TY - JOUR
T1 - Type II codes, even unimodular lattices, and invariant rings
AU - Bannai, Eiichi
AU - Dougherty, Steven T.
AU - Harada, Masaaki
AU - Oura, Manabu
N1 - Funding Information:
Manuscript received August 25, 1998; revised December 7, 1998. This work was supported in part by a grant from the Japan Society for the Promotion of Science. E. Bannai and M. Oura are with the Graduate School of Mathematics, Kyushu University, Fukuoka 812–8581, Japan. S. T. Dougherty is with the Department of Mathematics, University of Scranton, Scranton, PA 18510 USA. M. Harada is with the Department of Mathematical Sciences, Yamagata University, Yamagata 990–8560, Japan. Communicated by I. Blake, Associate Editor for Coding Theory. Publisher Item Identifier S 0018-9448(99)03174-0.
PY - 1999
Y1 - 1999
N2 - In this paper, we study self-dual codes over the ring Z2k of the integers modulo 2k with relationships to even unimodular lattices, modular forms, and invariant rings of finite groups. We introduce Type II codes over Z2k which are closely related to even unimodular lattices, as a remarkable class of self-dual codes and a generalization of binary Type II codes. A construction of even unimodular lattices is given using Type II codes. Several examples of Type II codes are given, in particular the first extermal Type II code over Z6 of length 24 is constructed, which gives a new construction of the Leech lattice. The complete and symmetrized weight enumerators in genus g of codes over Z2k are introduced, and the MacWilliams identities for these weight enumerators are given. We investigate the groups which fix these weight enumerators of Type II codes over Z2k and we give the Molien series of the invariant rings of the groups for small cases. We show that modular forms are constructed from complete and symmetrized weight enumerators of Type II codes. Shadow codes over Z2k are also introduced.
AB - In this paper, we study self-dual codes over the ring Z2k of the integers modulo 2k with relationships to even unimodular lattices, modular forms, and invariant rings of finite groups. We introduce Type II codes over Z2k which are closely related to even unimodular lattices, as a remarkable class of self-dual codes and a generalization of binary Type II codes. A construction of even unimodular lattices is given using Type II codes. Several examples of Type II codes are given, in particular the first extermal Type II code over Z6 of length 24 is constructed, which gives a new construction of the Leech lattice. The complete and symmetrized weight enumerators in genus g of codes over Z2k are introduced, and the MacWilliams identities for these weight enumerators are given. We investigate the groups which fix these weight enumerators of Type II codes over Z2k and we give the Molien series of the invariant rings of the groups for small cases. We show that modular forms are constructed from complete and symmetrized weight enumerators of Type II codes. Shadow codes over Z2k are also introduced.
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U2 - 10.1109/18.761269
DO - 10.1109/18.761269
M3 - Article
AN - SCOPUS:0032690017
SN - 0018-9448
VL - 45
SP - 1194
EP - 1205
JO - IRE Professional Group on Information Theory
JF - IRE Professional Group on Information Theory
IS - 4
ER -