Information Theory and Coding - Codes: The Guide to Secrecy from Ancient to Modern Times - page 456

G 24 Golaycode to obtain the
G 23 -code bymerelydeleting the last entryof each codeword in

The Perfect Golay Code : One can use the

G 24 . Thus, G 23 is

a linear [23 . 12 . 7]-code, which unlike the G 24 -code is perfect, which follows from

the Hamming bound on page 445.
There is a mechanism to more naturallyintroduce the Golaycodes, but with
more machinery necessary, namely, cyclic codes , which we will introduce shortly.
For now, we conclude with the comments of a more sophisticated nature.

Advanced Facts Concerning Golay and Related Codes : The following
are features of Golaycodes, and others studied thus far, for the reader with some
deeper mathematical background (or the desire to gain it).

1. The automorphism group (see page 600), of G 24 is the Mathieu group M 24 ,

one of the so-called sporadic groups , first discovered byMathieu in the late
nineteenth century. (For general group theory information, see [277].)

2. The G 24 -code can be employed to define the Leech lattice , which is one of the

most eLcient sphere-packing mechanisms known today. It was discovered
in 1964 byJohn Leech (see [143]), as the unique lattice with the properties
that it is unimodular (namely, it can be generated by the columns of a
distinguished 24

24 matrix with determinant 1); the length of anyvector
in the lattice is an even integer; and the shortest length of anyvector in
the lattice is 2, meaning that the unit spheres centered at points in the
lattice do not overlap. (For general lattice theoryinformation, see [114].)

×

3. The words of weight 8 in G 24 form a S (5 , 8 , 24) Steiner system , which is a

24-element set S together with a set, T , of 8-element subsets of S , with

the propertythat each 5-element subset of S is contained in exactlyone

T

of the subsets in . One means of constructing this Steiner system is to

form the matrix whose rows are the 2 12 = 4096 Golay24-bit codewords.
These rows form a group under addition modulo 2. In addition to the row
consisting of 0 , there are 759 rows having weight 8; 2576 having weight
12; 759 having weight 16; and one having weight 24. The 759 elements
of weight 8 form the aforementioned set

T , called octads . (For a studyof

Steiner systems in general, see [59].)

4. There exists the (11 , 6) ternaryGolaycode, which is the onlyknown perfect
nonbinarycode. This (11 , 6)-code over F 3 has minimum distance d =5

and can correct up to two errors. As with the Golay[23 , 12] binarycode,
the (11 , 6) ternaryGolaycode can be extended to the (12 , 6) ternaryGolay
code, and as with the extended

G 24 code, the (12 , 6)-code is also unique.

5. A result proved in the mid-1970s shows that the Hamming and Golay
codes are the only(nontrivial) perfect codes (up to equivalence). (Here
we consider the repetition codes, described on page 454, to be trivial.) In
other words, it is known that the onlynontrivial q -aryperfect codes are
those with parameters given bythe Golayand Hamming codes and so are
equivalent to one of: (1) the q -aryHamming codes (see page 596); (2) the