Certain self-dual codes over ℤ4 and the odd Leech lattice
Recently, alternative constructions of the Leech lattice and the shorter Leech lattice have been discovered using self-dual codes over ℤ4. In this paper, we provide a classification of length 24 double circulant Type I codes over ℤ4 with minimum Euclidean weight 12. These codes determine (via Construction A4) the odd Leech lattice, which is a unique 24-dimensional odd unimodular lattice with minimum norm 3.
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- 1.E. Bannai, S.T. Dougherty, M. Harada and M. Oura, Type II codes, even unimodular lattices and invariant rings, (in preparation).Google Scholar
- 2.A. Bonnecaze, P. Solé and A.R. Calderbank, Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory Vol. 41 (1995) pp. 366–377.Google Scholar
- 3.A. Bonnecaze, P. Solé, C. Bachoc and B. Mourrain, Type II codes over ℤ4, IEEE Trans. Inform. Theory, (to appear).Google Scholar
- 4.A.R. Calderbank and N.J.A. Sloane, Double circulant codes over ℤ4 and even unimodular lattices, J. Alg. Combin., (to appear).Google Scholar
- 5.J.H. Conway and N.J.A. Sloane, Self-dual codes over the integers modulo 4, J. Combin. Theory Ser. A Vol. 62 (1993) pp. 30–45.Google Scholar
- 6.J.H. Conway and N.J.A. Sloane, Sphere Packing, Lattices and Groups (2nd ed.), Springer-Verlag, New York, 1993.Google Scholar
- 7.S.T. Dougherty, T.A. Gulliver and M. Harada, Type II self-dual codes over finite rings and even unimodular lattices, (submitted to J. Alg. Combin.).Google Scholar
- 8.T.A. Gulliver and M. Harada, Extremal double circulant Type II codes over ℤ4 and construction of 5-(24,10,36) designs, (submitted to Discrete Math.).Google Scholar
- 9.M. Harada, New extremal Type II codes over ℤ4, (submitted to Des. Codes and Cryptogr.).Google Scholar
- 10.V. Pless, P. Solé and Z. Qian, Cyclic self-dual ℤ4-codes, Finite Fields and Their Appl. Vol. 3 (1997) pp. 48–69.Google Scholar