Abstract
In this paper, we investigate self-dual codes over finite rings, specifically the ring \(\mathbb{Z}_{2^m }\) of integers modulo 2m. Type II codes over \(\mathbb{Z}_{2^m }\) 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 \(\mathbb{Z}_{2^m }\) are also studied.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Bonnecaze, P. Solé, C. Bachoc, and B. Mourrain, “Type II codes over Z4,” IEEE Trans.Inform.Theory 43 (1997), 969–976.
A. Bonnecaze, P. Solé, and A.R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices,” IEEE Trans.Inform.Theory 41 (1995), 366–377.
A.R. Calderbank and N.J.A. Sloane, “Modular and p-adic cyclic codes,” Designs, Codes and Cryptogr. 6 (1995), 21–35.
A.R. Calderbank and N.J.A. Sloane, “Double circulant codes over Z4 and even unimodular lattices,” J.Alg.Combin. 6 (1997), 119–131.
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.
J.H. Conway and N.J.A. Sloane, “Self-dual codes over the integers modulo 4,” J.Combin.Theory Ser.A 62 (1993), 30–45.
J.H. Conway and N.J.A. Sloane, Sphere Packing, Lattices and Groups, 2nd edition, Springer-Verlag, New York, 1993.
A.R. Hammons, Jr., P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, and P. Solé, “The Z4-linearity of Kerdock, Preparata, Goethals and related codes,” IEEE Trans.Inform.Theory 40 (1994), 301–319.
M. Harada, “New extremal Type II codes over Z4,” Designs, Codes and Cryptogr. 13 (1998), 271–284.
M. Klemm, “Selbstduale codes über dem Ring der ganzen Zahlen modulo 4,” Arch.Math. 53 (1989), 201–207.
F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam 1977.
C.L. Mallows and N.J.A. Sloane, “An upper bound for self-dual codes,” Inform.Control 22 (1973), 188–200.
O.T.O. O'Meara, Introduction to Quadratic Forms, Springer-Verlag, New York, 1971.
V. Pless and Z. Qian, “Cyclic codes and quadratic residue codes over Z4,” IEEE Trans.Inform.Theory 42 (1996), 1594–1600.
V. Pless, P. Solé, and Z. Qian, “Cyclic self-dual Z4-codes,” Finite Fields and Their Appl. 3 (1997), 48–69.
M. Ventou and C. Rigoni, “Self-dual doubly circulant codes,” Discrete Math. 56 (1985), 291–298.
J.A.Wood, “Duality for modules over finite rings and applications to coding theory,” submitted.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dougherty, S.T., Gulliver, T.A. & Harada, M. Type II Self-Dual Codes over Finite Rings and Even Unimodular Lattices. Journal of Algebraic Combinatorics 9, 233–250 (1999). https://doi.org/10.1023/A:1018696102510
Issue Date:
DOI: https://doi.org/10.1023/A:1018696102510