Cyclic Codes and Self-Dual Codes Over

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1 1250 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 4, MAY 1999 Cyclic Codes and Self-Dual Codes Over A. Bonnecaze and P. Udaya TABLE I MULTIPLICATION AND ADDITION TABLES FOR THE RING F 2 + uf 2 Abstract We introduce linear cyclic codes over the ring F 2 + uf 2 = f0; 1;u;u = u +1g, where u 2 = 0and study them by analogy with the Z4 case. We give the structure of these codes on this new alphabet. Self-dual codes of odd length exist as in the case of Z4-codes. Unlike the Z4 case, here free codes are not interesting. Some nonfree codes give rise to optimal binary linear codes and extremal self-dual codes through a linear Gray map. Index Terms Codes over rings, cyclic codes, Gray map, self-dual codes. I. INTRODUCTION Among the four rings of four elements, the Galois field F 4, and more recently the ring of integers modulo four Z 4, are the most used in coding theory. Z 4-codes are renowned for producing good nonlinear codes by the Gray map, namely Kerdock, Preparata, or Goethals codes [11]. On the other hand, the ring F 4 admits a linear Gray map which does not give good binary codes. The ring R = F 2 + uf 2 shares some good properties of both Z 4 and F 4. This alphabet is given by all binary polynomials in indeterminate u of degree less than 2, and is closed under usual binary polynomial addition and multiplication modulo u 2. The set of elements of R is f0; 1;u;u = u +1g. It is easy to verify that R is a local ring with a maximal ideal given by f0;ug. The multiplication and addition table for the ring is given by Table I. The multiplication table coincides with that of Z 4, when u and u are replaced by, respectively, 2 and 3. In this sense, R is analogous to Z 4 and here u plays the role of 2. However, the addition table is different. The addition table is similar to that of the Galois field F 4 = f0; 1;; 2 = +1g, when u and u are replaced, respectively, by and 2. Note that from the definition, the characteristic of the ring is 2. Thus in the structure of alphabets, R lies between Z 4 and F 4. This ring can also be viewed as a vector space of dimension 2 over F 2. Moreover, the sets f0; 1g, f0;ug, and f0; ug form three subspaces in R and the subspace f0; 1g (= F 2) is a subring. Note that the ring R is isomorphic to the quotient ring Z[X]=(2; (X +1) 2 ), which was first used by Bachoc [1] in connection with constructions of modular lattices. In this correspondence, we describe the structure of cyclic codes and cyclic self-dual codes over R. Since F 2 is a subring of R, the minimum distance of a lifted cyclic code over R is not increased. However, the dimension is higher and the binary Gray image is linear. Unlike the Z 4 case, here, free codes are not interesting and the best codes are obtained when the minimum distance of the residue code is about the double of the minimum distance of the torsion code. The image by the linear Gray map of cyclic codes over R leads to a new representation of some class of hu; u + vi constructed codes. This is remarkable since certain good hu; u + vi constructed binary codes have a simple representation as cyclic R codes (see also [20]). Manuscript received October 8, 1997; revised October 13, The work of P. Udaya was supported by ARC under Grant A The material in this correspondence was presented in part at the IEEE Internationa Symposium on Information Theory, MIT, Cambridge, MA, August 16 21, A. Bonnecaze is with GECT, Université de Toulon et du Var, La Garde, France. P. Udaya is with the Department of Mathematics, City Campus, RMIT University, G.P.O. Box 2476V, Melbourne VIC-3001, Australia. Communicated by T. Kløve, Associate Editor for Coding Theory. Publisher Item Identifier S (99) Note that cyclic Z 4 codes reveal the structure of certain good binary nonlinear codes. Indeed Z 4 cyclic codes could also be viewed as a generalization of the hu; u + vi construction of codes (see [13]). The correspondence is organized as follows. Section II is devoted to the study of cyclic codes over R. All good cyclic codes of lengths 7 and 15 are given. Cyclic self-dual codes of odd length are introduced in Section III. They represent an interesting family since they produce modular lattices [1]. Cyclic self-dual codes of lengths 15, 21, and 31 are particularly interesting. The binary Gray images of some of the extended and augmented codes are extremal self-dual binary codes of parameters [44; 22; 8], [32; 16; 8], and [64; 32; 12]. II. CYCLIC CODES OVER F 2 + uf 2 We first establish some terminology. By a linear code C over a ring R (or an R-code), we mean an additive submodule of the R-module R n. Duality for codes is understood with respect to the form xy = xiyi, where x =(x1;x2; 111;xn) and y =(y1;y2; 111;yn). The i code C is said to be self-dual if C = C?. Two codes are equivalent if one can be obtained from the other by permuting the coordinates and, if necessary, exchanging 1 and u in certain coordinates. The Lee weight w L of x =(x 1 ; 111;x n ) is defined as n 1 (x)+2n 2 (x), where n 2(x) and n 1(x) are, respectively, the number of u symbols and the number of 1 or u symbols present in x. A nonzero linear code C over R has a generator matrix which, after a suitable permutation of the coordinates, can be written in the form G = I k A B 0 ui k ud where A and B are matrices over R and D is an F 2 matrix. The code C then contains all codewords [v 0 ;v 1 ]G, where v 0 is a vector of length k 1 over R and v 1 is a vector of length k 2 over F 2. Thus C contains 4 k 2 k codewords. The parameters of C are given by [n; 4 k 2 k ;d L ], where d L represents the minimum Lee distance of C. Following [8], we associate to the code C two binary codes. The residue code C 1 = fx 2 F n 2 j9y 2 F n 2 j x + uy 2 Cg and the torsion code C 2 = fx 2 F n 2 j ux 2 Cg. A cyclic code of length n over R is a linear code with the property that if (c 0 ;c 1 ; 111;c n01) 2 C then (c 1 ;c 2 ; 111;c 0 ) 2 C. We assume that n is odd and we represent codewords by polynomials. Then cyclic codes are ideals in the ring A. Galois Extension Ring of R R n = R[x]=(x n 0 1): The method of constructing Galois rings over R is similar to the construction of Galois rings over Z 4. The general case of such rings over R = F 2[u]=(w(u) k );k > 1, where w(u) is an irreducible polynomial of degree m 1 over F 2, has been studied in [21]. The ring F 2 + uf 2 is a special case of these rings when w(u) = u and k =2. Let R[x] be the ring of polynomials over R. We have a natural homomorphic mapping, from R to its residue field F 2. For any a 2 R, let ^a denote the polynomial reduction modulo u. Now, /99$ IEEE

2 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 4, MAY define a polynomial reduction mapping : R[x]! F 2 [x] in the obvious way f (x) = r a i x i i=0! r ^a i x i : i=0 A monic polynomial f over R[x] is said to be a basic irreducible polynomial if its projection (f ) is irreducible over F 2[x]. The Galois ring of R denoted as GR (R; r) is defined as R[x]=(f(x)), where f (x) is a basic monic irreducible polynomial of degree r over R. Hence the ring GR (R; r) is a module over R. The basic monic irreducible polynomial of degree r over R can be lifted from a monic irreducible polynomial over F 2. The trick is to consider a monic irreducible polynomial over F 2 which is a subring of R. For any polynomial f (x) 2 F 2[x], let f (x) denote the same polynomial viewed as an element of R[x]. Since F 2 is a subring of R, we will not make distinction between f and f if the context is clear. Any irreducible polynomial over the subring is obviously irreducible over the ring. Thus any monic irreducible polynomial f (x) over F 2 is a basic monic irreducible polynomial over R. Note that this is not the situation in the Z4 case where the polynomial lift from the ground field Z2 is nontrivial [11]. Like Galois fields, GR (R; r) is unique for a given r [14]. The group of units of GR (R; r) denoted by GR? (R; r) is given by a direct product of two groups GR? (R; r) =G C 2 G A where G C is a cyclic group of order 2 r 0 1 and G A is an Abelian group of order 2 r [21], [7]. Lemma 1: The set fg C ; 0g is isomorphic to the residue field F 2 and is also a subspace of GR (R; r). Thus the set is a subring over GR (R; r). Proof: Since G C is cyclic, the set fg C; 0g satisfies the multiplicative axiom of a field. For every, 1;2 2 G C ; (1 + 2) 2 = (1 + 2), since the characteristic of R is 2. This implies that (1 + 2) 2 01 = 1 and (1 + 2) 2 fg C ; 0g. Thus fg C ; 0g is closed under addition which proves the lemma. Using Lemma 1, the elements of G A are given by the set f(1 + u); 2 F 2 g: The zero divisors of the ring R are given by the elements of the maximal ideal generated by u, namely, f(u); 2 F 2 g. Thus we have Lemma 2: The only ideals of GR (R; r) are (0); (1); and (u). Thus any element of GR (R; r) can be uniquely represented as = 1 + u2; 1;2 2 F 2 : (1) This is analogous to the p-adic representation considered in [5]. The Galois automorphism group of GR (R; r) is cyclic of order r and is generated by the Frobenius map defined by where is as in (1). () =(1) 2 + u(2) 2 ; 2 GR (R; r) B. Cyclic Codes, Generators, and Idempotent Polynomials In order to define cyclic codes of length n over R, we need to know a factorization of the polynomial x n 0 1 over R and study the ideals in the polynomial algebra generated by x n 0 1. We first demonstrate that such a factorization exists by using additive properties of the ring. Since there exists a primitive element in GR (R; r), the polynomial (x n 01), where n =2 r 01, factors linearly over GR (R; r). We have (x n 0 1) = (x 0 )(x 0 2 ) 111(x 0 n ) where is a primitive element of fg C ; 0g = F 2. Define the minimal polynomial of ; 2 F 2 in GR (R; r) as m() =(x 0 )(x 0 ()) 111(x 0 t01 ()) where 2 F 2 of order 2 t 0 1; t divides r. It is easy to see that if (x 0 ) divides (x n 0 1), then (x 0 ()) = (x 0 2 ) also divides (x n 0 1). Similarly, the minimal polynomial of any nonzero element also divides (x n 0 1) and m() is a polynomial over R. Moreover, since belongs to F 2 of GR (R; r), m() is a polynomial over F 2 in R. Thus considering minimal polynomials of elements of G C belonging to distinct Frobenius classes gives the factorization of (x n 0 1) over R (x n 0 1) = m(1)m(2) 111m( t ) where 1;2; 111; t are generators of distinct Frobenius classes. The factorization of (x n 0 1) can also be obtained as follows: we know that F 2 is a subring of R and that x n 0 1 factors uniquely as a product of pairwise-coprime irreducible polynomials over the binary field. If any polynomial factors over a subring, it also factors over the ring. Thus the factorization x n 0 1 =f1f2 111f r carries over R. Note that this is not the situation in the Z4 case where the factorization has to be achieved by a nontrivial lift [11]. We have the following lemma. Lemma 3: If x n 0 1=f1f2 111f r, where f i are basic irreducible and pairwise-coprime, then this factorization is unique. The factorization is obtained from factorization of the binary polynomial x n 0 1. Proof: The factorization is demonstrated above. The ring R is a local ring with unique maximal ideal. By Hensel s Lemma [14], regular polynomials (polynomials which are not zero divisors) over R have a unique factorization. From [14] any zero divisor f (x) in R can be uniquely written as uf 0 (x), where f 0 (x) is a regular polynomial. In particular, (x n 01) is regular and hence the lemma is proved. We define f as a primary polynomial if (f) is a primary ideal [14]. The maximal ideal contains all zero divisors. In Lemma 2, we have given the structure of the prime ideals in R[x]=(f (x)) [17]. Now we give the structure of all ideals in R n along the lines of results of [17] for Z4-cyclic codes. Theorem 1: Let n be odd. Let x n 0 1 = f1f2 111f r, where the f i(1 i r) are basic irreducible and pairwise-coprime polynomials. Let ^f i denote the product of all f j except f i. Then any ideal in the ring is a sum of ( ^f i) and (u ^f j). Proof: The proof is similar to the proof given in [17, Theorem 1] for ideals in Z4[x]=(x n 0 1) as in this case also, ideals in R n can be written as I = I1 8 I I t where I i ;i =1; 111;t; is an ideal of the Galois ring R[x]=(f i ). The only ideals in R[x]=(f i) are (0); (1) or (u). The ideals (1) and (u) in I i = R[x]=(f i) correspond, respectively, to the ideals (f i) and ( ^f i ) in R n. In any case, the ideal I is a sum of (f i ) and ( ^f j ) since (f1f2 111f t)=(f1)(f2) 111(f t): As a consequence of the above theorem, the number of cyclic codes over R of length n is 3 t, where t is the number of basic irreducible polynomial factors in x n 0 1 over R. The following two theorems characterize cyclic codes and their duals over R by giving generator polynomial description. We omit the proofs as they are identical to the corresponding theorems in [17] for Z4 cyclic codes.

3 1252 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 4, MAY 1999 Theorem 2: Suppose C is a cyclic code of odd length n over R, then there are unique, monic polynomials f; g; h such that C = (fh;ufg), where fgh = x n 01 and jcj =4 deg (g) 2 deg (h). We have a) when h =1;C =(f) and jcj =4 n0deg (f ) ; b) when g =1;C =(uf) and jcj =2 n0deg (f ). Theorem 3: Suppose C = (fh;ufg) is a cyclic code of odd length n over R, where f;g; h are monic polynomials such that fgh = x n 0 1, and jcj =4 deg (g) 2 deg (h). Then, the dual of C is C? =(g? h? ;ug? f? ) and jc? j =4 deg (f ) 2 deg (h) : If h = 1;C = (f) and jc? j = (g? ).Ifg = 1;C = (uf) and jc? j =(h? ;uf? ), where f? ;g? ; and h? are, respectively, reciprocal polynomials of f;h; and g. Hence, with any cyclic code over R, the residue and torsion codes are given by 1) the residue code C 1 = (fh) of dimension deg (g); 2) the torsion code C 2 = (f ) of dimension deg (g) + deg (h). Note that the code C over R is completely determined by the residue and torsion codes, as the residue field obtained using the homomorphic mapping is a subring in R. A code is said to be free if and only if the dimension of the residue code is equal to the dimension of the torsion code. We define an idempotent in R[x] as a polynomial e(x) such that TABLE II WEIGHT DISTRIBUTION OF SIMPLEX CODES OVER R TABLE III ALL NONTRIVIAL CYCLIC CODES OF LENGTH 7. THE SYMBOL? MEANS THAT THE CODE IS OPTIMAL e(x) 2 = e(x)mod(x n 0 1): Let C =(f) be a free code (respectively, C =(uf)) over R, then C has an idempotent generator e(x) which is given by the idempotent generator of the binary residue (respectively, torsion) code. This is a straightforward consequence of the factorization of (x n 0 1) in R. Note that if the code is free the idempotent of its dual is given by 1 0 e(x 01 ). But this is not true for a code which has only a torsion part. In general, if C =(fh;ufg), where fgh = x n 0 1, then C also has a representation C = (e 1;ue 2), where e 1 and e 2 are the idempotent polynomials of the binary codes ((fh)) and ((fg)), respectively. C. Binary Codes Obtained from Cyclic Codes Over F 2 + uf 2 For any element of R expressed as x + uy, we let where x; y 2 F 2. (x + uy) =(y; x + y); Lemma 4: If a code C is linear or self-dual, so is (C). The minimum Lee weight of C is equal to the minimum Hamming weight of (C). In the next lemma, we relate minimum Lee weight of a code C to its component binary codes. Lemma 5: The minimum Lee weight of C is given by min (d 1 ; 2d 2 ), where d 1 and d 2 are, respectively, minimum distances of the residue and torsion code. When the code is free, its minimum distance is exactly the minimum distance of its residue code. Proof: Since the codes C 1 (residue) and uc 2 (code obtained by multiplying u with C 2) are completely included in C, the result is obvious. The above lemma gives us an idea of the minimum distance of the R-cyclic codes. In general it is difficult to find the exact minimum distance and weight distribution. But, for the simplex code (free code generated by (x n 0 1)=f, where f is a primitive irreducible polynomial of degree r), the weight distribution can be computed using the vector space structure of GR (R; r) [21]. The weight distribution depends on the ranks of the matrices M =[1;] formed by adjoining the element 1 with each belonging to F 2 [21]. It is easy to verify that there are 2 r 0 2 such matrices whose rank is 2. The rest will have rank 1. The weight distribution is given in Table II. The simplex code is the analog version of the Z 4 Kerdock code. The codewords of the simplex code share many properties of m-sequences over F 2 but are not field m-sequences. They form an interesting class of optimal sequences with respect to Hamming correlation [21]. The main implication of Lemma 5, is that the minimum distance of a free code is equal to that of its residue code. This is the reason why free codes are not interesting over R in the sense that the parameters of their binary images are not good. D. Examples 1) Length 7: Table III presents all the nontrivial cyclic codes of length 7. The factorization of x 7 +1 is f 1 f 2 f 3 where f 1 = x +1;f 2 = x 3 + x +1;f 3 = x 3 + x Optimal codes are indicated by the symbol? (by an optimal code, we mean a binary code having the maximal minimum distance for the given length and the dimension). The binary images of 13th and 14th codes are equivalent and isomorphic to D 14 of [15]. The binary images of extended augmented codes (16th and 22th codes) are equivalent to A 8 8 A 8 in the classification of Pless in [15]. Remark: The binary Gray image of the free codes considered in our correspondence can be equivalently expressed as repeated root cyclic codes in the sense of [6]. However, this is not true when the code is not free. For example, the codes corresponding to the second

4 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 4, MAY TABLE IV EXAMPLES OF CYCLIC CODES OF LENGTH 15. THE SYMBOL? MEANS THAT THE CODE IS OPTIMAL Theorem 4: Let C =(fh;ufg) be a cyclic code over R, where fgh = x n 0 1, n odd. Then C is self-dual if and only if f = g? and h = h?. Proof: Note that the factorization of x n 0 1 is the binary factorization. Hence [18, Theorem 2] holds with = 1, where = f=g? = h=h?. The condition of existence of a self-dual code of odd length n has been studied deeply in [18]. The following theorem is due to [18] which applies to codes over R. Theorem 5: Nontrivial cyclic self-dual codes of length n exist if and only if i (mod n) for any i. As a consequence of this theorem, nontrivial self-dual cyclic codes do not exist for lengths 17 and 19. Similarly to the Z 4 case, the conditions in the above theorem can be further refined by using a number-theoretical result of Moree [18, Appendix]. and fourth row are not equivalent to the code of parameters [14; 3; 8] of [6, Table I]. Similarly, the binary image of the code corresponding to the 13th row is self-dual and not equivalent to the self-dual cyclic code given in [19] (itself equivalent to D 14 [15]). 2) Length 15: The factorization of x is equal to f 1f 2f 3f 4f 5, where f 1 = x + 1;f 2 = x 2 + x + 1;f 3 = x 4 + x + 1; f 4 = x 4 + x 3 +1;f 5 = x 4 + x 3 + x 2 + x +1. There are 3 5 different cyclic codes for length 15. We give only codes which have best minimum Lee distance. For example, there are many codes whose binary images have the same dimension but have different minimum distances. We choose the best among these. Table IV presents all such good cyclic codes of length 15. The cyclic self-dual codes mentioned in the table are equivalent and give rise to a Type I self-dual binary code with parameters [30; 15; 6]. This code is equivalent to r 30 in the classification of Pless [16]. Note that the binary image of the corresponding Z 4 cyclic self-dual codes also gives rise to r 30 [18]. The binary Gray image of the extended augmented code is equivalent to the Reed Muller code r 32 in [16]. III. SELF-DUAL CODES OVER F 2 + uf 2 The extended quadratic residue Z 4-codes represent one of the most important classes of self-dual Z 4 -codes. For lengths 8, 24, 32, or48 these codes are extremal Type II codes [2], and are involved in the construction of even unimodular lattices, including the Leech lattice. But we must note that over this ring, the good constructions differ from that of Z 4 case where free codes are the most interesting. Over R, we are not going to consider quadratic residue codes. We will prefer to consider codes whose rates k 1 and k 2 are approximately the same. Recently, it has been shown in [18] that nontrivial cyclic selfdual Z 4-codes of odd length n exist. This property also holds over R. In this section, we study such cyclic codes and their extensions. Most of their structural properties match those of self-dual Z 4 -cyclic codes. The condition for a cyclic code of odd length n to be self-dual is given by the following theorem. A. Examples If C is a cyclic self-dual code of odd length n, it is always possible to construct another self-dual code of length n +1. Let C = (fh;ufg), then from Theorem 3, x +1 has to divide the polynomial h. Consider an R-cyclic code given by ^C =(f ^h; uf ^g), where ^g =(x +1)g and ^h = h=(x +1): Then, the code formed by extending ^C is a self-dual code of length n +1. This construction is equivalent to the method using shadow codes introduced in [8]. 1) Length 21: For length 21, there exist three cyclic self-dual codes with different symmetrized weight enumerators. Among them, just one has good parameters. The factorization of x 21 +1is equal to f 1 f 2 f 3 f 4 f 5 f 6, where f 1 = x +1 f 2 = x 2 + x +1 f 3 = x 3 + x +1 f 4 = x 3 + x 2 +1 f 5 = x 6 + x 4 + x 2 + x +1 f 6 = x 6 + x 5 + x 4 + x 2 +1: The most interesting code is C21 1 = (fh;ufg) where f = f 5 ; g = f 6;h = f 1f 2f 3f 4: It is a cyclic self-dual code of parameters [21; ; 6]. Its binary Gray image has parameters [42; 21; 6]. It is possible to extend this code as explained at the beginning of this section. We thus obtain an extremal binary self-dual code of parameters [44; 22; 8] whose weight enumerator corresponds in [9] to W 1 with =38. The order of the automorphism group of this code is = This order is different from those of codes with the same weight enumerator discovered in [10], [4]. Hence, this is a new self-dual code. The two other codes are less interesting since their minimum distance is only 4. Their binary Gray images are selfdual codes of parameters [42; 21; 4]. Their constructions are given in the following table: 2) Length 23: For length 23, there exists just one nontrivial cyclic self-dual code: C23. The factorization of x is equal to f 1 f 2 f 3, where f 1 = x +1 f 2 = x 11 + x 9 + x 7 + x 6 + x 5 + x +1 f 3 = x 11 + x 10 + x 6 + x 5 + x 4 + x 2 +1:

5 1254 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 4, MAY 1999 C23 = (fh;ufg) where f = f 2 ;g = f 3 ;h = f 1 : It is a cyclic self-dual code of parameters [23; ; 8]. Its binary image is a selfdual [46; 23; 8]-code. The code C23 can also be extended to obtain a [24; 12; 8] self-dual code with binary Gray image of parameters [48; 24; 8]. This code is the doubled Golay code which has some interesting combinatorial properties [3]. 3) Length 31: For length 31, there exist five inequivalent cyclic self-dual codes according to the criterion given in [12]. Among them, just two have good parameters. The factorization of x 31 +1is equal to (x +1)f 1f 1cf 2f 2cf 3f 3c, where f 1 =1+x 2 + x 5 f 1c =1+x 3 + x 5 f 2 =1+x + x 2 + x 3 + x 5 f 2c =1+x 4 + x 3 + x 2 + x 5 f 3 =1+x + x 2 + x 4 + x 5 f 3c =1+x 4 + x 3 + x + x 5 : The most interesting code is C31 1 =(fh;ufg) where f = f 1 f 2 ; g = f 1c f 2c ;h = (1 + x)f 3 f 3c : It is a cyclic self-dual code of parameters [31; ; 10]. Note that the minimum distances of its residue and torsion code are, respectively, 10 and 5. Its Gray image is a self-dual binary (Type I) code with parameters [62; 31; 10]. As explained in Section III-A, it is possible to construct a selfdual code of length 32 from C31 1. We obtain a [32; ; 12] code. Its Gray image is an extremal binary self-dual Type II code with parameters [64; 32; 12]. The second interesting code, C31 2, has the same parameters as C31 1. We have C31 2 = (fh;ufg) where f = f 1 f 1c ;g = f 2 f 2c ;h =(1+x)f 3 f 3c : Similarly, this code gives self-dual codes of lengths 32 over R, and 62 and 64 over F 2 with same weight enumerators. The order of the automorphism group of the above two codes is = They have an automorphism of order 31. Hence each of these codes must be equivalent to one of the 38 extremal codes constructed in [22]. The three other cyclic self-dual codes correspond to the case where the difference between the rates k 1 and k 2 is large. Hence, they have bad parameters in the sense that their minimum distance is less than or equal to 8. Their constructions are given in the following table. 4) Higher Lengths: It is possible to obtain cyclic self-dual R-codes in higher lengths. For lengths 35; 39; 45; 47; and 51 they are not extremal in view of Lemma 5. For length 55, there exists a cyclic self-dual code leading to binary codes with parameters [110; 55; 10] and [112; 56; 12]. The code with parameters [112; 56; 12] is of Type II and is nearly a code with parameters [112; 56; 16] as it has only 11 codewords with weight 12. Note that there exist Type II self-dual codes with parameters [112; 56; 16], even though it is not certain that these are extremal [9]. Similarly, at lengths 63 and 127 the cyclic R self-dual codes lead respectively to Type II binary self-dual codes with parameters [128; 64; 16] and [256; 128; 28]. IV. CONCLUSION We studied cyclic codes over F 2 + uf 2 and constructed some interesting cyclic self-dual codes over this ring. The generalization to F p + uf p u k F p may be of interest in coding theory. These rings have already been used in the construction of optimal frequency-hopping sequences [21]. We think that codes over F 2 + uf 2 are of importance because of two reasons. 1) They can lead to optimal linear and extremal self-dual binary codes. We have given such good codes for lengths 32; 42; 44; 62; and 64. Furthermore, a Type II [24; 4 12 ; 12] R-code has also been obtained in [8]. 2) These codes are interesting from a decoding point of view. The binary Gray image of a code over F 2 + uf 2 can be decoded in the ring [20]. This means that the decoding problem for a code of length 2n changes to one of length n. Since the characteristic of the ring is two, we obtain considerable advantage in decoding complexity. The next length to consider is 63. In this length, we may get a binary code whose performance could be compared to the [127; 78; 15] BCH code used in the industry to convert data rates from 9.6 to 16 kbits/s. The advantage of this ring compared to Z 4 is mainly due to the fact that our codes can be easily decoded and implemented. Even though the minimum distance of the Gray images of these codes are not as good as some exceptional Z 4 cyclic codes, they can correct some extra burst errors along with random errors [20]. ACKNOWLEDGMENT The authors wish to thank Serdar Boztaş, Masaaki Harada, Vera Pless, and Patrick Solé for their comments on the manuscript. They also thank the referees for their useful comments which improved the presentation of the material. The calculations were mainly done using the computer algebra package Magma. REFERENCES [1] C. Bachoc, Application of coding theory to the construction of modular lattices, J. Combin. Theory Ser. A, vol. 78, pp , [2] A. Bonnecaze, P. Solé, C. Bachoc, and B. 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6 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 4, MAY [18] V. Pless, P. Solé, and Z. Qian, Cyclic self-dual Z 4 -codes, Finite Fields Their Applic., vol. 3, pp , [19] N. J. A. Sloane and J. G. Thompson, Cyclic self-dual codes, IEEE Trans. Inform. Theory, vol. IT-29, pp , [20] P. Udaya and A. Bonnecaze, Cyclic codes over a linear companion of Z 4, in Proc IEEE Int. Symp. Inform. Theory (MIT, Cambridge, MA, 1998), p [21] P. Udaya and M. U. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomials residue class rings, IEEE Trans. Inform. Theory, vol. 44, pp , [22] V. Y. Yorgov, Doubly-even extremal codes of length 64, Probl. Inform. Transm., vol. 22, pp , Further Results on Generalized Hamming Weights for Goethals and Preparata Codes Over Tor Helleseth, Fellow, IEEE, Bo Hove, and Kyeongcheol Yang, Member, IEEE Abstract This correspondence contains results on the generalized Hamming weights (GHW) for the Goethals and Preparata codes over Z 4. We give an upper bound on the rth generalized Hamming weights d r(m; j) for the Goethals code G m(j) of length 2 m over Z 4, when m is odd. We also determine d 3:5 (m; j) exactly. The upper bound is shown to be tight up to r =3:5. Furthermore, we determine the rth generalized Hamming weight d r (m) for the Preparata code of length 2 m over Z 4 when r =3:5 and r =4. Index Terms Generalized Hamming weights, Goethals codes, linear codes over Z 4, minimum support size, nonlinear code, Preparata codes, weight hierarchy. I. INTRODUCTION Let R m = GR (4;m) be a Galois ring with 4 m elements and let Rm 3 be the set of units of R m. Rm 3 has a multiplicative cyclic subgroup of order 2 m 0 1. Let T m = f0; 1;;111; 2 02 g, where 2 Rm 3 is an element of order 2 m 0 1. Any element z 2 R m can be expressed uniquely as z = A +2B for A; B 2T m. Let denote the modulo 2 reduction map. Note that = () is a primitive element in the finite field F 2 with 2 m elements, thus (T m )=F 2 (see [2] and [6] for details). A. The Goethals Code Over Z 4 Let j be an integer relatively prime to m, that is, gcd (j; m) =1. Let G m(j) be the code of length 2 m over Z 4, whose parity-check Manuscript received December 15, 1997; revised October 29, This work was supported in part by the Korea Science and Engineering Foundation (KOSEF) under Grant , and The Norwegian Research Council under Grants /410 and /420. T. Helleseth is with the Department of Informatics, University of Bergen, N-5020 Bergen, Norway. B. Hove is with the Department of Mathematical Sciences, Aalborg University, DK-9220 Aalborg Ø, Denmark. K. Yang is with the Department of Electronic and Electrical Engineering, Pohang University of Science and Technology, Pohang , Korea. Communicated by A. Barg, Associate Editor for Coding Theory. Publisher Item Identifier S (99) matrix is given by H = (2 +1) (2 +1)(2 02) : The quaternary code G m(1) is called the Goethals code of length 2 m over Z 4. In Hammons, Kumar, Calderbank, Sloane, and Solé [2] it is shown that if m is odd, then G m (1) has minimum Lee weight 8 and the image of G m(1) under the Gray map gives a (2 m+1 ; m02 ; 8) binary nonlinear code and so it is optimal. In [5], Helleseth, Kumar, and Shanbhag showed that G m (j) has the same Lee weight distribution as the Goethals code G m (1). Hence, the code G m(j) is called the Goethals code of length 2 m over Z 4 in this correspondence. B. The Preparata Code Over Z 4 Let P m be the code over Z 4 whose parity-check matrix is given by H = : The quaternary code P m is called the Preparata code of length 2 m over Z 4. In Hammons, Kumar, Calderbank, Sloane, and Solé [2] it is shown that if m is odd, then P m has minimum Lee weight 6 and its image under the Gray map gives a (2 m+1 ; m02 ; 6) binary nonlinear code. The binary image of P m under the Gray map has the same weight distribution as the original Preparata code, so it is optimal. It can be easily checked that if m is even, then P m has minimum Lee weight 4. The generalized Hamming weights (GHW) for linear codes over Z 4 is defined as follows (see [1]). Let C be an [n; k] linear code over Z 4, and let B Cbe a subcode. Then the support of B is defined as follows: (B) =fi j c i 6= 0for some (c1; 111;c n) 2Bg: For any r, where 0 r k and 2r is an integer, the rth generalized Hamming weight of C is defined as d r (C) := minfj(b)j : B is a submodule of C with jbj =4 r g: Conventionally, d 0 is assumed to be 0. The weight hierarchy of C is the sequence given by fd r (C)g k r=0:5, where 2r is an integer. C. Main Results Let d r (m; j) be the rth generalized Hamming weight for G m (j). In this correspondence, we consider only the case when m 3 is an odd integer. Recently, Yang and Helleseth determined d r (m; j) for r =0:5; 1; 1:5; 2; 2:5; and 3 [9]. In this correspondence (Section III), we determine d 3:5(m; j) exactly, and give an upper bound on d r (m; j), which is shown to be tight up to r =3:5. Let d r (m) be the rth generalized Hamming weight for P m. Recently, Yang and Helleseth determined d r(m) for r = 0:5; 1; 1:5; 2; 2:5; and 3 [8]. Here, in Section IV, we determine d 3:5 (m) and d 4 (m) exactly. II. PRELIMINARIES From now on, we will normally use the capital letters X; Y; A; B, etc., to denote elements in T m, and the small letters x; y; a; b to represent their corresponding projections modulo 2 in F 2. For example, we use a = (A); x i = (X i ) 2 F 2 for A; X i 2T m, /99$ IEEE