Efficient bounded-distance decoding of the hexacode and associated decoders for the Leech lattice and the Golay code

1996 ◽  
Vol 44 (5) ◽  
pp. 534-537 ◽  
Author(s):  
O. Amrani ◽  
Y. Beery
Keyword(s):  
Golay Code ◽  
Leech Lattice ◽  
Bounded Distance

The Leech lattice and the Golay code: bounded-distance decoding and multilevel constructions

1994 ◽  
Vol 40 (4) ◽  
pp. 1030-1043 ◽  
Author(s):  
O. Amrani ◽  
Y. Be'ery ◽  
A. Vardy ◽  
Feng-Wen Sun ◽  
H.C.A. van Tilborg
Keyword(s):  
Golay Code ◽  
Leech Lattice ◽  
Bounded Distance

scholarly journals Some Design Theoretic Results on the Conway Group $\cdot$0

10.37236/290 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ben Fairbairn
Keyword(s):  
Automorphism Group ◽  
Steiner System ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Golay Code ◽  
Full Automorphism Group ◽  
Leech Lattice ◽  
Necessary And Sufficient ◽  
Binary Golay Code

Let $\Omega$ be a set of 24 points with the structure of the (5,8,24) Steiner system, $\cal{S}$, defined on it. The automorphism group of $\cal{S}$ acts on the famous Leech lattice, as does the binary Golay code defined by $\cal{S}$. Let $A,B\subset\Omega$ be subsets of size four ("tetrads"). The structure of $\cal{S}$ forces each tetrad to define a certain partition of $\Omega$ into six tetrads called a sextet. For each tetrad Conway defined a certain automorphism of the Leech lattice that extends the group generated by the above to the full automorphism group of the lattice. For the tetrad $A$ he denoted this automorphism $\zeta_A$. It is well known that for $\zeta_A$ and $\zeta_B$ to commute it is sufficient to have A and B belong to the same sextet. We extend this to a much less obvious necessary and sufficient condition, namely $\zeta_A$ and $\zeta_B$ will commute if and only if $A\cup B$ is contained in a block of $\cal{S}$. We go on to extend this result to similar conditions for other elements of the group and show how neatly these results restrict to certain important subgroups.

Sign in / Sign up

Export Citation Format

Share Document