

1 + ana; + a2-lag = o (5 )

The monomial X" occurs only in (B') (for i = 3) and its coefficicnt is a;. This implies U; = 0. The monomial X14 occurs only in (C') (for i = 3) and its coefficient is a3. This implies a3 = 0. Thus (5) becomes 1 = 0. Our hypothesis is false. It is impossible that only one b, be nonzero. Now slippose that all the b, are zero. Then f ( X ) = E:;' &X2'

f (XI3 = c t , , E t n , ,m-1)

For j $! { i - l , i , i + l}, 2z + 2' $! T, the coefficient of the monomial is zero:

a,a:-, + a,aT-l = 0, for all j { i - 1, i , i + 1)

These equations are the same as the ones obtained in lemma 2 for the b,. Only three possibilities may occur:

,911 the a, are zero: this is impossible, since U is a permutation. ,911 the a, are nonzero, then a, = (a = ala,2). Using similar arguments, we can suppose that a = 1 and a, = a:'. Thus f ( X ) = T r ( a o X ) and U is not a permutation. Only one a, is nonzero, then o(g) = a,g2', U E H .

We have then proved that H = Per(&). 0 Corollary 2 : For m > 4, the automorphism group of the extended

double-error-correcting BCH codes of length 2'" is the semi-affine group ofGF( 2'"):

GSA = { U E S(G) /o(g) = ag2' + b,

a E G', b E G,i E { O , . . . , m - 1))

Pro08 iit is a direct consequence of the preceding theorem and paragraph 1.4 p. 5 .

542 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 40, NO. 2, MARCH 1994

B. The exceptional case m = 4

15 = Z4 - 1 over GF(2). Its defining-set is:

From (8) and (lo), it follows ala: = a2u;‘.

Then f ( X ) = X + a l X 2 + a ; X 4 + a : X X k t B4 be the BCH code of designed distance 5 and length If a1 # 0 and a2 # 0, we deduce a2 = a?, and a3 = ala; = a:

1 f l l

T = T U 3 = {1,2,4,8,3,6,9,12} f ( X ) = -(ad- + a;x2 + a?X4 + a y )

Let E be the dual of B4. The defining-set of is (cf. [3]): f ( X ) = & T r ( a l X ) , U is not a permutation.

Theorem 4: The automorphism group of the double-error- correcting BCH code of length 15 is the semi-linear group of GF(16) over GF(4):

A

T = { ~ / 1 5 - s 4 T } = {0,1,2,4,8,5. lo}

Let U be a permutation 0fB4, and f ( X ) = CItT X , X z its associated polynomial.

Since Per(&) = P e r ( g ) , f ( X ) = x I E p X L X L . Then we obtain:

f ( X ) = X , X t = c a , X 2 ’ and

Per(&) = GSLGF(,)(GF(~G))

3 Proof: Let U be an element of the semi-linear group of GF( 16) over GF(4) and f(x) its associated polynomial. Then f ( X ) is a linear polynomial over GF(4) or ~2 o f ( X ) is a linear polynomial .ET& 1=0

3 3

f ( X ) 3 = a,a;-1X2’+23 ,=o J=o

over GF(4)(cf. [7] ch.3 4). Hence:

f (xX) = a X + b X 4 or f ( X ) = a X 2 + b X R . 0

Corollary 3: The automorphism group of the extended double-

If j a,a:-, +

{ i - 1, i , i + l}, the coefficient of X2’f23 must be zero. Then = 0. That means

(6) error-correcting BCH code of length 16 is the semi-affine group of GF(16) over GF(4) .

(7)

Four possibilities may occur: 1) Only one a, is nonzero implies that u ( g ) = a ,g2’ . 2) a0 and a2 are nonzero and a1 = a3 = 0: f(2Y) = a o X +

a 2 x 4 . The polynomial f ( X ) is a linear permutation polyno- mial over GF(4) (cf. [7] ch.3 5 4) The permutation U is linear over GF(4), i.e., 0 is an element of G L G F ( ~ , ( G F ( I ~ ) ) , which is the linear group of GF(16) over GF(4).

3) a1 and a3 are nonzero and a0 = a2 = 0: f (X) = a1-X’ + a3XR. If 42 : g -+ g 2 , then @ p ( f ( X ) ) = a3X + a l X 4 ; that means $2 o U E G L G F ( ~ ) ( G F ( ~ ~ ) ) .

4) All the a, are nonzero. Lemma 3 If all the a, are nonzero, then U is not a permutation. Proof: Without loss of generality, we can suppose that a0 = 1.

Then from (6) and (7) we obtain

a3 + a l a ; = 0 (8)

Remark: Xecently at the International Symposium on Information Theory,’ Dr. C.-C. Lu2 claimed that he had obtained separately similar results on the automorphism group of BCH codes in his Ph. D. dissertation.

ACKNOWLEDGMENT

The authcr would like to thank one of the reviewers for his remarks and suggestions about Theorem I , and both reviewers for their assistance in improving the readability of the text. The author wishes also to thank Dr. A. Montpetit of the University of Sherbrooke (Quebec) for his helpful remarks.

REFERENCES

[I] T. Bergw, “Sur les groupes d’automorphismes des codes cycliques ttendus primitifs affine-invariants,” These de Doctorat, Universitk de Limoges, France, 1991.

[2] P. Charpin, “Codes cycliques ttendus invariants sous le groupe affine, Thkse de Doctorat d’Etat, Universitt Paris VII, LITP, 1987.

[3] -, “’The extended Reed Solomon codes considered as ideals of a modular algebra,” Ann. Discrete Math., vol. 17, pp. 171-176, 1983.

[4] P. Delsarte, J. M. Goethals, and F. .I. MacWilliams, “On generalized Reed-Muller codes and their relatives,” Inform. Conrr.. vol. 16, 1974.

[5] T. Kasarni, S. Lin, and W. W. Peterson, “Some results on cyclic codes which are invariant under the affine group and their applications,” Inform. {Zontr., vol. 11 , pp. 475496, 1967.

[6] F. Laubie, “Definition intrins\‘(e)que de certains codes cycliques et de leur extension,” dkpartement de Mathtmatiques, Rapport de recherche, Limoges, France, 1991.

[7] R. Lid1 i nd H. Niederreiter, Finife Fields. Cambridge, England: Cam- bridge Llniversity Press, 1983.

[8] F. J. MacWilliams, “Codes and ideals in group algebras,” in Combina- torial M,ithematics and its Applications, R. C. Bose and T. A. Dowling, Eds. Chapel Hill, NC: Univ. of North Carolina Press, 1969.

[9] F. J. Ma:Williams and N. J. A. Sloane, The Theory of Error Correcting Codes. Amsterdam. The Netherlands: North Holland, 1977.

A

Sinc_e 5 E T we can apply Theorem 1 to for s = 5. Since 12 T, the corresponding coefficient is equal to zero: I San Antonio, TX, January 17-22, 1993.

’Department of Electrical Engineering, National Tsing Hua University. a3 + a,a; = 0 (10) Hsinchu, Taiwan 300, China.