Ternary Code Construction of Unimodular Lattices and Self-Dual Codes over \Bbb Z 6
Masaaki Harada1
, Masaaki Kitazume2
and Michio Ozeki1
1Department of Mathematical Sciences Yamagata University Yamagata 990-8560 Japan
2Department of Mathematics and Informatics Chiba University Chiba 263-8522 Japan
2Department of Mathematics and Informatics Chiba University Chiba 263-8522 Japan
DOI: 10.1023/A:1021185314365
Abstract
We revisit the construction method of even unimodular lattices using ternary self-dual codes given by the third author (M. Ozeki, in Théorie des nombres, J.-M. De Koninck and C. Levesque (Eds.) (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 772-784), in order to apply the method to odd unimodular lattices and give some extremal (even and odd) unimodular lattices explicitly. In passing we correct an error on the condition for the minimum norm of the lattices of dimension a multiple of 12. As the results of our present research, extremal odd unimodular lattices in dimensions 44, 60 and 68 are constructed for the first time. It is shown that the unimodular lattices obtained by the method can be constructed from some self-dual 
6-codes. Then extremal self-dual 6-codes of lengths 44, 48, 56, 60, 64 and 68 are constructed.
6-codes. Then extremal self-dual 6-codes of lengths 44, 48, 56, 60, 64 and 68 are constructed.Pages: 209–223
Keywords: ternary self-dual code; extremal self-dual $Zopf _{6}$-code; extremal unimodular lattice
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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 45 (1999), 257-269.
2. R. Chapman, private communication, November 4, 2000.
3. J.H. Conway and N.J.A. Sloane, “A note on optimal unimodular lattices,” J. Number Theory 72 (1998), 357-362.
4. J.H. Conway and N.J.A. Sloane, Sphere Packing, Lattices and Groups, 3rd edition, Springer-Verlag, New York, 1999.
5. E. Dawson, “Self-dual ternary codes and Hadamard matrices,” Ars Combin. 19 (1985), 303-308.
6. S.T. Dougherty, M. Harada, and P. Solé, “Self-dual codes over rings and the Chinese remainder theorem,” Hokkaido Math. J. 28 (1999), 253-283.
7. M. Harada, “On the existence of extremal Type II codes over Z6,” Discrete Math. 223 (2000), 373-378.
8. H. Koch, “The 48-dimensional analogues of the Leech lattice,” Proc. Steklov Inst. Math. 208 (1995), 172-178.
9. C.L. Mallows, V. Pless, and N.J.A. Sloane, “Self-dual codes over G F(3),” SIAM. J. Appl. Math. 31 (1976), 649-666.
10. P.S. Montague, “A new construction of lattices from codes over GF(3),” Discrete Math. 135 (1994), 193-223.
11. M. Ozeki, “Ternary code construction of even unimodular lattices,” in Théorie des nombres, J.-M. De Koninck and C. Levesque (Eds.) (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 772-784.
12. E. Rains and N.J.A. Sloane, “The shadow theory of modular and unimodular lattices,” J. Number Theory 73 (1998), 359-389.
13. E. Rains and N.J.A. Sloane, “Self-dual codes,” in Handbook of Coding Theory, V.S. Pless and W.C. Huffman (Eds.), Elsevier, Amsterdam, 1998, pp. 177-294.
14. N.J.A. Sloane and G. Nebe, “Unimodular lattices, together with a table of the best such lattices,” in A Catalogue of Lattices, published electronically at http://www.research.att.com/~njas/lattices/.
2. R. Chapman, private communication, November 4, 2000.
3. J.H. Conway and N.J.A. Sloane, “A note on optimal unimodular lattices,” J. Number Theory 72 (1998), 357-362.
4. J.H. Conway and N.J.A. Sloane, Sphere Packing, Lattices and Groups, 3rd edition, Springer-Verlag, New York, 1999.
5. E. Dawson, “Self-dual ternary codes and Hadamard matrices,” Ars Combin. 19 (1985), 303-308.
6. S.T. Dougherty, M. Harada, and P. Solé, “Self-dual codes over rings and the Chinese remainder theorem,” Hokkaido Math. J. 28 (1999), 253-283.
7. M. Harada, “On the existence of extremal Type II codes over Z6,” Discrete Math. 223 (2000), 373-378.
8. H. Koch, “The 48-dimensional analogues of the Leech lattice,” Proc. Steklov Inst. Math. 208 (1995), 172-178.
9. C.L. Mallows, V. Pless, and N.J.A. Sloane, “Self-dual codes over G F(3),” SIAM. J. Appl. Math. 31 (1976), 649-666.
10. P.S. Montague, “A new construction of lattices from codes over GF(3),” Discrete Math. 135 (1994), 193-223.
11. M. Ozeki, “Ternary code construction of even unimodular lattices,” in Théorie des nombres, J.-M. De Koninck and C. Levesque (Eds.) (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 772-784.
12. E. Rains and N.J.A. Sloane, “The shadow theory of modular and unimodular lattices,” J. Number Theory 73 (1998), 359-389.
13. E. Rains and N.J.A. Sloane, “Self-dual codes,” in Handbook of Coding Theory, V.S. Pless and W.C. Huffman (Eds.), Elsevier, Amsterdam, 1998, pp. 177-294.
14. N.J.A. Sloane and G. Nebe, “Unimodular lattices, together with a table of the best such lattices,” in A Catalogue of Lattices, published electronically at http://www.research.att.com/~njas/lattices/.