Namespaces
Variants
Actions

Difference between revisions of "Octacode"

From Encyclopedia of Mathematics
Jump to: navigation, search
(TeX)
m (link)
 
Line 1: Line 1:
 
{{TEX|done}}
 
{{TEX|done}}
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,
+
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,
  
 
$$\begin{pmatrix}3&3&2&3&1&0&0&0\\3&0&3&2&3&1&0&0\\3&0&0&3&2&3&1&0\\3&0&0&0&3&2&3&1\end{pmatrix},$$
 
$$\begin{pmatrix}3&3&2&3&1&0&0&0\\3&0&3&2&3&1&0&0\\3&0&0&3&2&3&1&0\\3&0&0&0&3&2&3&1\end{pmatrix},$$
Line 9: Line 9:
  
 
====References====
 
====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>
+
<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>

Latest revision as of 17:32, 17 April 2016

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
How to Cite This Entry:
Octacode. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Octacode&oldid=38582
This article was adapted from an original article by P. Solé (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article