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 $\mathbf Z_4$ code of length $8$ and minimum Lee distance $6$. Seeing the parity-check matrix of that code,

$$\begin{pmatrix}3&3&2&3&1&0&0&0\\3&0&3&2&3&1&0&0\\3&0&0&3&2&3&1&0\\3&0&0&0&3&2&3&1\end{pmatrix},$$

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 $A_3^8$. The Leech lattice, the conjecturally densest sphere packing in $24$ dimensions, can be built up from a product of eight copies of the face-centred cubic lattice $A_3$, the conjecturally densest sphere packing in three dimensions. The quotient of $A_3$ in its dual lattice $A_3^*$ is a cyclic group of order $4$, and so to get the Leech lattice from $A_3$ one needs a code of length $8$ over $\mathbf Z_4$.

The preceding matrix shows that the octacode is an extended cyclic code with parity-check polynomial $\widetilde M(x)=x^3+2x^2+x-1$, which reduced modulo $2$ yields $M(x)=x^3+x+1$, which is the generator matrix of the $[7,4,3]$ 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 $8$; the attached lattice is $E_8$, the unique even unimodular lattice in dimension $8$ [a1]. Its residue code modulo $2$ is the doubly even binary self-dual code $[8,4,4]$.

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 $4$" 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 $\mathbf Z_4$-linearity of Kerdock, Preparata, Goethals, and related codes" IEEE Trans. Information Th. , 40 (1994) pp. 301–319