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BULLETIN OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 80, Number 6, November 1974
BINARY SELFDUAL CODES OF LENGTH 24
BY VERA PLESS
1
AND N. J. A. SLOANE
Communicated by Olga Taussky Todd, February 28, 1974
ABSTRACT. There are 26 distinct indecomposable selfdual codes of
length 24 over GF(2), including unique codes of minimum weights 8 and 6,
whose groups are, respectively, the Mathieu group M
2
4 and the maximal sub
group of index 1771 in M^> For each code we give the order of its group,
the number of equivalent codes, and its weight distribution.
1.
Introduction. An [n, k] code C is a ^dimensional subspace of
the vector space of all «tuples of O's and l's with mod 2 addition. The
dual code C
1
= {u: u
•
v = 0 for all v G C} is an [n, n  k] code. C
is selforthogonal if C C C
1
, selfdual if C
—
C
1
. Selfdual codes exist when
ever the length n is even. The weight of a vector is the number of its non
zero components, and the minimum weight of C is the minimum weight of
any nonzero codeword. The weight distribution of C is the set {a
0
,
ot
l9
• • • ,
0L
n
}
9
where a
f
is the number of codewords of weight I
The group G(C) of a code C is the set of all permutations of the
coordinates which send C into itself setwise. Two codes are equivalent if
there is a coordinate permutation sending one into the other. The number of
codes equivalent to C is «!/order of G(C). The direct sum of codes C'
and C", written C' 0 C", is {(w, v): u GC',v e C"}. If C = C' 0 C",
where C' and C" are nonzero, then C is decomposable. Otherwise C is
indecomposable.
Pless [4] classified all selfdual codes of length < 20, Conway
(unpublished) found the 9 selfdual codes of length 24 in which the weight
of every codeword is a multiple of 4, and Niemeier [2] found the 24 even
unimodular lattices in dimension 24, 9 of which correspond to the codes
found by Conway.
AMS (MOS) subject classifications (1970). Primary 94A10; Secondary 05A15, 15A03.
1
The work of the first author was supported in part by Project MAC, an MIT
interdepartmental laboratory sponsored by the Advanced Research Projects Agency, De
partment of Defense, under Office of Naval Research Contract N0001470A03620001.
Copyright © 1974, American Mathematical Society
1173
1174
VERA PLESS AND N. J. A. SLOANE
[November
We have found that there are 8 inequivalent, indecomposable selfdual
codes of length 22, and 26 of length 24. The latter are shown in Table I,
which gives for each code a basis, the order of its group, the number of codes
equivalent to it (written as a multiple of v = 1 • 3 • 5 • 7 • ...
•
23 = 316,
234,
143, 225), and the weight distribution a
4
, a
6
,* • • , a
12
(omitting a
0
— 1, a
2
=
a
odd
= 0, a
t
=
a
2
4z
f°
r
*
> 12). Full details of the enumera
tion will appear in [5].
2.
Selforthogonal codes of minimum weight 4. Table I was obtained by
classifying the codes according to minimum weight. A selfdual code of mini
mum weight 2 is decomposable. For minimum weight 4 we use
THEOREM 1. Let C be an indecomposable selforthogonal code
gener
ated by codewords of weight 4. Then C is one of the codes d
n
(n = 4,
6, 8,
•
• •), e
7
, or E
s
, generated by the rows of the following matrices:
"till
1 1 1 1
1
' i L
,
I i I i
I I i I
i i i i
£
8
:
1 1 1 1
1 1 1 1
1 1 1 1
1111
Furthermore a selfdual code containing E
s
as a subcode is decomposable.
Let C be an indecomposable selfdual code of length 24, and let C
r
be the subcode generated by codewords of weight 4. By Theorem 1, C' has
the form d„ ©
•
• • © d„ © e
n
©
•
• • © e
n
. We considered all such C'
and all ways of extending C to a selfdual code. For each code we com
puted the order of its group. In this way all the codes of minimum weight 4
were obtained.
The notation used to specify the basis vectors is best illustrated by an
example. The code /
24
generated by the rows of (1)
k
0
0
b
b
_
a
0
e
i
0
c
0
0
0
0
e
i
0
c
o
00
00
00 j
10
01
11J
where a = 101010... 10, b = 110000.. .00, c =
111...
1, is written d
%
e
2
n
1974]
BINARY SELFDUAL CODES OF LENGTH 24
1175
4
2/bcol0/boc0l/ao
2
l
2
, where the + 2 indicates two coordinates which do
not meet any codeword of weight 4. a' denotes a 4 b = 011010... 10.
We omit the full details of W
24
, X
24
, Y
24
.
3.
Minimum weight 6 and 8. It is known [3], [1] that the [24, 1.2]
Golay code is the unique code of minimum weight 8, and that its group is the
Mathieu group M
24
.
We determined that there is a unique selfdual code of minimum weight 6,
which is generated (in Todd's [6] notation) by the set of 64 nonspecial hexads
associated with any set of 6 mutually complementary tetrads in the Golay code.
Its group is a maximal subgroup of index 1771 in M
24
.
TABLE I
Indecomposable SelfDual Codes of Length 24 (Page 1)
{
Generator Matrix
Order of Group Number * v a
4
<*
6
a
8
a
10
a
12
( d
2
Jab/ba
A
24
j
(2
5 \ '
2
.
2
1,848 30 0 639 0 2756
^fXnP.i
18
'
102
f
24
°
663
°
272
°
c^)'%tT
a
46
'
200 18
°
687
°
2684
^ )d*(a)/baao/obaa/aoba/aaob
24
j(2
2
• 3!)
4
4! '
40
°
l2
0 711 0 2648
24
(2
11
12 2 66 0 495 0 2972
\ d%(a)/boa
3
o/oboa
3
/aoboa
2
/a
2
oboa/a
3
obo/oa
3
ob
24
(4
6
• 6!3 221,760 6 0 735 0 2612
„
\
Golay code
rt
21
C
MJ2».3».5.7.11.23
M13
23
° °
759
°
2576
H
U
%
d
ï6
lablba
24
J2
3
• 4!2
7
• 8! 1,980 34 64 239 960 1500
!
d.dod^/b
3
/a
2
o/oa
2
2*2!2
3
4!2»6!
110
'
88
°
22 64 28? %0 M28
1176
VERA PLESS AND N. J. A. SLOANE
[November
TABLE I
Indecomposable SelfDual Codes of Length 24 (Page 2)
Î
Generator Matrix
Order of Group Number f v
a
4
<x
6
a
8
a
x
\d
8
e
2
+ 2/bcol0/boc01/ao
2
l
2
(see (1))
24
(2
3
• 4!168
2
• 2 181,028^ 20 64 295 960 1416
!
d
6
d
l0
e
1
+ l/b
2
cl/oao\labol
2
2
• 3!2
4
• 51168 253,440 20 64 295 960 1416
\d
3
(b)/b
3
/a
2
o/oa
2
^
24
(2
3
• 4!)
3
• 3! 46,200 18 64 303 960 1404
M
\dl(c)la
3
/ba'o/boa'
24
(2
3
• 4!)
3
• 2 138,600 18 64 303 960 1404
\d\d
XQ
+ 2lb
3
\l/oa
2
nlabo01lbao\0
24
j(2
2
• 3!)
2
2
4
• 5'2 887,040 16 64 311 960 1392
Î
d ld \/ab
2
o/boao/oboa/baob
(2 • 2')
2
(2
3
• 4!)
2
• 2
1,663,200
14 64 319 960 1380
\d
4
dle
1
+ \/ob
2
c\/ab
2
o0loaa'o0/boaol
24
(2 • 2!(2
2
• 3!)
2
168 • 2 2,534,400 14 64 319 960 1380
!
d^(b)faoao/boa
2
joaoajobaa
(2
2
• 3!)
4
• 8 739,200 12 64 327 960 1368
\d
2
6
d
8
+4/b
2
ol
4
/bobl
2
0
2
/o
2
a0l
2
0/ao
2
01
3
/oao\
3
0
24
((2
2
• 3!)
2
2
3
• 4! • 2 8,870,400 12 64 327 960 1368
24
(2 • 2!(2
2
• 3!)
3
• 2 17,740,800 10 64 335 960 1356
4.
General enumeration theorems. The following theorems, and others,
were used to check Table I.
THEOREM 2. Let
OL
C
(X)
= SJLQ a
t
x
l
be the weight enumerator of C.
Then
1974] BINARY SELFDUAL CODES OF LENGTH 24 1177
z
oc
c
(
X
)
=
"jnV
+
o
•
p
2
^
+
x
n
)
+
L
(")*'j,
where
the sum
extends
over
all
selfdual
codes
C of
even
length
n.
THEOREM
3. If n is
even,
the
number
of
selfdual
codes
with
length
n
and
minimum
weight
> 4 is
nil
(— \\iy.\ n/2i l
£6
2H\(n

20!
A
i
TABLE I
Indecomposable Self Dual Codes of Length 24 (Page 3)
~  /Generator Matrix
VT
,
Code <_ , ._ Number v i> a* a
A
a
R
a
in
a
\Order of Group
4
"6
w
8 "10 "12
rfjâf
8
/babab/ba
2
oa/oab
2
a'/aoba
2
/b
2
oaa '
'4
a
8.
*
24
4
4
• 2
3
• 4! • 8 4,989,600 10 64 335 960 1356
\d\d\ + 4lob
2
ol
2
0
2
/oa
2
o0
3
llobob0
2
l
2
/oaoa0l0
2
/b
2
o
2
1
4
/a
2
o
2
1010
24
(4
2
(2
2
• 3!)
2
• 4 53,222,400 8 64 343 960 1344
[dl(b)lbabo
3
/obabo
2
/o
2
babo/o
3
bab/bo
3
ba/abo
3
b
24
\4
6
• 6 • 8 9,979,200 6 64 351 960 1332
(
4
3
• 2
2
• 3! • 3! • 2 106,444,800 6 64 351 960 1332
d\ + 8/
•
•
4
4
• 4! • 2 159,667,200 4 64 359 960 1320
Y
)d
2
+ l6o/ • •
**
2
u
3
2
106,444,800 2 64 367 960 1308
z
see §3
24 
2l0
.
3
3 . 5 14,192,640 0 64 375 960 1296
Total: 556,041,557 ~ • v
REFERENCES
1.
E. F. Assmus, Jr. and H. F. Mattson, Perfect codes and the Mathieu groups,
Arch.
Math. (Basil) 17 (1966), 121135. MR 34 #4050.
2.
H.V. Niemeier, Definite quadratische Formen der Dimension 24 und Diskrimin
ante 1, J. Number Theory 5 (1973), 142178.
1178
VERA PLESS AND N. J. A. SLOANE
3.
Vera Pless, On the uniqueness of the Golay codes, J. Combinatorial Theory 5
(1968),
215228.
4.
, A classification of self orthogonal codes over GF(2), Discrete Math. 3
(1972),
209246. MR 46 #3200.
5. Vera Pless and N. J. A. Sloane, On the classification and enumeration of
self
dual codes J. Combinatorial Theory.
6. J. A. Todd, A representation of the Mathieu group M^ as a collineation group,
Ann.
Mat. Pura Appl. (4) 71 (1966), 199238. MR 34 #2713,
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