Duality for modules over finite rings and applications to coding theory

over nite rings. The nite Frobenius rings are singled out as the most appropriate for coding theoretic purposes because two classical theorems of MacWilliams, the extension theorem and the MacWilliams identities, generalize from nite elds to nite Frobenius rings. It is over Frobenius rings that certain key identi cations can be made between the ring and its complex characters.
1991 Mathematics Subject Classi cation. Primary: 94B05, 16L60, 16P10. Key words and phrases. Characters, Frobenius rings, Morita duality, equivalence, MacWilliams identities. Partially supported by NSA grants MDA904-91-H-0029, MDA904-94-H-2025, and MDA904-96-1-0067, and by Purdue University Calumet Scholarly Research Awards. Copyright c The Johns Hopkins University Press. The article rst appeared in the American Journal of Mathematics, 121 1999, 555 575. 1
DUALITY FOR MODULES
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Acknowledgments. I thank: Dave Benson for suggesting the form of Theorem 3.10 and for numerous examples, Vic Camillo for introducing me to the ideas behind Proposition 5.1, Ed Assmus for extensive discussions and guidance, and the referee for placing this work in the context of Morita duality and other helpful suggestions. In addition, I am grateful for advice, suggestions, and references from D. D. Anderson, R. R. Bruner, G. D. Forney, I. Herzog, Th. Honold, W. C. Hu man, R. G. Larson, J. L. Massey, H. F. Mattson, M. May sj, A. A. Nechaev, N. J. A. Sloane, S. Valdes-Leon, and H. N. Ward. Finally, I thank Vera Pless for originally suggesting a re-examination of MacWilliams' work on the extension problem. Conventions. All rings are assumed to be associative with 1 6= 0. All units are assumed to be two-sided. In nite rings, one-sided units are necessarily two-sided. In any module, the unit element of the base ring is assumed to act as the identity. We denote the ring of integers modulo m by Z=m and the number of elements in a nite set S by jS j.
Introduction
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JAY A. WOOD
is the extension theorem of MacWilliams. It is the analogue of the theorems of Witt 40 and Arf 3 on the extension of isometries for non-degenerate bilinear and quadratic forms. There are other proofs of the extension theorem for Hamming weight over nite elds: 7 , 39 . In the latter, Ward and the author use an argument based on the linear independence of complex characters. It is this character theoretic argument which generalizes to Frobenius rings, because Frobenius rings allow for certain key identi cations between a ring and its character module. Klemm 22 and Claasen and Goldbach 9 provided initial ideas in this direction. Hirano 20 and Xue 44 have independently arrived at results similar to those given here. For other types of weight functions, extension theorems still hold. For the case of symmetrized weight compositions over nite elds, see 17 ; over nite Frobenius rings, see 41 . For homogeneous weight functions over Z=mZ, see 11 , and for general weight functions over nite commutative chain rings, see 42 and 43 . The MacWilliams identities relate the weight enumerator of a code to that of its dual code. The most common proof of the MacWilliams identities is Gleason's proof using the Poisson summation formula for Fourier transforms 4, x1.12 , 25, Chapter 5 . Klemm 23 proved the identities for nite commutative Frobenius rings, and Delsarte 13 proved them for additive codes. The results here encompass those of Klemm and Delsarte. Nechaev 33 , 34 has results similar to ours. Gleason's argument extends to nite Frobenius rings because, once again, a key identi cation can be made: this time, between the dual code and the character theoretic annihilator. Because of the importance of characters in our proofs, we treat character theory as a duality functor and place it in the context of Morita duality. We thank the referee for this idea. Here is a brief outline of the paper. Sections 1 3 discuss quasiFrobenius and Frobenius rings, duality functors, and Morita duality. Section 4 summarizes some key identi cations for Frobenius rings that are needed for coding theory. The reader interested primarily in the coding theoretic results may wish to begin with Section 4. Section 5 contains a technical result on partial orderings. The coding theory begins in earnest in Section 6 with a review of essential de nitions and a proof of the extension theorem. In Section 7 the orthogonals associated to a duality functor are discussed and another key identi cation for Frobenius rings is established. The MacWilliams identities are proved in Section 8, and essential results about characters are summarized in Appendix A.
DUALITY FOR MODULES OVER FINITE RINGS AND APPLICATIONS TO CODING THEORY
JAY A. WOOD
In memory of Edward F. Assmus, Jr.
Abstract. This paper sets a foundation for the study of linear
Since the appearance of 8 and 19 , using linear codes over Z=4Z to explain the duality between the non-linear binary Kerdock and Preparata codes, there has been a revival of interest in codes de ned over nite rings. This paper examines the foundations of algebraic coding theory over nite rings and singles out the nite Frobenius rings as the most appropriate rings for coding theory. Why are Frobenius rings appropriate for coding theory? Because two classical theorems of MacWilliams|the extension theorem and the MacWilliams identities|generalize to the case of nite Frobenius rings. The extension theorem of MacWilliams 26 , 27 deals with the notion of equivalence of codes. Two codes are equivalent if there is a monomial transformation taking one to the other. This extrinsic description has an intrinsic formulation: if two linear codes are isomorphic as abstract vector spaces via an isomorphism which preserves Hamming weight, then this isomorphism extends to a monomial transformation. This