Type II Self-Dual Codes over Finite Rings and Even Unimodular Lattices
Steven T. Dougherty1
, T.Aaron Gulliver2
and Masaaki Harada3
1University of Scranton Department of Mathematics Scranton PA 18510
2University of Canterbury Department of Electrical and Electronic Engineering Private Bag 4800 Christchurch New Zealand
3Yamagata University Department of Mathematical Sciences Yamagata 990 Japan
2University of Canterbury Department of Electrical and Electronic Engineering Private Bag 4800 Christchurch New Zealand
3Yamagata University Department of Mathematical Sciences Yamagata 990 Japan
DOI: 10.1023/A:1018696102510
Abstract
In this paper, we investigate self-dual codes over finite rings, specifically the ring \mathbb Z 2 m \mathbb{Z}_{2^m } of integers modulo 2m. Type II codes over \mathbb Z 2 m \mathbb{Z}_{2^m } are introduced as self-dual codes with Euclidean weights which are a multiple of 2m +1. We describe a relationship between Type II codes and even unimodular lattices. This relationship provides much information on Type II codes. Double circulant Type II codes over \mathbb Z 2 m \mathbb{Z}_{2^m } are also studied.
Pages: 233–250
Keywords: self-dual code over finite ring; type II code; double circulant code; even unimodular lattice
Full Text: PDF
References
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3. A.R. Calderbank and N.J.A. Sloane, “Modular and p-adic cyclic codes,” Designs, Codes and Cryptogr. 6 (1995), 21-35.
4. A.R. Calderbank and N.J.A. Sloane, “Double circulant codes over Z4 and even unimodular lattices,” J. Alg. Combin. 6 (1997), 119-131.
5. J.H. Conway, V. Pless, and N.J.A. Sloane, “The binary self-dual codes of length up to 32: A revised enumeration,” J. Combin. Theory Ser. A 60 (1992), 183-195.
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7. J.H. Conway and N.J.A. Sloane, Sphere Packing, Lattices and Groups, 2nd edition, Springer-Verlag, New York, 1993.
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10. M. Klemm, “Selbstduale codes \ddot uber dem Ring der ganzen Zahlen modulo 4,” Arch. Math. 53 (1989), 201-207.
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14. V. Pless and Z. Qian, “Cyclic codes and quadratic residue codes over Z4,” IEEE Trans. Inform. Theory 42 (1996), 1594-1600.
15. V. Pless, P. Solé, and Z. Qian, “Cyclic self-dual Z4-codes,” Finite Fields and Their Appl. 3 (1997), 48-69.
16. M. Ventou and C. Rigoni, “Self-dual doubly circulant codes,” Discrete Math. 56 (1985), 291-298.
17. J.A. Wood, “Duality for modules over finite rings and applications to coding theory,” submitted.