Octacode

Led by modulation considerations, G.D. Forney and M.D. Trott discovered in October 1992 that the Nordstrom–Robinson code was obtained by Gray mapping (cf. also Gray code) a certain code of length and minimum Lee distance Seeing the parity-check matrix of that code,

NJ.A. Sloane identified this code with the octacode [a4], which had turned up already in one of the "holy constructions" of the Leech lattice [a2], Chapt. 24, in particular in the construction based on . The Leech lattice, the conjecturally densest sphere packing in dimensions, can be built up from a product of eight copies of the face-centred cubic lattice , the conjecturally densest sphere packing in three dimensions. The quotient of in its dual lattice is a cyclic group of order , and so to get the Leech lattice from one needs a code of length over .

The preceding matrix shows that the octacode is an extended cyclic code with parity-check polynomial , which reduced modulo yields , which is the generator matrix of the binary Hamming code. It is indeed both the first quaternary Kerdock code and the first quaternary Preparata code [a5] (cf. also Kerdock and Preparata codes), and as such it is self-dual [a3]. It is indeed of type II, i.e. the Euclidean weight of its words is multiple of ; the attached lattice is , the unique even unimodular lattice in dimension [a1]. Its residue code modulo is the doubly even binary self-dual code .

References

[a1]	A. Bonnecaze, P. Solé, A.R. Calderbank, "Quaternary construction of unimodular lattices" IEEE Inform. Th. , 41 (1995) pp. 366–376
[a2]	J.H. Conway, N.J.A. Sloane, "Sphere packings, lattices and groups" , Springer (1992)
[a3]	J.H. Conway, N.J.A. Sloane, "Self-dual codes over the integers modulo " J. Combin. Th. A , 62 (1993) pp. 31–45
[a4]	G.D. Forney, N.J.A. Sloane, M.D. Trott, "The Nordstrom Robinson code is the binary image of the octacode" R. Calderbank (ed.) G.D. Forney Jr. (ed.) N. Moayeri (ed.) , Coding and Quantization , Amer. Math. Soc. (1993) pp. 19–26
[a5]	A.R. Hammons, P.V. Kumar, A.R. Calderbank, N.J.A. Sloane, P. Solé, "The -linearity of Kerdock, Preparata, Goethals, and related codes" IEEE Trans. Information Th. , 40 (1994) pp. 301–319