Complete joint weight enumerators and self-dual codes

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 5, MAY 2003 1275 Complete Joint Weight Enumerators and Self-Dual Codes YoungJu Choie, Steven T. Dougherty, and Haesuk Kim Abstract—We define the complete joint weight enumerator in genus for codes over 2 and use it to study self-dual codes and their shadows. These weight enumerators are related to the theta series of the associated lattices and Siegel and Jacobi forms are formed from these series. Index Terms—Jacobi and Siegel forms, self-dual codes, unimodular lattices. I. INTRODUCTION A CODE of length over is a subset of , and if it is an additive subgroup then we say that it is a linear code. We shall use the following weights on this ambient space. be a vector in , then the Hamming Let is the number of nonzero components and the weight Euclidean weight The minimum Hamming and Euclidean weights are denoted by and , respectively, and are the smallest Hamming and Euclidean weights among all nonzero codewords of . and in , We use the standard inner product of specifically where of and . The dual code is defined as for all We say that a code is self-orthogonal if and self-dual if . are defined as self-dual In [1], Type II codes over , and those codes with Euclidean weights divisible by self-dual codes which are not Type II are said to be Type I. This definition is natural because Type II codes produce Type II lattices and Type I codes produce Type I lattices under the usual construction; see equation (1) later. Manuscript received November 22, 2000; revised October 6, 2001. The work of Y. Choie was supported in part by Com MaC-KOSEF and KOSEF2000-015DP0009. The work of H. Kim was supported by Com MaC-KOSEF. Y. Choie and H. Kim are with the Department of Mathematics, Pohang University of Science and Technology, Pohang, 790-784, Korea (e-mail: [email protected]). S. T. Dougherty is with the Department of Mathematics, University of Scranton, Scranton, PA 18510 USA (e-mail: [email protected]). Communicated by P. Solé, Associate Editor for Coding Theory. Digital Object Identifier 10.1109/TIT.2003.810649 We shall take the standard definition of a shadow of a be a Type I code over . self-dual code. Namely, let as the subcode of vectors whose Euclidean weight Define . Then is index in and is congruent to where is the shadow and and . be -dimensional Euclidean space with inner product Let . An -dimensional lattice in is a free -module spanned by linearly independent vectors. The dual lattice is defined by for all A lattice is integral if and unimodular if .A unimodular lattice is Type II if the norms of all elements are even and the lattice is Type I otherwise, where the norm of a . vector is , by Define the reduction modulo , by Given a code over we construct a lattice by (1) is a Type It is shown in [1] that if is a Type I code then I unimodular lattice, and that if is a Type II code then is a Type II unimodular lattice. As for codes we make an analogous definition of the shadow denote the subset of a lattice. Let be a Type I lattice and let is index in and of vectors with even norms. Then where is the shadow and and . It is shown in [8] that (2) and for up to labeling. In particular, where is the shadow of the unimodular lattice for . II. WEIGHT ENUMERATORS Throughout the remainder of the paper we shall often denote by . The following definition is a generalization the ring of the complete weight enumerator in genus introduced in [1] and of the standard joint weight enumerator. Definition 1 (Complete Joint Weight Enumerator in Genus ): Let , be codes in . The complete joint 0018-9448/03$17.00 © 2003 IEEE 1276 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 5, MAY 2003 weight enumerator for codes defined as of length over is with (3) The joint weight enumerator in genus was introduced in [7] and is the natural generalization of the joint weight enumerator. be codes over of length . Then if Let , , , , where a subset of is a monomial and represents vectors , in in different orders then by counting the weights of we have where and and . Note that we do not assume that the codes are linear, in fact, we are interested in determining complete joint weight enumerators where some of the codes are shadows, which are nonlinear codes. We shall often denote the complete joint weight enumerator . Notice that by where is the complete weight enumerator for a code given in [1]. Notice, however, that the MacWilliams relations given in [1] apply only in this specific case. MacWilliams relations for the complete weight enumerator in genus must be able to determine from , where is for is for other . These MacWilliams relation some and will be given in Section II-A. The complete joint weight enumerator in genus is also related in a natural way to the following weight enumerators. Let be the set of equivalence classes of where the elements are equivalent if or . Then we can make the following definition. Definition 2 (Symmetrized Joint Weight Enumerator in Genus ): Let be codes in . The symmetrized joint weight enumerator for codes of length over is defined as Of course, a similar theorem is true for the symmetrized joint weight enumerator and the Euclidean weight and for the joint weight enumerator and the Hamming weight. It follows then that for all monomials in we have Again, similar results can be made for the other weight enumerators. A. MacWilliams Relations We shall describe the following notation to describe the action be an by of a matrix on a polynomial ring. Let matrix and a polynomial in then be a primitive th root of unity, that is, Let The matrix is given by with (7) (4) Definition 3 (Joint Weight Enumerator in Genus ): Let be codes in . The joint weight enumerator of length over is defined as for codes where and are in . Theorem 2.1 (MacWilliams Relations): Let be codes in and let denote either or with . Then (8) (5) where if with where an element (6) . if if and only if . , . Proof: The proof of this theorem follows exactly as the proof given for the joint weight enumerator in [7], except that the function is changed and the matrix is changed accordingly. CHOIE et al.: COMPLETE JOINT WEIGHT ENUMERATORS AND SELF-DUAL CODES Note that the matrix is an by matrix and is a polynomial in variables. that Corollary 2.2: Let denote either or be codes in 1277 Proof: The proof proceeds by induction on the cardinality of . Let and let . Then (13) (9) since any monomial such that representing vectors has where and otherwise. Next assume that (12) holds for with and (14) where if (10) where if . for Then notice that have the result. Let Proof: The result follows from the previous theorem. denote the shadow of the Type I code . Theorem 3.2: Let , and we , i.e., be Type I codes over Note that the MacWilliams relation for were given in [7]. These are not the only possible weight enumerators that can be obtained from the complete joint weight enumerator. For example, two elements can be related in a variety of ways, see [13] for examples. , then (15) where if III. JOINT WEIGHT ENUMERATORS OF SHADOWS AND CODES In this section, we show how the complete joint weight enumerator relating shadows and Type I codes can be determined. be Type I codes in . Let denote Let consisting of those vectors that have Euthe subcode of . clidean weight congruent to , with For a code , define its shadow by . . Define as follows: Let be a subset of (11) for . Theorem 3.1: Let be Type I codes over , then (12) where if if . if and if if . Proof: Again we proceed by induction on the cardinality and let with of . Let if and only if 1278 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 5, MAY 2003 Next assume that (15) holds for Specifically Proof: The code . so we have (21) (16) where if and only if Then we have . Then and this gives the result. A. Generalized Shadows was A generalized version of the shadow for codes over introduced in [5]. Specifically, if is a self-dual code over and is a constant vector not in , such that , then we define and the result follows as above. for all A less general version of this theorem was stated incorrectly in [1]. Specifically, the constant was missing. Other weight enumerators can be computed in similar ways, where is either , , or . for example, Moreover, each of these weight enumerators must have nonnegative integral coefficients for all monomials in order for the code to exist. Corollary 3.3: Let be Type I codes over , then (17) where is defined as in Theorem 3.1, and (18) Then so that . In general, is chosen to be a constant vector so that all the respective weight enumerators can be computed. These shadows are useful for construction is in of larger self-dual codes as in (19). Since for all self-dual codes over it is not difficult to produce such an by simply dividing this vector by , until we have a vector that is not in . Of course, it is possible that all of these vectors are in in which case such a shadow is simply equal to the code as the usual shadow is for a Type II code. be self-dual codes in and let Let be vectors such that , and . We choose as described earlier so divides for all . Let denote the subcode of that consisting of those vectors orthogonal to . Define as above. Let be a subset of . Define as follows: (22) is defined as in Theorem 3.2. where Proof: Follows from the previous two theorems noting in . that In [9] it was shown that if and are Type I codes of length and , respectively, with either or with , then for . be self-dual codes over Theorem 3.5: Let then (19) and are the subcodes of vectors is a self-dual code where and the other cosets are with weight congruent to defined as usual. The code is said to be formed by the shadow sum of and . be self-dual codes of Theorem 3.4: Let be self-dual code of length length and with and satisfying the above conditions so that is the self-dual code formed from the shadow sum of and . Then (23) where if if and (24) where if (20) if CHOIE et al.: COMPLETE JOINT WEIGHT ENUMERATORS AND SELF-DUAL CODES 1279 for and if , group , is invariant under the action of the , where if Proof: Similar to the proof of Theorems 3.1 and 3.2. Proof: Similar to the proof of Theorem 3.6. B. Invariant Rings for Shadow Codes Let be a subgroup of defined as is an even integral symmetric matrix where Theorem 3.6: Let be self-dual codes in . , the complete joint weight enumerator For codes is invariant under the action of the group . Proof: The fact that the weight enumerator is invariant , , follows from Theorem 2.1. The under can be shown in a similar manner as in invariance of [1, proof of Theorem 5.3]. Notice that runs over all even integral symmetric matrices. The matrices the form In this subsection, we consider the invariant rings for generalized shadow codes. From the fact that the complete joint weight enumerator for the generalized shadow codes can be obtained from the complete joint weight enumerator for codes, we easily obtain the following theorems. be self-dual codes over Theorem 3.9: Let and be as in Theorem 3.5, specifically, is either the code or its generalized shadow depending on . For codes , the complete joint weight enumerator is invariant under the action of the group , where Proof: Similar to the proof of Theorem 3.7. , generate all matrices of In a similar manner to Theorem 3.8, we have the following theorem. with if and only if . Thus, the group also . These matrices repcontains all matrices of the form resent the action of the MacWilliams relations on any subset of the indexes. be Type I codes over and Theorem 3.7: Let be as in Theorem 3.2, specifically is either the code or its shadow depending on . Then, the complete joint weight enu, , is invariant under the acmerator for tion of the group C. Invariant Rings for Generalized Shadow Codes , where Proof: The result follows from Theorems 3.2 and 3.6. Similarly, we define the invariant ring for the symmetrized be a subgroup of joint weight enumerator in genus . Let defined by is an even integral symmetric matrix be self-dual codes in Theorem 3.10: Let . For codes , the symmetrized joint weight enuis invariant under the action of the group merator , where Proof: Similar to the proof of Theorem 3.8. IV. JOINT WEIGHT ENUMERATORS, JACOBI FORMS, AND SIEGEL MODULAR FORMS In this section, we show the connection between the complete joint weight enumerators of codes over and Jacobi forms. First, we recall the definition of Jacobi forms of higher genus given in [14]. be the Siegel upper half space of genus Let Let be a character on the sympletic group . For a and any nonnegative holomorphic function integer and , the slash operators with respect to are defined as where and Theorem 3.8: Let be self-dual codes in . , the symmetrized joint weight enumerFor codes ator is invariant under the action of the group . be Type I codes over and be as in Let Theorem 3.2. Then the symmetrized joint weight enumerator where For a finite index subgroup Jacobi form as follows. of we define a 1280 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 5, MAY 2003 Definition 4 (Jacobi Form): Let and be nonnegative integers and be a character of . A holomorphic function satisfying for all (25) for all (26) and having a Fourier expansion of the following form: where (30) Proof: Let be a natural homomorphism from to . Let be a -tuple of codewords of . Then (27) with only if (31) is half integral positive semidefinite, is called a Jacobi form of is the weight and index with respect to on . Here, trace of . We denote the space of Jacobi forms of weight and index with respect to on by . In the case when is . trivial, we denote the space of Jacobi forms by Remark 1: For form of weight of genus , is a Siegel modular with respect to on . be a subset of and be Let self-dual codes over . We denote the lattice induced from the as , where code if if so that if if . Note that the lattice formed from the shadow of a Type I code is the shadow of the formed unimodular lattice (see (2)). Define on as follows: a theta series (28) is a Type II code over then . Also note that if The next theorem gives a connection between the theta series defined over the lattices induced from codes and their weight enumerators. be self-dual codes over . Theorem 4.1: Let For the complete joint weight enumerator for where is a preimage of all of whose entries are nonnegand ative integers less than . Let . The sums in (31) are then equal to the expressions given in the equation at the bottom of the page. The conclusion follows from the fact that the number of in which are equal to is exactly . Corollary 4.2: Let and be as in Theorem 4.1. with be a symmetrized joint Let weight enumerator in genus . Then (32) where Proof: Notice that cially, the result. . Espe. From Theorem 4.1 we obtain Now we show that the theta series defined over the lattices induced from codes are Jacobi forms on some subgroup of . For simplicity, we only consider the case when the weight of Jacobi forms (or Siegel modular forms) is integral in this paper. From now on, we assume that the length of each code is even. One can obtain a similar result for the case when the length of the codes is odd. Specifically, Jacobi forms (or Siegel modular forms) of half integral weights are obtained from the complete weight enumerators (or symmetrized weight enumerators) of codes of odd length. be a subgroup of such that Let It is known (see [10]) that three types of matrices: , we have and (29) . is generated by the following CHOIE et al.: COMPLETE JOINT WEIGHT ENUMERATORS AND SELF-DUAL CODES where and and over run over any symmetric integral matrices . Let be a character on defined by 1281 Finally, the condition on a Fourier expansion of follows from the following note (or see [2, proof of Theorem 3.2]): and Theorem 4.3: Let be self-dual codes over of length , and denote the lattice in induced from . Then, the theta series defined as is a Jacobi form in . Moreover, if is a Type II code for all then is in . Proof: To check the first transformation formula of , we need only to check it for three types of and of . First, the following generators functional equation, derived from the Poisson summation formula: (33) , implies that Thus, if each . on is trivial since is Type II, then is in is a Jacobi form of weight and index with respect to on if are self-dual codes over of length . , we obtain that (34) Furthermore, since, for any matrix and In this case, the character . Second, the implies that . we check Second, from the definition of since , Poisson summation formula in (33) with Remark 2: a) Note that the character is also trivial if the length of each code is divisible by . So, if the length of is divisible is a Jacobi form in . by then b) Theorems 4.1 and 4.3, imply that implies that, for any symmetric integral matrix For the unimodularity of So, with the fact that This claims that is in . Next, we recall that if every code is Type II then are [1]. even unimodular for all and the length is generated by , integral symmetric, Again, since , we only need to check the transformation formula and for two types of generators of . First , it follows that Now, the second transformation formula can be checked directly (see also [2, proof of Theorem 3.2] for more detailed information) using and be as in Theorem 4.3. Corollary 4.4: Let is a Siegel modular form of weight Then with respect to on . Moreover, if all code are Type II, then is a Siegel modular form on . is a Similarly, we see that Siegel modular form of weight with respect to on if are self-dual codes over of length . In general, we have the following theorem. be self-dual codes over Theorem 4.5: Let of length and denote the lattice in induced from . Then is a Jacobi form of weight and index with respect to of where and if otherwise. Proof: First, note that for each Here, is given in (11). Then the result follows from Theorems 3.7, 4.1, and 4.3. 1282 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 49, NO. 5, MAY 2003 Corollary 4.6: Let be self-dual codes over of length and denote the lattice in induced from . is a Siegel modular form of weight Then [4] J. H. Conway and N. J. A. Sloane, Sphere Packing, Lattices and Groups, 2nd ed. New York: Springer-Verlag, 1993. [5] S. T. Dougherty, “Shadows of codes,” presented at the Yamagata Conf., Oct. 2000. [6] S. T. Dougherty, T. A. Gulliver, and M. Harada, “Type II self-dual codes over finite rings and even unimodular lattices,” J. Alg. Combin., vol. 9, no. 3, pp. 233–250, 1999, to be published. [7] S. T. Dougherty, M. Harada, and M. Oura, “Note on the biweight enumerators of self-dual codes over ,” paper, submitted for publication. [8] S. T. Dougherty, M. Harada, and P. Solé, “Shadow lattices and shadow codes,” Discr. Math., vol. 219, pp. 49–64, 2000. [9] S. T. Dougherty and P. Solé, “Shadows of codes and lattices,” in Proc. 3rd Asian Mathematics Conf., T. Sunada, P. W. Sy, and Y. Lo, Eds., 2000, pp. 139–152. [10] T. Ibukiyama, “On Jacobi forms and siegel modular forms of half integral weights,” Commun. Math. Univ. Sancti Pauli, vol. 41, no. 2, pp. 109–124, 1992. [11] F. J. MacWilliams, C. L. Mallows, and N. J. A. Sloane, “Generalizations of Gleason’s theorem on weight enumerators of self-dual codes,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 794–804, Nov. 1972. [12] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977. [13] J. Wood, “Duality for modules over finite rings and applications to coding theory,” Amer. J. Math., vol. 121, no. 3, pp. 555–575, 1999. [14] C. Ziegler, “Jacobi forms of higher degree,” Abh. Math. Sem. Univ. Hamburg, vol. 59, pp. 191–224, 1989. with respect to on . We remark that only the invariance properties of the weight enumerators have been used to prove the previous few theorems. One can derive a map from an invariant space under the action (or, ) to a ring of Jacobi forms (or, a of the group ring of Siegel modular forms). REFERENCES [1] E. Bannai, S. T. Dougherty, M. Harada, and M. Oura, “Type II codes, even unimodular lattices, and invariant rings,” IEEE Trans. Inform. Theory , vol. 45, pp. 1194–1205, May 1999. and Jacobi forms of genus n,” [2] Y. Choie and H. Kim, “Codes over J. Combin. Theory, Ser. A, vol. 95, no. 2, pp. 335–348, 2001. [3] Y. Choie and N. Kim, “The complete weight enumerator of Type II code and Jacobi forms,” IEEE Trans. Inform. Theory, vol. 47, pp. over 396–399, Jan. 2001.