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Orbit and Coset Analysis of the Golay and Related Codes

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  • The OEIS Foundation Inc.

Abstract and Figures

Let b be a code of length n over a field F , with automorphism group G ; b <sub>w</sub> denotes the subset of codewords of weight w . The goal is to classify the vectors of F <sup>n</sup> into orbits under G and to determine their distances from the various subcodes b <sub>w</sub>. This is done for the first-order Reed-Muller, Nordstrom-Robinson, and Hamming codes of length 16, the Golay and shortened Golay codes of lengths 22, 23, 24, and the ternary Golay code of length 12
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1038
IEEE
TRANSACTIONS
ON
INFORMATION
THEORY,
VOL. 36,
NO.
5,
SEPTEMBER
1990
Orbit and Coset Analysis
of
the
Golay and
Related Codes
Abstract-Let
8
be
a code
of
length
n
over a field
F,
with automor-
phism group
G;
8,
denotes the subset of codewords of weight
w.
Our
goal is
to
classify the vectors of
IF"
into orbits under
G
and
to
determine
their distances from the various subcodes
8,.
We do this for the
first-order Reed-Muller, Nordstrom-Robinson, and Hamming codes
of
length
16,
the Golay and shortened Golay codes
of
lengths
22,
23,
24,
and the ternary Golay code of length
12.
I. INTRODUCTION
ET
t
be one of the following codes: the first-order
L
Reed-Muller, Nordstrom-Robinson, or Hamming
codes
of
length 16, the Golay and shortened Golay codes
of lengths 22, 23, or 24 (all these are binary), or the
ternary Golay codes
of
lengths
11
or 12. The main results
of this paper are the graphs in Figs. 1-5, which classify
the vectors of
5"
(where
n
is the length
of
t
and
[F
=
IF,
or
5,
is the appropriate field) into orbits under the action
of the automorphism group of
B.
The groups considered
are
M,,,
2.M,,,
M,,,
M,,:2,
M,,,
M24
(where
M,,
de-
notes a Mathieu group [41,
[SI),
and the subgroups of MZ4
isomorphic to
Z4:
A,
and Z4:
A,.
Other properties of the
orbits are summarized in Tables
I,
IV, V, VII, VIII,
XI,
XIII,
and Fig. 6.
The circled nodes in the graphs indicate the constant
weight subcodes
8,
of each code. Since distances in
these graphs (measured by number of edges) coincide
with Hamming distances between orbits, these graphs also
classify the vectors
of
5"
according to their distances from
the constant weight subcodes.
Tables
11,
VI,
IX,
X,
XII,
and
XIV
show how the cosets
of
these codes are decomposed into orbits under the
groups. These tables are expanded versions
of
the usual
coset weight distribution tables. The final table, Table
XV,
gives the weight distributions of the cosets of the [ll,
6, 51 perfect ternary Golay code.
Orbits of binary vectors under
M24
(the case when
t
is
the Golay code of length 24) were classified in ([2], [8],
Chap. 10). In the present paper we introduce a new
parameter, the specification number (or spec), to describe
these orbits-see Fig.
1
and Table
I.
This makes it easy to
Manuscript received September
22,
1989; revised March 10, 1990.
J.
H. Conway
is
with the Mathematics Department, Princeton Univer-
N.
J.
A.
Sloane is with AT&T Bell Labs, Room 2C-376, Murray Hill,
IEEE
Log Number 9036392.
sity, Princeton,
NJ
08540.
NJ
07974.
determine the distance of an orbit from the code and to
tell when one orbit is contained in another.
11.
THE
[24, 12,
81
GOLAY CODE
The automorphism group of the [24, 12,
81
Golay code
9
is the Mathieu group
M24
(see [4], [8]).
As
described in
([21,
[8],
Chap. 101, there are 49 orbits of vectors in
5z4
under the action of
M24,
denoted by
SJOr
w
1241,
T,(8
I
w
I
161, U,(6
I
w
I
181,
PI,
and
XI,,
where the
subscript gives the weight
of
the vectors. These orbits are
displayed in Fig.
1
and their properties are summarized in
Table
I.
In Fig.
1
two orbits
A,B
are joined by an edge if a
vector in
B
can be obtained from some vector in
A
by
complementing a single bit. The edge joining
A
and
B
is
labeled near
A
with the number of choices for this bit.
The Golay code
9
itself consists
of
the orbits
9,
=
So
=
{O},
9,
=
S,
(the 759
special octads,
forming the Steiner
system S(5,
8,
2411,
9,,
=
U,,
(the 2576
umbral dodecads),
9,,
=
SI,
(the 759
special I6-sets)
and
~9,~
=
S,,
={I}.
These nodes are circled in Fig. 1. The vectors of
S,
for
w
<
12
contain or are contained in a special octad and are
called
special w-sets;
the vectors
of
U,
for
w
<
12 are
contained in an umbral dodecad and are called
umbral
w-sets;
the vectors of T, are called
transverse w-sets;
while
the vectors of
XI,
(called
SA
in [ll, [21) and
P,,
(called
U;
in [l], [2]) are the
extraspecial
and
penumbral do-
decads,
respectively. (This terminology was introduced in
[2], [13].) The vectors in
S,,
T,,
U,
are the complements
of
the vectors in
S24-,,
T24-w,
U24-,,
respectively, while
the types
PI,
and
XI,
are self-complementary.
Fig.
1
has the convenient property that the minimal
Hamming distance between two orbits is given by the
minimal number of edges joining the corresponding nodes
of
the graph. In other words, distance in the graph is the
same as Hamming distance.
The orbits in Fig.
1
are positioned according to their
weight (increasing downwards) and
specification number
or
spec
(increasing across). For a vector
of
weight
w
I
12
not in
TI,
or
XI,,
the specification number is defined to
be the number
of
points in its support that lie in a nearest
octad, minus the number of points outside that octad,
while for vectors in
TI,
or
XI,
it is 3 and 5, respectively.
The specification number
of
a vector of weight greater
001 8-9448/90/0900- 1038$01
.OO
0
1990 IEEE
CONWAY AND SLOANE:
ORBIT
AND COSET ANALYSIS
OF
GOLAY AND RELATED CODES
0
‘1
Fig.
1.
Orbits
1039
downwards)
1040
IEEE
TRANSACTIONS
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INFORMATION
THEORY, VOL.
36,
NO.
5,
SEPTEMBER
1990
than 12 is defined to be the same as that of its comple-
ment.
The specification number has two useful properties.
a) A vector
of
weight
W
and spec
S
contains a vector
of
weight
w
and spec
s
just
if
W
-
w
2
IS
-
SI.
b) The distance
of
a vector of spec
s
from the Golay
code is at least min{s, 8
-
s),
and is equal to this
except when the parity is wrong; that is to say,
except for the vectors
of
T,,
and
XI2,
which are at
distance 4 (not 3) from the code.
We also record some other properties of Fig.
1.
The
sum of the labels on edges upwards from an orbit of
weight
w
is equal to
w,
while the sum
of
the labels on
downward edges is
n
-
w,
where
n
is the length of the
code. Furthermore if there is an edge from orbit
A
to
orbit
B
labeled
a
(at
A)
and
p
(at
B),
then
alAl=
PIBI.
(1)
Before describing Table
I
we introduce our notation for
Golay codewords. We shall write Golay codewords in the
4x6 MOG (or miracle octad generator) array, as de-
scribed in [3]-[6], [8]-[10]. We follow the version given in
[SI, Chaps. 10,
11,
and first define the
hexucode
to be the
[6, 3, 41 code over
F4
with generator matrix
001111
O1OloG]
[
1001wo
(see [8], pp. 300-301). Then the 124, 12, 81 Golay code
consists of all 4
X
6 binary arrays with the properties that
the weights of the columns and the top row have the same
parity, and the
six
inner products of the columns with the
vector (0,1,
o,
W)
forms a word
of
the hexacode ([8], pp.
303-304). We order the coordinates of the MOG’s by
reading down the columns, from left to right. When the
Golay code defined by MOG coordinates is read in this
way it coincides with the lexicographic version of this
code ([71, [SI, p. 327).
Table
I
begins by giving (in column 2) the number of
vectors in each orbit. These numbers are easily calculated
from Fig. 1, using (11, and an alternative enumeration is
given later in this section. The next column describes the
subgroup of
M24
fixing a vector in the orbit. We use the
ATLAS notation (see [4], [SI) for these groups. In particu-
lar,
A
X
B
indicates a direct product,
A.B
or
AB
is a
group with a normal subgroup isomorphic to
A
for which
the corresponding quotient group is isomorphic to
B,
A:
B
denotes the case of
A.B
which is a split extension
(or semidirect product), and
i(S,
X
S,)
indicates the even
permutations
of
the group
S,
X
S,
acting on
m
+
n
ob-
jects.
The fourth column gives the action
of
this group on the
24 coordinates, with the action on the 1-coordinates and
on the 0-coordinates separated by a vertical bar. Orbits
are separated by commas,
so
for example 6,5, 2 indicates
three orbits of sizes 6,
5,
and 2. A symbol such as
2’
indicates an orbit
of
14 points having an invariant parti-
TABLE
1
ORBITS
UNDER
M24
Orbit Size Stabilizer Action Spec
Error
Pattern
S”
1
M24
0124
0 0,)
SI
24
M23
1123
1
11
s2
276
M2,:2
2122 2 22
s3
2024
M,,
:S3
3121
3
33
S,
42504 24:gS3X
S,)
5116,3 5 3,
S,
21252 z4:S, 6116,2
6
2
0
U,
113344
3S,
s7
6072
Z4:
A,
7116,l 7
1n
U7 340032
’6
6,1115,2 5
31
Sn
759 24:
A,
8116 8
00
Tn
97152
‘47
7,1115,l
6
2,
U,
637560
z4.S4
24142,24 4 42222,
S,
12144
A8
8,1115
7
11
32
T,
728640
L
(7).2
7,2127,1 5
U,
566720 3’:2s4
9134,3
3 3,
S,,
91080 2’:LL,(2).2
8,2127 6 2,
S,
10626 2h:i(S3XSs) 4145 4 44~mm
613‘ 4 41,,11,
Ti,,
1700160
S3xS4
32,414X3,2 4 43311
I1
U,,
170016
S,.2
1016’,2 2 2,
SI,
425040
i(S4XS4).2
42,3143,1 5
33
T,,
2040192
S,
10.116.5.2
3 3,
.
.,,
U;;
30912
Mi, 11112,l
1
1
XI, 35420
2‘.33.S:.2
4’14s 5 4444,
SI,
1275120
2..S4
24,4124,4 4 4,22220
TI,
1020096 (2X A,).2
2612‘ 3 4222222
PI,
370944
L2(11)
11,1111,l 2 21
U12
2576
MI2
12112
0
00
tion (or system of imprimitivity) into seven sets of 2, while
4x3 indicates an orbit of 12 points having invariant
partitions into four sets of 3 and three sets
of
4.
The fifth column gives the specification number (de-
fined earlier).
The last column gives the distance
d
from the code,
with a subscript describing the minimal error pattern(s). If
U
is a vector in the orbit, and
d
is at most 3, there is a
unique closest codeword
c
E
9.
Then
e
=
U
+
c
is the
error pattern and the entry in the last column is
d,,
where
i
=
wt(v
n
e).
On the other hand if
U
is at distance 4 from
the code then there are
six
codewords
c,;
e,
c5
(say) all
at distance 4 form
U,
and
six
equally likely minimal error
patterns,
e,
=
U
+
c,
(0
I
r
I
5).
In this case the entry is
4,”,,
I,,
where
i,
=
wt(u
n
e,).
The
six
vectors
e,,
. . .
,
e5
all have weight 4, with their
1’s in disjoint sets of coordinates, and any sum
e,
+
e,(r
#
s)
is a codeword of weight 8. In this situation the individ-
ual 4-sets are called
tetrads
and the set of
six
tetrads is
called a
sextet
([8], Chap. 10). Any 4-set belongs to exactly
one sextet, and there are
+(
7)
=
1771 distinct sextets.
The
six
columns of the MOG form a sextet, and we shall
usually take this as our typical example. We see that the
S,,
(the “deep holes” in the Golay code) are at distance 4
from the code and reduce modulo the code to any of the
six
tetrads of some sextet.
Table
I
describes only orbits of weight
WI
12. The
entries for
S24-w,
T24-w,
U24-,,,
(w
I
11) are the same as
those for
S,,,,
T,,,,
U,,,,
respectively, except that the “Ac-
tion” column is reversed, and in the final column
d,
vectors in
S4,
U69
U,,
TI,,
E12,
S12, T12,
Ul8,
and
CONWAY AND SLOANE:
ORBIT AND COSET ANALYSIS
OF
GOLAY AND RELATED CODES
1041
TABLE
I1
COSETS
OF
[24, 12,8]
GOLAY
CODE
9
No.01234567 8 9 10
11
12
11
159 2576
Sn
Sn
U12
24
1
253 506 1288
SI
s7
SY
U,
I
276
1
77 352 330
+
616 1344
2024
1771 6 64 360 960 20
+
720
+
576
s2
S6
Tn
Sin
U
TIO PI2
1
21 168 360
+
280 210+ 1008
s3
s5
U7
TY
U
U9 SI,
U
TI,
TABLE
111-A
How
MANY
SPECIAL
OCTADS?
759
506 253
330 176 77
210 120 56 21
130 80
40 16 5
78 52 28 12 41
46 32 20
8
4 01
30 16
16
4
4 001
30
0
16
0
4
0
001
TABLE 111-B
How
MANY
UMBRAL
DODECADS?
2576
1288 1288
616 672 616
280 336 336 280
120 160 176 160 120
48 72 88 88 72 48
16 32
40
48 40 32 16
0
16 16 24 24 16 16
0
00
16
0
24
0
16
00
becomes
ddPa
and
4,,).
15
becomes
4,,
.
,”
where
j,
=
4
-
i,.
For example, for
TI,
and
TI,
the actions are
4
x
3, 2 132, 4
and
2’, 117, 2,
respectively, and the minimal error pat-
terns are described by
4333311
and
3,,
respectively.
From Fig.
1
and Table I we may obtain a complete
analysis of the cosets of the Golay code, as displayed in
Table 11. This is an expanded version of the usual coset
weight distribution table (as found for example on p.
69
of
[ll]),
and is more-or-less obtained by folding Fig.
1
about
a vertical line through its center (and transposing).
We next show how to construct and enumerate the
vectors in each orbit. For orbits at distance
I
3
from the
code (belonging to the first four rows of Table 111, there is
It then follows that the numbers in the ith row of Table
I1 for
i
I
3
are found by multiplying the ith row of each
Leech triangle by the ith row of Pascal’s triangle! For
example the numbers
77 352 330
+
616 1344 616
in row
3
of Table I1 are obtained from row
3
of Tables
111-A, 111-B:
77x1 176x2 330x1
+616x1 672x2 616x1.
a unique description that can be read off Fig.
1.
For
example, any vector of type
T9
is obtained by adding
two
Similarly the fourth row
points to a special octad and deleting one point from that
21 168 360 210
+280 1008 1008 280
octad. To count such vectors we make use of the familiar
“Leech triangles” of numbers shown in Tables 111-A,
111-B (cf.
[8],
p.
278,
1111,
p.
68).
If
{a,,a2;
.
e,
a8)
is the (support of) a special octad,
then the number of special octads intersecting
{a,;
*,a,)
21x1 56x3 120x3 210x1
in exactly
(a,,.
.
-,a,}
is the
(j
+
1)th
entry in the
(i
+
11th
+280x1 336x3 336x3 280x1.
row of Table 111-A. Similarly Table 111-B gives the num-
ber of umbral dodecads meeting
(a,;
.
.,a,)
in exactly
{a,,.
.
*,a,).
follows from
The vectors in the final row of Table 11, the deep holes
in
9,
may also be enumerated in this way, but (because
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1990
TABLE
IV
DEEP
HOLES
IN
THE
[24, 12,8]
GOLAY
CODE
Name Error pattern Example Number+ 1771
s4
4,,, Column
of
MOG
U6
4111111 H(word)
VU
4,,,,, H(weight 4 word)
+top row
Tin
4331
11
I
H(word)
+
2 columns
XI,
4,,,, 3 columns
of
MOG
SI,
4,,,,,, H(weight 4 word)
+top row
+column
+top row
TI
2
4,,,,,, H(weight 6 word)
1.6=6
64.1=64
45.2,
=
360
64.(6)=960 2
146)= 3 20
45.(2,.2)= 720
18.2,
=
576
the representatives modulo
9
are no longer unique), it is
simpler to enumerate them from their error patterns
(given in the last column of Table I). The results are
shown in Table IV.
Consider for example a vector of type
S,,
which, since
its error pattern is described by 4400000, consists of one
tetrad from a sextet. Since there are 1771 sextets, each
containing six tetrads, the number of
S,
vectors is 1771
x
6=10626. As an example we may take any of the
six
columns of the MOG.
Vectors of type
U,
have error pattern 4111111, and
typical examples consist of 4
X
6 MOG arrays with a single
1
in each column, chosen
so
that the positions of the 1’s
(when the rows of the array are labeled 0,1,
o,
0)
form a
word
w
in the hexacode. We call this vector
H(w).
The
number of such vectors is 1771 (for the choice of sextet)
times 64 (for the choice of a hexacodeword). In the
column headed “Number” in Table IV, the first factor is
the appropriate number of hexacodewords, and the sec-
ond factor gives the number of other choices that must be
made.
We omit details of the remaining entries in Table IV.
(Readers familiar with Chap.
11
of [81 will have no diffi-
culty in verifying these enumerations, and the numbers
are in any case available in Table I.)
Finally, Fig.
1
makes it easy to find the vectors at a
specified distance from the code. For example, in con-
structing constant weight codes in [ll it was necessary to
determine the vectors of length 24, weight 12 and having
distance 6 from the 2576 words of
gI2
=
U12.
From Fig.
1
and Table I we see that there are exactly 35420 such
vectors, those of the orbit
XI,.
111.
THE
[23, 12, 71 GOLAY CODE
The [23, 12, 71 perfect Golay code
9’
is obtained by
deleting one fixed coordinate (which we label
03)
from
every word
of
9,
and Aut
(9’)
is the Mathieu group
M,,.
Of
course the dual code to
9‘,
the [23, 11,
81
even weight
subcode of
9‘,
has the same group.
Let
U
be a vector of length 23 and weight
w,
and let
x
and
y
be the vectors of length 24 obtained from
U
by
adjoining a
0
or
1
respectively in the coordinate. If
x
belongs to the orbit
A,
of Fig. 1, and
y
to the orbit
B,+I,
then
U
corresponds to the
edge
in Fig.
1
from
A,
to
Bw+I.
We describe
U
by saying it is of type
AwB.
Its
complement
b
is
of
type
BwfA,
where
w’
=
23
-
w.
It is not difficult to verify (we omit the details) that
M,,
is transitive on vectors of each type. We conclude that
orbits of vectors in
[F;,
under
M,,
are in one-to-one
correspondence with the edges of Fig. 1. There are there-
fore 72 orbits.
These orbits are shown in Fig. 2, which uses the same
conventions-except for specification number-as Fig.
1.
The edge labels and the sizes of the orbits (given in Table
V) can be determined from the information in Fig.
1
and
Table 1, as we now demonstrate.
TABLE
V
SIZES
OF
ORBITS
UNDER
M2,
Sns
1
U,,
28336
SIOS
53130
SI,
23
U,,
212520
Tlos
141680
s2,
253
s,,
506
TI,,
850080
S,,
1771
T,,
4048
U,,,
85008
S,,
8855
T,,
60720
U,,,
14168
S,,
5313
U,T
212520
S,,,
17710
S,,
1771
Sy,
7590
TI,,
425040
U6”
85008
Ty,
425040
TI,,
170016
S7,
253
U,,
283360
U,,,
15456
S,,
28336
U,,
212520
Slls
212520
S,,
14168
Ty,
30360
7‘11,
510048
S7,
4048
Uy,
70840
U,,,
1288
Consider for example the edges in Fig.
1
at the node
T,.
There is an edge from
T,
to
TI,
(labeled 14 at
T,),
and an edge from T9 to
SI,
(labeled 1). Since there are
728640 vectors of type T, (from Table
I),
there are
14
-
X
728640
=
425040
24
vectors of type
T,,,
and
1
24
-
X
728640
=
30360
vectors of type
T9s.
The calculation of the edge labels in Fig. 2 is only
slightly more complicated. Consider for example a vector
L’
E
ffi3
of type
T,,,
so
that
x
(U
with a
0
adjoined) is of
type
T,
and
y
(U
with a
1
adjoined) is
of
type
Tlo.
From
the edge labels in Fig.
1
we see that complementing a
0
in
x
leads in one way to a vector
of
SI,
and in 14 ways to a
vector of
TI,
(one
of
which is
y).
In Fig. 2, therefore,
there is one edge from
T,,
to a node of type
SI,,
and
13
edges to nodes of type
TI,.
(where the stars indicate
unknown letters). On the other hand, complementing a
1
in
y
leads in
two
ways to a vector of
SI,
and in 12 ways to
a vector of
Til.
This tells us that in Fig. 2 there are
two
edges to nodes of type and 12 edges to nodes
of
type
The possible nodes that T,, can be joined to are
therefore
SI,,, SI,,,
TI,,
and
TI,,.
However, from Fig.
1
we see that
SI,,
is not joined to
TI1,
so
a node of type
SI,,
*
10,‘
1044
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SEPTEMBER
1990
TABLE
VI
COSETS
OF
[23, 12,7]
GOLAY
CODE
9’
No.
0
12 3 4
5
6 7
8
9
10
11
11
253 506 1288
SOS
SI,
Sns
U1
IU
23
1
77 176 176 330 616 672
SI,
’6,
slT
TKS
SYS
uinu
UIIP
253
1
21 56 112 240 120+280 210+336 672
S2S
s5S
s6U
ulT
T8T TYS
sll)S
uulllT
s3S
s4S
s5U
u8TuuXU TYTUUYT T~0.5‘uTIOT
1771
1
5
16 48 120 120+120 240+160 80+480
**
**:
10+120+240+288 corresponding to
Sll,USl,suTl,sUT,l~.
is impossible. We conclude that a vector of type
T9,
transforms in one way to type
SI,,,
in
12
ways to type Tlo,
and in one way to type TI,,. The labels at the bottom
ends of these edges are then found from (1) and Table
V.
From Fig.
2
and Table
V
we obtain a complete analysis
of the cosets of
9’,
as shown in Table
VI.
IV.
By shortening
9
to length
22
we obtain
[22,
10, 81,
E22,
11, 71, and
[22,
12,
61 codes. The automorphism group of
the first and third of these is
M,,
:
2,
while the automor-
phism group of the
[22,
11, 71 code (obtained from the
words of
J
that begin
00
or 01)
is
M2*.
Without giving any details we mention that the orbits of
M,,
are in one-to-one correspondence with the edges of
Fig.
2.
There are therefore 130 orbits, which can be
named in the following way.
An
edge in Fig.
2
directed
from
A,,
to
Cw+l,D
indicates that there is a vector
uEIFi2
of weight
w
such that uOOEA,,,, ~01
E
B,+,,
u10
E
Cw+,,
ull
E
D,+,.
The appropriate name for the
orbit of
U
under
M,,
is then
A,BCD.
Under the action of
M,,
:
2,
however, the orbits
A,,,BCD
and
AWcBD
fuse, and the composite orbit should be
named For example the
M,,
orbits
U,,,,
and
U,,,,
fuse under
M,,
:
2
to give the orbit
U,(,,),.
There
are
105
distinct orbits under
M,,.2.
THE SHORTENED
GOLAY
CODES
OF
LENGTH
22
V.
THE FIRST-ORDER REED-MULLER
AND
HAMMING CODES
OF
LENGTH 16
The [16,
5,
81 first-order Reed-Muller code
9
and the
[16, 11,
41
Hamming code
&?
are duals and both have
automorphism group
GI
24:
A,,
where
A,
is the alter-
nating group of order 8 ([81, p.
277).
To define these
codes and the Nordstrom-Robinson code of Section
VI
we divide the coordinates of the MOG into three “bricks”
of eight coordinates each, and label the left-hand brick as
follows:
CO
0
3
2
5
1
6
4
(cf. [81, p. 316).
Then
9
consists of the codewords of the
[24, 12,
81
Golay code
9
that vanish on the left-hand brick (with
this brick deleted), while
2Y
is the projection
of
9
onto
the last two bricks.
To study how vectors
L;
E
F:6
of weight
w
I
8 fall into
orbits under
G
we shall adjoin the left-hand brick (a
special octad) to
6,
obtaining a vector
U
of weight 8+
w,
belonging to one of the orbits of Fig. 1. Conversely, each
orbit in Fig. 1 that contains a special octad arises in this
way. To classify vectors of
F:6
under
G
we must therefore
take the orbits in Fig.
1
that contain a special octad and
study them acc2rding to the special octads they contain.
We denote by
X,,,
the type of vector formed by removing
a special octad from a vector
of
type
X,,,.
It turns out (as
usual
we
omit the details) that
G
ispansitive on vectors
of each of Jhese types, except for
U,,,
which splits into
two
orbits
U;
and
U;,.
So
there are
32
orbits under
G,
as
displayed in Fig.
3,
whose properties are summarized in
Tables
VI1
and
VIII.
A
S8
15
Fig. 3. Orbits of vectors
of
length 16 under action of automorphism
group
~2~:
A,)
of
Reed-Muller code
9
and Hamming code
2.
Words in
9
have
two
circles, words in
2
have one
or
two circles.
Weight
is
8
less than subscript. Omitted lower half
of
graph can be
obtained by taking mirror image of top half.
CONWAY AND SLOANE: ORBIT AND COSET ANALYSIS
OF
GOLAY AND RELATED CODES
1045
TABLE
VI1
PROPERTIES
OF
ORBITS
UNDER
AUT(.&)
=
AUT(X')
Weight Name Size
nx
n,,
nlh
Orbits
0
1
2
3
4
4
5
5
6
6
6
7
7
7
8
8
8
8
$8
11
00
8
16 1
00
8
$0
120 1
00
8
S) 2
1680
3
00
8
TI2
140 1
00
8
TI3
2688
1
00
8
:I
4
840 7
00
8
VI
4
448
2 1
0
2+6
SI,
240
15
0
0
8
TI,
6720
7
00
8
tls
4480
6 1
0
2+6
s,l
h
30
30
0
1
8
TI,
1920 15
0 0
8
q5
840
1+12 2
0
8
U;,
10080
12+1 2
0
4+4
:y
:I
I
560
1
00
8
s,I
3
1680
3
00
8
TI4
6720
3
00
8
Note that now the weight of any type of vector is
8
less
than the subscript
on
its symbol. The vectors of
9
are
marked with double circles, the remaining vectors of
2
with single circles. The omitted lower half of the graph in
Fig. 3 can be obtaine; by, taki?g th,e mirror image of the
top half. The types
SI,,
T,,,
U:,
U,',
of weight
8
vectors
are self-complementary.
In
Table
VII,
the columns headed
n8,
nI2,
and
nI6
give
the numbers of special octads, umbral dodecads and
special 16-ads contained in
U,
while the last column shows
how the stabilizer of
L:
acts
on
the
8
coordinates of the
left-hand brick. To explain the last two rows of Table
VII,
we note that if
c'
is of type
U,,
the it contains 13 special
octads, which fall into orbits of sizes
1
and 12 under the
stabilizer of
U.
Thus the left-hand brick can be chosen in
two
e!sentially different ways, producing the orbits
and
U,',.
Table
VI11
contains samples
of
the vectors
U;
TABLE
VIII*
59
512
U14
U15
11111
11111
@6
*Omitting the left-hand
8
coordinates from these pictures produces
samples from the orbits
of
Aut(%)= Aut(3).
orbit representatives
C:
for Aut
(9)
=
Aut
(2)
are ob-
tained by omitting the left-hand brick.
The cosets
of
9
and
2
are analyzed in Tables
IX
and
X,
respectively. (The weight distributions of the cosets of
9
were originally given in [121.)
TABLE
IX
COSETS
OF
[16,5,8]
REED-MULLER
CODE
9
No.
0
1 2
3
4
5
6
7 8
TABLE
X
COSETS
OF
[16, 11,41
HAMMING
CODE
2
No.01
2
3
4
5
6 7
8
1046
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THEORY,
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36.
NO.
5,
SEPTEMBER
1990
VI.
THE
NORDSTROM-ROBINSON CODE
OF
LENGTH
16
We use the notation of the previous section. Let
9,
(0
si161
denote the words
of
the Golay code
9
that
have
1’s
in coordinates
w
and
i,
and
0’s
elsewhere in the
first
8
coordinates, with the first
8
coordinates deleted.
Each
si
is a translate of
9
containing
16
words of
weight
6
and
16
of
weight
10,
and
M=9u90u~lu
...
u9,
is the Nordstrom-Robinson code. Thus
M
consists of the
words
of
9
that begin with one of
with these first
8
coordinates deleted, and Aut
(M)
=
Again we study vectors
D
E
Fi6
by adjoining the left-
hand octad (consisting of
8
“ghostly” points), one of
which
(03,
or the “focus”) is special, obtaining a vector
U
E
Ff”.
We classify
L;
by saying what
U
reduces to modulo
9,
i.e., its minimal error pattern. This is either a vector
e
of weight at most
3,
or
six
vectors
e,;
.,
e5
of weight
4,
all mutually congruent modulo
9,
i.e., a
sextet
(see Sec-
tion 11). These minimal error patterns
(e
or
{e,;.*,e,})
are described in the fourth column of Table
XI,
using the
symbols
F
for the “focus” (or coordinate),
G
for a
“ghostly” point (one
of
the other seven points in the
left-hand brick),
0
for a coordinate out
of
the last
16
where
U
is
0,
and
1
for a coordinate where
U
is
1.
24:
A,.
$9
15
la
Fig. 4. Orbits
of
vectors
of
length
16
under action
of
automorphism
Vectors in
&’
are group (Z4:
A,)
of
Nordstrom-Robinson code
circled. Weight is
8
less
than subscript.
It turns out that the minimal error pattern is enough to
distinguish the orbits
of
Fi6
under Aut(M), and further-
more that Aut(M) is transitive on vectors
of
each type.
Once again we omit the proof. There are therefore
39
orbits under Aut(M), those of weight at most
8
being
shown in Fig.
4
and Table XI.
In Fig.
4,
as in Fig.
3,
the weight is
8
less than the
subscript. Again the FottFm $alf of the graph has been
omitted; The types
SI,,
TI,,
v:
are self-complementary,
while
U&
complements to
U,.
Fig.
4
closely resembles
Fig.
3,
except that certain nodes and edges have been
split.
The sizes and error patterns for the orbits are given in
Table
XI.
TABLE
XI
PROPERTIES
OF
ORBITS
UNDER AUT(&’)
Weight Name Size
Error
Patterns under
d
0
$8
1
2
$,,
120
1
S,
16
3
SI,
560
4
SJ2
1680
4
XI2
140
5
SIX
1680
5
TA
612
5
fi
2016
6
SI,
840
6
TI,
6720
6
t:,
112
6 336
1
SI5
240
7
r,,
6720
7
3360
8
TI,
1920
8 840
8 5040
8
U,
5040
7
v;,
1120
8
SI,
30
-
1
12
1’
(FG’, G4, 14,
04,
04,04)
0’
FG
0
G2O
O2
(FG10,G210,G210,G210,
10’,lOX)
FG
G*
0
102
FG 1
G21
(~~02,~202,~~0~,~202,14,04}
-
10
(FG3,G4,
1202,
1202,
1202,
1202)
(FGI2,G2l2,G2O2, G202, 1202, 1202}
(G212,G212, FG02,G202, 1202,
I2O2}
Although the Nordstrom-Robinson code
M
is nonlin-
ear, it has the property that certain of its translates
partition the whole space (see Table XII). The union of
M
and the seven translates described by the last row of
Table XI1 is the Hamming code
A?.
VII.
THE
TERNARY
GOLAY CODES
OF
LENGTH
11
AND
12
The automorphism group
of
the
[12, 6, 61
ternary Golay
code
9-
is the group
2.M,,
(see
[41,
[SI).
In this section we
classify orbits of
F:’
under the action of this group.
There is an essential difference between the binary and
ternary classifications. In the binary case there is only one
way to change a bit,
so
edges in the graphs of Figs.
1-4
link
pairs
of orbits.
An
edge linking orbits
A,
and
B,-l
indicates that any vector in can be obtained by
changing a
1
in some vector of
A,
to a
0.
CONWAY
AND
SLOANE:
ORBIT
AND COSET
ANALYSIS
OF
GOLAY
AND
RELATED
CODES
1047
TABLE
XI1
TRANSLATES
OF
LENGTH
16
NORDSTROM-ROBINSON
CODE THAT PARTITION THE WHOLE SPACE
No.01
2
3
4
5
6
7
8
In the ternary case we take the components of the
vectors
u
E
F:
to be
O’s, +’s
(or
+
1’s)
and
-3
(-
1’s).
Consider the pair of vectors
U,U’
at Hamming distance
1
from
u
that are obtained by changing a particular nonzero
component of
U.
One
(U
say), obtained by changing the
sign
of
this component, has the same weight as
U;
the
other
(U‘
say), obtained by changing this component to a
0,
has weight one less. This process links the words of
F,”
in
triples.
If
U,U,U‘
belong to different orbits
A,,
B,,C,-l,
re-
spectively, we indicate this by a “trident”:
It turns out that
two
different
U’S
obtained from
U
in this
way are in the same orbit under
2.M,,
just if the corre-
sponding
~1”s
are. We may therefore label the trident with
the numbers
a,p,y,
where
a
is the number of ways to
choose the nonzero component of
u
E
A,
that leads to a
U
E
B,
when its sign is changed and to a
U’
E
C,-
when
it is replaced by a
0.
Similarly
y
is the number
of
zero components of
U’
E
C,-
that when replaced by one sign lead to a
u
E
A,
and when replaced by the other sign to a
U
E
B,.
We then
have
alA,I
=
PIB,I
=
7IC,-ll.
(2)
Of course it may happen that
U
and
U
are in the same
orbit, in which case we make the top arms of the trident
coincide:
Now
(3)
There are
48
orbits in
Fi2
under
2.MI2,
displayed in
Figs.
5
and
6,
and Table XIII. Unfortunately the graph in
Fig.
5
(strictly speaking a hypergraph, since the nodes are
linked in triples) is too complicated to be conveniently
drawn in one piece. We have therefore broken it up into
five sections, giving the orbits of weights
12-10,
9,
8,
7,
and
6-0
separately.
As
in the binary case, Hamming
distance between orbits is measured by the distance in the
graph, only now one must remember that following two of
the three arms of a trident takes one unit
of
Hamming
distance. The Golay code itself is indicated by double
circles.
We shall write words in the ternary Golay code
7
in
3
x
4
MINIMOG arrays; the reader is referred to
[4]
and
[SI
for the definition. (Note the erratum at the end of this
section.)
The second column in Table XI11 gives the number of
vectors in each orbit. The third column gives the distance
d
from the code, with a subscript describing the minimal
error pattern($. Fig.
6
gives an example
of
a vector from
each orbit. If
U
is a vector in the orbit and
d
is at most
2,
there is a unique closest codeword
c
E
7.
Then the error
pattern
e
=
U
-
c
is given (for the particular
U
of
the
example) in Fig.
6,
and the third column in Table XI11
gives
di,
where
i
is the number of coordinates where
U
and
e
are both nonzero. (In Fig.
6
we give only the
left-hand one or
two
columns of the MINIMOG array for
e.
The rest
of
this array is zero.)
On the other hand
if
L’
is at distance
3
from
7
then
there are four codewords
c,;
.
-,
c4
all at distance
3
from
U,
and four equally likely minimal error patterns
e,
=
U
-
c,
(0
5
r
5
3).
The four vectors
e,,
e,
e3
all have weight
3
and have disjoint supports, and any difference
e,
-
e,
(r
#
s)
is a codeword of weight
6
in
7.
In this situation
the four
er’s
are called a
quartering
(analogous to a
sextet
in the binary case). Modulo the code,
U
is congruent to
any of
e,;
e,
e3.
The simplest example
of
a quartering
occurs when
e,, . . .
,
e3
are the successive columns of
1048
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TRANSACTIONS
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INFORMATION
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VOL.
36,
NO.
5,
SEPTEMBER
1990
(e)
Fig.
5.
Orbits
under
2.M,,,
separated in five pieces.
(a)
Weights
12-10,
(b)
9,
(c)
8,
(d)
7,
and (e)
6-0.
CONWAY AND SLOANE: ORBIT AND COSET ANALYSIS
OF
GOLAY AND RELATED CODES
1049
jiriil
PI2
+--
QI2
liiI_I
RI2
Ipp',:I=N
WSE
ipT..."
o++o
-+++
00
o+-0
00
p;
p7
Q7 R7
s7 p6 06 R6
RzFFlsN
m=N
+-00
0-+o
00
0-++
s6
U:
U,
Q5
m.E
Inin3.E
BEN
m=E
0000
RS
s5
R4
s4
s3
s2
SI
SO
Fig. 6. Example
of
vector from each orbit
of
2.MI2, and minimal error
pattern(s) modulo ternary Golay code.
N
indicates any column
of
array (4).
TABLE
XI11
ORBITS
UNDER
2.M1
Orbit Size Error Pattern Orbit Size
Error
Pattern
012
440
PI,
1760
Q12
1584
RI, 288
SI2
24
PI,
5280
QII 15840
RI, 3168
SI,
288
PI,
2640
Q& 23760
Q, 23760
RIO 15840
SI,
1584
Py
440
Q, 3960
Ry
15840
S:
5280
S;
31680
TT 31680
T; 23760
Pl
3960
P;
3960
Q, 31680
47520
7920
23760
7920
15840
15840
47520
19008
3168
2640
3960
1584
264
19008
3 1680
15840
7920
1584
3960
3960
1760
264
24
1
The symbol
N
in Fig.
6
stands for any of the columns
of
this array. If
U
is at distance
3
from
F
the entry in the
third column of Table
XI11
is 3r0r,r2r3, where
i,
is the
number of coordinates where
U
and
e,
are both nonzero
(Osrs3). However,
if
i,=3 and
U
and
e,
have the
opposite sign on each
of
these three coordinates, then we
put a bar over
i,.
This information is sufficient to determine the signs in
e,;
a,
e3.
For each column of
U
adds up to the same
number
(a
say) modulo 3, and
(+
=
-
wt(u)
(mod3).
So
we can determine the signs of the coordinates where
U
and
e,
intersect, except that three agreements in sign are
indistinguishable from three disagreements. The bar then
enables us to distinguish these
two
cases.
The cosets of
F
are analyzed in Table
XIV.
TABLE
XIV
COSETS
OF
[12,6,61
TERNARY
GOLAY
CODE
7
No.012345 6 7 8 9
10
11
12
11
24
1 66
SI
SS
264 1
1.5
30
s2
s4 Rs
440 4 9 36
S3
R,
Qs
SO
264
66
15
+
72
6+72
'6
Rti
Q.5
U
u6+
Ph
U
U,-
440
132 165 165
60+72 120+30+30
60+90
P;
VR, QBUSl UT, R,UT;
36+108
9+108+54 12+72+72
P:
vQ,
Pl
U
R,uS;
S:
US;
UT:
PY
s,
p;
Q,
24
s12
110
12 12
Pin
SI1
RI2
90+6 20+12 6
Q&Usio PIIURII
Q12
54+36 36 1+4
QGURio
QII
Oi2uP12
TABLE
XV
WEIGHT
DISTRIBUTION
OF
COSETS
OF
[11,6,5]
GOLAY
CODE
No.0123 4 5 6
7
8 9
1011
1
1
0
0
0
0
132 132
0
330
110
0
24
22
0
1
0 0
30 66
108 180
165 135 32 12
220
0 0
1
6
21 60
123 174
174
114 48
8
1050
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INFORMATION
THEORY, VOL.
36,
NO.
5,
SEPTEMBER
1990
Finally, we briefly mention the
[11,
6,
51
perfect Golay
[21
J.
H. Conway, “Three lectures on exceptional groups,” in
Finite
Simple Groups,
M. B. Powell and G. Higman, Eds. New York:
Academic Press, 1971,
pp,
215-247.
[31
-,
“The miracle octad generator,”
in
Topics in Group Theory
and Computation,
M.
P.
J.
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code, whose automorphism group
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there are
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... Chaps. 10, 11]. For example, [CS90] classifies all vectors of 16 2 into orbits under the symmetry group Aut(N ) of the NR code. ...
... One property that is not stated in [CS90] but which is very easy to establish is that NR is geometrically uniform. As in [CS90], [CS92] we describe words in the Golay code by coordinates arranged in 4 × 6 arrays consisting of three 4 × 2 arrays called 'bricks' (these are the Miracle Octad Generator or MOG coordinates for G). ...
... One property that is not stated in [CS90] but which is very easy to establish is that NR is geometrically uniform. As in [CS90], [CS92] we describe words in the Golay code by coordinates arranged in 4 × 6 arrays consisting of three 4 × 2 arrays called 'bricks' (these are the Miracle Octad Generator or MOG coordinates for G). N is formed by taking all words of G whose left brick is one of ...
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... Moreover, Snover [40] proved that any binary (16,256,6) code is equivalent to the Nordstrom-Robinson code. Analogous properties also hold for the punctured Nordstrom-Robinson code, a non-linear (15,256,5) code. In this paper, we prove that the Nordstrom-Robinson codes have other exceptional properties. ...
... Theorem 1.1. Any binary completely regular code of length m with minimum distance δ is equivalent to the Nordstrom-Robinson code, respectively the punctured Nordstrom-Robinson code, if (m, δ) = (16,6) or (15,5). Moreover, such a code is completely transitive. ...
... It is known that completely transitive codes are necessarily completely regular [21]. A consequence of Theorem 1.1 is that the converse holds for binary codes with (m, δ) = (16,6) or (15,5). This is similar to a result in [19] in which the authors proved that a binary completely regular code with (m, δ) = (12,6) or (11,5) is unique up to equivalence, and that such codes are completely transitive. ...
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... Chaps. 10, 11]. For example, [CS90] classifies all vectors of 16 2 into orbits under the symmetry group Aut(N ) of the NR code. ...
... One property that is not stated in [CS90] but which is very easy to establish is that NR is geometrically uniform. As in [CS90], [CS92] we describe words in the Golay code by coordinates arranged in 4 × 6 arrays consisting of three 4 × 2 arrays called 'bricks' (these are the Miracle Octad Generator or MOG coordinates for G). ...
... One property that is not stated in [CS90] but which is very easy to establish is that NR is geometrically uniform. As in [CS90], [CS92] we describe words in the Golay code by coordinates arranged in 4 × 6 arrays consisting of three 4 × 2 arrays called 'bricks' (these are the Miracle Octad Generator or MOG coordinates for G). N is formed by taking all words of G whose left brick is one of ...
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... The Any vector in F 24 2 can be categorized into 49 orbits w.r.t. G 24 (see Figure 1. in [49]). These orbits are denoted as S w (0 ≤ w ≤ 24), T w (8 ≤ w ≤ 16), U w (6 ≤ w ≤ 18), P 12 and X 12 , where the subscript w denotes the weight of the vectors in that orbit. ...
... ofFigure 1in[49])) depicts some of these orbits. An edge between orbitsA and B indicates that a vector in orbit B can be obtained from some vector in orbit A (and vice versa) by complementing a single bit. ...
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... Because of self-duality we have equality: π(G) is the all-even code [8,7,2] 2 and the projection of D to the last 16 coordinates is a [16, 5,8] 2 -code, a Reed-Muller code R. Goethals defines the code N R as the union of 8 cosets of R. More precisely a vector x ∈ F 16 2 is in N R if and only if it is the projection onto the last 16 parameters of a codeword of G whose projection onto the first 8 parameters is either the 0-word or a word of weight 2 with a 1 in the first coordinate. ...
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In this paper, we define M24 from scratch as the subgroup of S24 preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent.(Received June 15 1974)
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The Mathieu group M24 can be represented as a collineation group in space of 11 dimensions over the field of two elements. This paper discusses the geometry associated with this representation.