View source for Octacode

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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|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 [[#References|[a4]]], which had turned up already in one of the  "holy constructions"  of the [[Leech lattice|Leech lattice]] [[#References|[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|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 [[#References|[a5]]] (cf. also [[Kerdock and Preparata codes|Kerdock and Preparata codes]]), and as such it is self-dual [[#References|[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 $8$ [[#References|[a1]]]. Its residue code modulo $2$ is the doubly even binary self-dual code $[8,4,4]$.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Bonnecaze,   P. Solé,   A.R. Calderbank,   "Quaternary construction of unimodular lattices"  ''IEEE Inform. Th.'' , '''41'''  (1995)  pp. 366–376</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.H. Conway,   N.J.A. Sloane,   "Sphere packings, lattices and groups" , Springer  (1992)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.H. Conway,   N.J.A. Sloane,   "Self-dual codes over the integers modulo $4$"  ''J. Combin. Th. A'' , '''62'''  (1993)  pp. 31–45</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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</TD></TR></table>