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Oura, M.Osaka J. Math.34 (1997), 53-72
THE DIMENSION FORMULA FOR THE RING OFCODE POLYNOMIALS IN GENUS 4
MANABU OURA
(Received March 15, 1996)
1. Introduction
The purpose of this paper is to study the dimension formula for the invariantring C[/α for aeF^]1**, which may be considered as the ring of code polynomialsin genus 4. We also give all characteristic polynomials of elements in H4. Themain ingredient is the determination of the conjugacy classes of the symplecticgroup 5/7(8,2). Our result will be useful for the investigation of the Siegel modularforms in genus four.
We recall from [10, 11] that the finite group Hg is (up to ±1) just the imageof the modular group Γg = Sp(2g,Z) under the theta representation (of index 1)and that the ring of modular forms of even weight is given by
Λ(Γg)(2) = Θ [Γr*] = (qyj*- / {relation})",2|fc
where N denotes the normalization in its field of fractions and "relation" are thetheta relations. However, the generators and the dimension formulas for A(Γg) areknown only for genus g<3.
On the other hand, the invariant ring C[/α]Hg may be considered as the ring
of code polynomials in genus g. In [9], there is the definition of the g-th weightpolynomial for codes (codes mean the binary linear codes) and the connectionsamong codes, lattices, the invariant rings of the finite groups, and the theory ofmodular forms were studied (cf. [1], [4], [5], [8], [16]). In particular, it wasshown that the invariant ring of the group <//g,£8>, which is the subring ofC[/α]
H«, is generated by the g-th weight polynomials for self-dual doubly-evencodes, where £8 is the primitive 8-th root of unity. This invariant ring correspondsto the ring of the modular forms of weights divisible by 4.
The author would like to thank Prof. Bannai for suggesting this work andfor his encouragement. He owes thanks to Prof. Runge for his critical commentson the original manuscript. Furthermore, the author would like to thank thereferee for providing him with many improvements thoughout the whole paper.
54 M. OURA
2. On the group Hg
In this section, we study the group Hg. In addition to Runge, the group Hg
has been studied by several authors, for example, see [4], [7].
Let V be the g-dimensional vector space over the field of two elements, i.e.,K=Ff. For x,ye V, let x-y denote the usual dot product. Set
g'~ v 2
and for a symmetric g x g matrix S9
/)s:=diag(/S[Λ] with aeV),
where S[ά]:=aS*a. Let
7/^:=<7^,Ds|S runs over all symmetric matrices in Matgxg(Z)>
be the subgroup of Gl(28
9C) generated by the elements Tg and the Ds.
We get for the invarint ring C[/a]Hg the dimension formula
Φn.(0= Σ (dιmC[/Jd^ =_^___ ,
series of Hg .
where C[/fl]?g is the rf-th homogeneous part of C[/α]
Hg, ΦHg(0 is called the Molien
s of Hg .The following lemma gives the simplification we use.
Lemma 2.1 (Lemma 2.1 [9]). One has an exact sequence
where Ng:=(i,Dl,T~lDlTgy and the homomorphism φ is given by the conjugation
of Hg on the F2-vector space Λ^/</>. Π
Therefore we have the obvious formula
where {Q} are the conjugacy classes of Sp(2g,2) and zi is some element of Hg
with φ(zi)eCi. Our computation is done using (2.1).Finally we list the sizes of groups.
CODE POLYNOMIALS IN GENUS 4 55
In our case fe = 4), \N4\ = 1,024 = 210, \Sp(8,2)\ = 47,377,612,800 = 2163552 7 17,
\H4\= 48,5 14,675,507, 200 = 2263552 7 17.
REMARK. Hi9 <//1?C8>, //2
are the reflection groups No.8, No.9, No.31 in[14], respectively (cf. Proposition 2.6 [10]).
3. On 5/7(8,2) and its conjugacy classes
In this section, we study the symplectic group Sp(8,2). This is identified
with the Chevalley group of type (C4) over the field of two elements. We give all
the characteristic polynomials of elements in //4 . We remark that we cannot read
off representatives of the conjugacy classes of S/?(8,2) from Atlas [3] although it
is the good reference for the finite simple groups.
As is said, 5X8,2) is one of the Chevalley groups. The properties of such
groups are known and we describe what we need.
Let Δ = {±2f£, ±ξi±ξj (/</)! 1 </',./< 4} be the root system of type (C4), andchoose a = ξί — ξ2, b = ξ2 — ξ^ c = ξ3 — ξ4, d=2ξ4 for a fundamental system Π of
roots. We denote by Δ"1" the set of positive roots with respect to Π. We write an
element αα -f βb + yc + δd of Δ + as aβyδ. For example, we write 1023 for a + 2c + 3d.
Eij is the elementary matrix of size 8 x 8 with 1 in the (/J)-entry. For 1 < ij < 4,
xξi -</=!+ E^ + E4+jA + i (i <y ),
Then 5X8,2) is generated by xr (reΔ) and is known to be one of the finite simple
groups (cf. [3]).
Let Xr be the group generated by xr and put ̂ = AΓ
0100Ar
0010Ar
0110Ar
0001Ύ0011
^o 1 1 1^002 i^o 1 2 1^0221^1 ooo^i i oo^i 1 1 o^i 1 1 i^i 12 i^i 22 1^222 1 Then B is a Sylow2-subgroup of S/?(8,2) and is normal in 5/?(8,2).
Put nr = xrx_rxr and 7V=<« r | reΔ>. Λf is isomorphic to the Weyl group of
type (C4). We fix the following correspondence:
~ 2 :=[0,1,0,0,0,0,0,0],
56 M. OURA
<E3 ~ 3:= [0,0,1,0,0,0,0,0],
£4 ~ 4:= [0,0,0,1,0,0,0,0],
-^ <->-_!:= [0,0,0,0,1,0,0,0],
-ί2~:z2:= [0,0,0,0,0,1,0,0],
- £3 ̂ ^-3:= [0,0,0,0,0,0,1,0],
-ξ4^-4:= [0,0,0,0,0,0,0,1].
An element of N is uniquely determined by its natural action on 1, 2, 3, 4. IfnεN satisfies In = a9 ?« = /?, 3« = y, 4n = δ, then we denote n by w(α,/?,y,(5). Forexample, we have
01000000
00100000
00001000
00000001
00000100
00000010
10000000
00010000
In the following, we give the conjugacy classes of S/?(8,2) and all characteristicpolynomials of elements in 7/4. To determine the conjugacy classes of S/?(8,2),we have to show
(i) No two elements in the list are conjugate in S/?(8,2),
(ii)
where CSp(8>2)(c,) denotes the centralizer group of c{ in 5/?(8,2). These statementsare proved using GAP [13]. Then we compute 1024x81 determinants of size16 x 16. Since H4 is a subgroup of SU(2\Z\^^ (Proposition 2.6 [10]), thepolynomials have the type Σ0:δl^16έi/ with α0 = α16 = l, άi =α1 5, ά2 = α14, ά3 = α1 3,
ά4 = 012, ^5 = ̂ !!, ά6=α1 0, άΊ—ag, and άs = as (A bar denotes complexconjugation).
There are 81 boxes below and the ί'-th box (0<z<80) gives the characteristicpolynomials of elements in z(7V4 . In each box, the first column gives the multiplicityof each polynomial. The next columns give the values of α1,α2, ,α8,respectively. For example, in the first box, the 2 in the first column means thatthere are 2 occurences of the polynomial in the form 1+ (8 — 8/)ί — 64/f2 + (— 168
CODE POLYNOMIALS IN GENUS 4 57
Table.
order = 1, co = 1, \CSp(Λ,2)M\ = 473776128002
5042
256256
22
8-8:0
8 + 8:'00
-8 - 8:-8 + 8:
-64:0
64:8t
-8i64i
-64*
-168- 168:0
-168+168:00
168 - 168i168 + 168:
-644-4
-644-28-28
-644-644
-952 + 952:0
-952 - 952:00
952 + 952:'952 - 952:
2240:0
-2240:-56:
56i-2240:
2240:'
2136 -|- 2136:0
2136 - 2136:00
-2136 + 2136:-2136 - 2136:
33346
33347070
33343334
order = 2, GI = roooi, \CSp(e..2)(cι)\ = 185794560
2
504
2
256
256
2
2
8-8:
0
8 + 8:'
0
0
-8 - 8:
-8 + 8t
-64:
0
64:
8:'
-8:'
64:
-64:
-168- 168:
0
-168 + 168:
0
0
168 - 168:
168 + 168:
-644
-4
-644
-28
-28
-644
-644
-952 + 952:
0
-952 - 952:'
0
0
952 + 952:'
952 - 952:'
2240:
0
-2240:
-56:
56:
-2240:
2240:'
2136 + 2136:
0
2136 - 2136:'
0
0
-2136 + 2136:'
-2136 - 2136:
3334
6
3334
70
70
3334
3334
order = 2, 02 = #0100 » \Csp
4120
768120
444
8000
-8-8i
Si
24-8
08
24-24-24
24000
-2424t
-24i
-3628
-428
-36-36
-36
f 8 ι 2 )(c2)|= 8847360
-120000
120120t
-120i
-88-56
056
-8888
88
88000
-88881
-88i
19870
670
198198198
order = 2, c3 = α?oιoo*oi2i, |CsP(8,2)(c3)l = 29491204
240128128
4512
44
8
000
-80
-8i8ι
320
-8
8320
-32-32
88
000
-880
88i
-88i
188
-428
28188
4188188
328
00
0-328
0
-328i328i
480
0-56
56480
0-480-480
600
000
-6000
600ι-600i
646
67070
6466
646646
1632
38416
51232
1616
order = 2, c4 = 2:0100*0121 x i i i o , \CsP(8,2)(c*)\ = 4915240
0
-400
-4i4ί
8-8
08
08
-8-8
120
0-12
00
12i-12i
1228
-412
428
1212
40
0
-400
-4ι4i
-8-56
0-8
056
88
-200
020
00
-20«20z
-2670
6-26
670
-26-26
58 M. OURA
order = 2, c6 = xoioo^mo, \CSp(B.2)M\ = 73728016
96016
1616
404
-4t4i
000
00
-20
020
-20i20i
-20
-4-20
-20-20
36
0-36
-36ι36t"
640
64
-64-64
-20
020
-20i20ι'
-90
6-90
-90-90
order = 2, cβ = r0ιoo*oooι, \CSp(8 ι2)(c6)| = 147456
8480
864
3846488
4- 4i0
4 + 4i000
-4 + 4i
-4 - 4i
-I6i
0
I6i
Si
0
-Si
-16i
I6i
-20-20t'0
-20+20i0
0
0
20 + 20i
20 - 20t
-36
-4
-36
-28
4
-28
-36
-36
-28 + 28t'
0
-28 - 28t
0
0
0
28 - 28i
28 + 28ι
4Sι
0
-48i
-56i
0
56i
48t'
-48i
44 + 44t
0
44 - 44ι
0
0
0
-44 - 44i
-44 + 44i
70
6
70
70
6
70
70
70
order = 3, c7 = n(2,3,l,4), \C
Sp(6>2}(c
7)\ = 12960
16
480
480
16
16
16
4
0
0
-4
-4i
4t
6
-2
2
6
-6
-6
8
0
0
-8
Si
-Si
17
1
1
17
17
17
24
0
0
-24
-24i
24ι
22
-2
2
22
-22
-22
28
0
0
-28
28i
-28ι
36
4
4
36
36
36
order = 3, cβ = (xooιm(-4, -1, -2, -3))5, |C5p(8|2)(c8)| = 77760
256256256256
— 1
i
1-1
0000
5ι-5i
5-5
55
55
0000
-10-10
1010
lOi-10i
10-10
0000
order = 3, c9 = xoooιn(l, 2,3, -4), |C
ί5p(8(2)(c
9)| =4354560
4
504
4
504
4
4
8
0
-8ί
0
-8
Si
36
4
-36
-4
36
-36
112
0
112i
0
-112
-112i
266
10
266
10
266
266
504
0
-504i
0
-504
504i
784
16
-784
-16
784
-784
1016
0
1016z
0
-1016
-1016ι
1107
19
1107
19
1107
1107
order = 3, CJQ = xoooιn(-2, 3, -1, -4), |C'Spί8|2)(cιo)| =3888
64
384384
6464
64
2i
00
2-2i
-2
-3
-11
3-3
3
2i
00
-2-2i
2
-7
11
-7-7-7
-12i00
-1212i12
2
2-2
-22
-2
-8i00
8Si
-S
18
-2-2181818
CODE POLYNOMIALS IN GENUS 4 59
order = 4, cn = 2:0001*1110, l^5P(8.2)(cιι)| = 921608
240
8
240
8
8
512
4 - 4i0
4 + 4t'0
-4 +- 4i-4 - 4t
0
-12z4i
12t-4t
-12t12t
0
-4 - 4t0
-4 + 4«0
4 + 4t4-4i
0
24
-8
24
-8
24
24
4
36 - 36i0
36 + 36t0
-36 + 36t-36 - 36t
0
-28ι'-12t
28t
12t
-28i28t
0
28 + 28t0
28 - 28t0
-28-28t-28 + 28t
0
7814
78
14
78
78
6
order = 4, 012 = #000 1*1000 £1110^1111 , |C'sp(8,2)(ci2)| =921608
240240
8512
88
4- 4»00
4 + 4i0
-4 - 4t-4 + 4i
-20i4i
-4i20t
020t
-20t
-36 - 36t00
-36 + 36t0
36 - 36t36 + 36t
-104-8-8
-104-4
-104-104
-124+ 124:00
-124 - 124t0
124 + 124ί124 - 124t
252t-12t
12t-252t
0-252t
252i
220 + 220i00
220 - 220t0
-220+220t-220 - 220t
3341414
3346
334334
order = 4, 013 = #0100^0121^1100^2221 » I^Sj16
16
16
96
96
16
768
4
4i
-4t
0
0
-4
0
12
-12
-12
4
-4
12
0
28
-28t'28i
0
0
-28
0
52
52
52
4
4
52
4
84
84t
-84t0
0
-84
0
pίβ,2)(ci3)| = 36864116
-116-116
-4
4
116
0
140
-140t'140t
0
0
-1400
150
150
150
-10
-10
150
6
order = 4, cι4 = ^oiooa7oi2ixiioo, |C'sP(8,2)(ci4)| = 1228816
16
96
96
256
512
16
16
4
-4
0
0
0
0
-4i
4t
4
4
-4
4
0
0
-4
-4
-4
4
0
0
0
0
-4t
4t
-12
-12
4
4
4
-4
-12
-12
-12
12
0
0
0
0
12t
-12t
-4
-4
4
-4
0
0
4
4
12
-12
0
0
0
0
12t
-12t
22
22
-10
-10
6
6
22
22
order = 4, 015 = #0100^0001*1110, |C'sp(8,2)(ci5)| = 6144
32
384
32
32
32
512
2 -2z0
2 + 2i-2 + 2i-2 - 2i
0
0
0
0
0
0
0
6 + 6i0
6-61
-6 - 6i
-6 + 6t
0
8
0
8
8
8
4
-6 + 6t
0
-6 - 6i
6-6t
6 + 6t0
16t
0
-16t16i
-16ι0
-2 - 2i0
-2 + 2t2 + 2t2- 2t
0
-18
-2
-18
-18
-18
6
order = 4, GIG = xoιooa:oooι^ιιιo^ιιιι, |C'sp(8,2)(ci6)| = 614432
384323232
512
2 - 2 t0
2 + 2t-2 - 2t-2 + 2i
0
-8t0
8t8t
-8t0
-10- lOt0
-10+ lOt10 - lOt10 + lOi
0
-240
-24-24-24-4
-22 + 22i0
-22 - 22t22 + 22t22 - 22i
0
40i0
-40ι-40i
40i0
30 + 30t0
30 - 30t-30 + 30t-30 - 30t
0
46-24646466
60 M. OURA
order = 4, c17 = 270100*0121^1000, |Csp(8,2)(ci7)| = 307216
128192
12816
256256
1616
4
00
0-4
00
-4i4i
8
-40
48
00
-8-8
12
00
0-12
00
12ί-12z
16
80
816
4-4
1616
20
00
0-20
00
-20t20i
24-12
0
1224
00
-24-24
28000
-28
00
28t-2Si
30
14-2
1430
66
3030
order = 4, ciβ = #oιoo^oooι#Π2i , \C §p(B ,2)(cι*)\ = 3072
323232326464
384384
2- 2i-2 + 2i
2 + 2i-2 - 2i
0000
-4i-4i
4i4i4i
-4i00
-2 - 2i2 + 2i
-2 + 2i2- 2i
0000
-4-4-4-4-4-4-4
4
-6 + 6t6-6ί
-6-6t6 + 6t
0000
12i12t
-12t-12t
4i-4i
00
6 + 6t-6 - 6t
6-6t-6 + 6t'
0000
6666
-10-10
66
order = 4, cig = :roioo#ooio^ooii^222i» I^5p(8,2)(ci9)| = 1°243232
2563232
2562566464
2- 2i2 + 2i
0-2 + 2t-2 - 2i
0000
-4i4i0
~4i4i00
4t-4i
-2- 2i
-2-f 2t0
2 + 2t2-2t"
0000
000004
-4-8-8
2- 2i2 + 2t
0-2 + 2i-2 - 2i
0000
-4t4t0
-4t4t00
-12t12t'
-2 - 2t
-2 -I- 2t
02 + 2i2- 2i
0000
-2-2-2-2-2
66
1414
order = 4, c2o = a oioo^ooio^iooo, I^Spίβ,64
76864
6464
2
0-2
-2i2t
0
00
00
-2
02
-2t
2t
-4
0-4
-4-4
-6
06
6t
-61
2)(c2θ)| = 7680
00
00
6
0-6
6t-6t
6
-26
66
order = 4, 021 = a7oιoo*ooιo*oon*ιιιo> |C*5p(8,2)(c2i)| = 51264
64512
25664
64
2
-200
-2t2t
0
00
00
0
-2
200
-2t
2*
0
00
-400
2
-200
-2i2t
0
00000
-2
20
0-2t
2i
-2
-2-2
6-2
-2
CODE POLYNOMIALS IN GENUS 4 61
OΓdeΓ = 4, C22 = ̂ 0100^0010^0011^1110^2221, !
C'5pί8,2)(
c22)| = 512
64
64
64
64
512
256
2
2i
-2i
-2
0
0
4
-4
-4
4
0
0
6
-6t
6i
-6
0
0
8
8
8
8
0
4
10
lOt
-lOt
-10
0
0
12
-12
-12
12
0
0
14
-I4i
I4i
-14
0
0
14
14
14
14
-2
6
order = 5, c23 = *oooi^oonn(l, 2, -4,3), |C
f
Sp(8>2)(c23)| = 3600
16
480
16
480
16
16
-4t
0
4
0
4i
-4
-10
-2
10
2
-10
10
20i
0
20
0
-20i
-20
35
3
35
3
35
35
-52i
0
52
0
52i
-52
-68
-4
68
4
-68
68
80t
0
80
0
-80t
-80
85
5
85
5
85
85
order = 5, C24 = #ooio2'oiiin(-4, -3, -1, -2), |Cl
Sp(8(2)(c24 )| =300
256
256
256
256
— i
i
-1
1
0
0
0
0
0
0
0
0
0
0
0
0
3i
-3
3
-3
-3
3
3
0
0
0
0
0
0
0
0
order = 6, c25 = n(-4, -2, -1,3), |C5p(8i2)(c25)| = 864
256
32384
25632
3232
0
2 + 2i0
0-2 - 2i
-2 -f 2i2- 2i
-2i
4i0
2i4i
-4i-4i
0
00
00
00
-1
7
-1
-1777
0
8 + 8ι'0
0-8 - 8i
-8 -|- SiS-Si
2i
4i0
-2i4i
-4i-4i
0
6- 6t0
0-6 -|- 6i
-6 - 6i6 + 6t
4160
4161616
order = 6, c2β = n(-4, -1, -3, -2), \C Sp(
128512
6464
6464
128
00
-2i2ί2
-2
0
20
-2-2
22
-2
00
00
00
0
1
1
-3-3
-3-3
1
00
4t-4i
-4
4
0
8|2)(c26)| =9620
22
-2-2
-2
00
2»-2t
2-2
0
40
44
44
4
order = 6, c27 = n(-4,2,l,3), |Cr
Sp(8)2)(c27)| = 28825632
38432
2563232
02-2i
0-2 -|- 2i
0-2 - 2z
2 + 2i
2i-4i
0-4ι-2i
4i4i
0-4 - 4i
0
4 + 4i0
4- 4i-4 + 4;
-1-9-1-9-1-9-9
0-8 + 8i
08-8ι
08 + 8t
-8 - 8i
-2t12t
012Ϊ
2i-I2i-I2i
010 -f lOt
0-10- lOi
0-10-1- Wi
10 - lOi
4160
164
1616
62 M. OURA
order = 6, c28 = zooιι^oθ2in(2, -4,3, -1), |C
5p(
8|2)(c
28)| = 288
64
128
64
128
64
512
64
-2
0
2ι
0
-2t
0
2
2
-2
-2
2
-2
0
2
-4
0
-4i
0
4t'
0
4
5
1
5
1
5
1
5
-4
0
4i
0
-4i
0
4
6
-2
-6
2
-6
0
6
-6
0
-6i
0
6i
0
6
4
4
4
4
4
0
4
order = 6, c29 = ^oooi #0011 #0021 n(l, -2, -4,3), |Cr5p(8>2)(c29)| = 4608
128
51216
1921616
12816
0
0-4
04
4i
0-4t
-40808
-84
-8
0
0-8
08
-Si
08t
10
-22222
102
0
040
-4
-4t0
4t
-1600000
160
0
0-12
012
-12t0
12t
19
3193
19191919
order = 6, c30 = a:oo2in(l,4
l-3,2), |C
Sp(8ι2)(c
30)| = 13824
16
96
768
16
96
16
16
-4t
0
0
-4
0
4t
4
-12
-4
0
12
4
-12
12
24z'
0
0
-24
0
-24t
24
42
10
2
42
10
42
42
-60ι
0
0
-60
0
60i
60
-80
-16
0
80
16
-80
80
92i
0
0
-92
0
-92ϊ
92
99
19
3
99
19
99
99
order = 6, c3ι = :r 0ooiffooιι*oθ2irι(-4, -3, -2, 1), |C*sp(8ι2)(c3ι)| = 288
7686464
6464
0
2
2i-2
-2i
000
00
0
-4
4t4
-4t
-1
-5-5
-5-5
0
00
00
0
4-4
4-4
0
2-2i
-2
2t
0
00
00
order = 6, c32 = :rooιo#oooι*oi2in(l, 4, 3, -2), |Cί
Sp(8)2)(c32)| = 4320480
16
48016
1616
0-4
0
-4»
4i4
26
-2-6
-66
00
00
00
1-15
1-15
-15-15
024
024t
-24i-24
2-10-210
10-10
0-20
020t
-20t20
436
436
3636
order = 6, c33 = xoi2in(-l, -3, 2, -4), |C5p(8t2)(c33)| = 1152384384
643232323264
000
2 + 2t-2 - 2t-2 -|- 2t
2 - 2i0
00
-4t4i4i
-4i-4i
4i
000
-4 + 4i4 - 4 t4 + 4i
-4 - 4t0
2-2
-10-6-6-6-6
-10
000
-2-2t2 + 2t2 - 2 t
-2 -f 2t0
00
16t0000
-16t
000
-2 + 2i2 - 2t2 + 2i
-2 - 2i0
33
19-5-5-5-519
CODE POLYNOMIALS IN GENUS 4 63
order = 6, 034 = xoooι^oon^oιιι^oo2i^oi2i^(l» 2,3, -4),480
8888
256256
04 - 4 t4 + 4t
-4 - 4i-4 + 4t
00
0-16t
I6iIβi
-16t4t
-4i
0-24 - 24t-24 -f 24t
24 - 24ι24 + 24i
00
2-62-62-62-62-10-10
0-68 -|- 68t-68 - 68t
68 -f 68t68 - 68t
00
<?Sp(β,2)(c34)| = 69120
0128t
-128t-128t
128t-16t
16i
0108 + 108t108 - 108t
-108+ 108i-108- 108t
00
31631631631631919
order = 6, c35 = 0:0001^0111^0021^0121^(3, -4, -2,1), |C5p(8,2)(c35)| = 86476864
6464
64
0-2
22i
-2i
00
000
0000
0
-1
3
33
3
00
00
0
0-4-4
44
02
-22t
-2t
00
00
0
order = 6, 035 = ^oooι^oon^oιιι^oo2i^oi2i^(4, —2, — 1, -3), \CSp
128256
128256128128
-1 - t0
1 - t0
l + i-1 + f'
ti
— i— I
i— i
-1 + i0
1 + i0
1 - t-1 - i
3-1
3-1
33
-2 - 2i0
2-2 i0
2 + 2i-2 + 2t
0
2i0
-2t00
8.2)(c3β)| = 144-2 + 2t
0
2 + 2i0
2 - 2 t-2 - 2i
4-2
4-2
44
order = 6, c3? = (acoooιn(-4 f -2, -1, -3))3, |Cr
Sp(8|2)(c37)| = 1296256256128128
128
128
00
-1 - t
-1 + i1 + i1 - t
t— i
i— i
i
— i
00
3-3i3 + 3i
-3 + 3i-3 - 3i
-1-1
-5-5
-5
-5
00
2 + 2t2 - 2i
-2 - 2i-2 + 2i
2i-2i
-8i
Si-Si
Si
00
-6 + 6t-6 - 6i
6 -6i6 + 6i
-2-2
444
4
order = 6, c38 = (a"oooiaOoii^oii i^oi2in(l, -4, 2,3))2, |C'5p(8(2)(c38)| = 43264
38464
38464
64
20
-2i
0-2
2i
3
1-3-1
3
-3
60
6t0
-6
-6i
9
191
99
12
0-12t
0-12
12t
14-2
-142
14
-14
16
0I6i
0-16
-16t
18-218
-218
18
order = 6, c3g = (̂ oooin(4, 1, -2,3))
2, |C
5p(8|2)(c39)| = 288
256
256
256
256
-1
i
1
— t
0
0
0
0
-1
— t
1
t
1
1
1
1
0
0
0
0
-2
2
-2
2
2
2t
-2
-2t
0
0
0
0
order = 6, 040 = (a7ooιo^ooii^oin^oo2ia:oi2in(-2, -4,1,3))2, |C
f5p(8f2)(c4θ)| = I
728
256
256
256
256
1
— t
t
-1
0
0
0
0
-3
-3i
3i
3
-3
-3
-3
-3
0
0
0
0
2
-2
-2
2
2
2i
-2i
-2
0
0
υ0
64 M. OURA
order = 7, c4J = α:0oiin(-4, 2, 1, -3), |CSp(8ι2)(c4ι)| =42
64
38464
38464
64
-2
02
0-2i
2i
1
-1
1
1-1
-1
0
00
00
0
0
00
00
0
0
00
00
0
0
00
00
0
2
02
02i
-2i
4
04
044
order = 8, c42 = ^oioo^ooio^oooi, |C'sp(8,2)(c42)| = 384
32
19232
19232
32512
2- 2i
02-1- 2i
0-2 + 2i
-2 - Ίi0
-2i2t2i
-2t
-2f2t0
2 + 2ι
02- 2t
0-2 - 2i
-2 + Ίi0
80808
80
6-6t0
6 + 6i
0-6 + 6t-6-6t
0
-2i
2i2i
-2i-2i
2i0
6 + 6t
06-6«
0-6 - 6z
-6 + 6t0
14
-214-214
142
order = 8, 043 = oroioo^ooio^oooi^iiiij |^5p(8,2)(c43)| — 128
64
6464
64512128128
2
-2i2i
-2000
2
-2-2
20
-22
2
2t-2i
-2000
4
444000
6
-6t6t
-6000
6-6-6
602
-2
6
6i-6z-6
000
6
66
62
-2-2
order = 8, c4* = xoioo^ooio^oooi^iooo, |C'sp(8,2)(c44)| = 32128128
128128512
1 - ί
1-H
-! + «'-1 - t
0
00000
1 + i1 - i
-I -i
-1+f0
22220
1 - t
1 + f
-1 + f-1 - f
0
00
00
0
1 + f1 -i
-1 -f
-1+i0
22
22
-2
order = 8, c45 = ^oioo^ooio^oooi^ooii , I^Spίβ^ί^s)! = 38432
19232
1923232
512
2 - 2»0
2 + 2t'0
-2 - 2t-2 -f 2i
0
-6ί-2t'
6t2i6»
-6t0
-6 - 6i0
-6 + 6i0
6-6t6-|-6t'
0
-12_ 4
-12-4
-12-12
0
-10 -f Wi0
-10 - lOt0
10 -f Wi10 - 10ί
0
18t6t
-18i-6i
-I8i18i
0
14 -f 14<0
14 - 14t0
-14-f 14ι'-14- 14t
0
226
226
2222
2
order _ 8, c4β = #oιoo^ooιo^oooι#ooιι#ιooo> |C^5p(8f2)(c46)| = 32128
128128
128512
1 - t
1 + i-1 - t
-1 + i0
-2i
2i2i
-2i0
-1 -i
-1 + ί1 - i
1 + i0
-2
-2-2
-20
-1 + i-1 - ί
1 + i1 - ί
0
2f
-2ί-2ί
2f0
1+i1 -ί
-1 + i-1 - ί
0
2
22
22
CODE POLYNOMIALS IN GENUS 4 65
order = 8, 047 = ίCθl00^0010^0001ίPθ011^1110^1111^2221» I^Sρ(8,2)(c47)| = 128
6464
6464
512128
128
2-2t
2t-2
00
0
2-2
-22
0-2
2
2
2t-2t-2
00
0
00
00
04
4
-22t
-2t2
00
0
-22
2-2
0-6
6
-2-2i
2t2
00
0
-2-2
-2-2
26
6
order = 9, c48 = :roooira(-4, -l,3,-2), \CS
64
6464
384384
64
2t2
-2t
00
-2
o
3-3-1
1
3
-4t4
4t00
-4
5
55
11
5
6t
6
-6t00
-6
Pί8,2)(c48)| = 54
-7
7-7
-11
7
-8t
8
8i00
-8
9
991
1
9
order = 9, 049 = #ιιii^ii2iπ(-4, -1, -3, -2), \CSp(St
256
256256
256
— t
t1
-1
000
0
— tt
-1
1
-1
-1-1
-1
0
00
0
-1
-11
1
2)(C49)| = 27
I
—i1
-1
0
00
0
order = 10, c50 = xooιm(-4, -2,-lf-3), |C'sp(8,2)(c5θ)| = 8051264
12864
64
12864
0-2t
0
2t2
0-2
0-2
2-2
2
-22
0
2ι0
-2t
2
0-2
-1333
333
0-2ι
02i
2
0-2
0040
0
-4O
0000
000
1
151
151
order = 10, c5ι = xooiιn(-4, -1, -3, -2), |CSp(8t2)(c5ι)| = 20
256256256
256
I
— i
1-1
000
0
000
0
000
0
— t
t-1
1
1
1-1
-1
000
0
0
000
order = 10, c52 = x0oiin(-4, 2, -1, -3), \CSp(Bι2)
32
384256
32256
3232
2 + 2i
00
-2 - 2i
0-2 + 2i
2 - 2ί
4i
0-2i
4i2i
-4t
-4t
-2 + 2t00
2- 2i0
2 + 2t-2 - 2ι
11
-3
1-3
1
1
2 + 2i00
-2 - 2i
0-2 + 2i
2- 2i
(c52)| = 2400
04t
0
-4i00
4-4t00
-4 + 4t0
-4 - 4t
4 + 4i
9
15
95
99
66 M. OURA
order = 10, c5s = ̂ oooiΛ oonn(2, 1,4, -3), |C
Sp(8(2)(c53)| = 240
64
768
64
64
64
-2
0
-2i
2i
2
4
0
-4
-4
4
-6
0
6t'
-6i
6
9
1
9
9
9
-10
0
-lOi
lOt'
10
12
0
-12
-12
12
-12
0
12«
-I2i
12
13
1
13
13
13
order = 12, c54 = x0ooin(-4, -3, 2, 1), |CsP(8,2)(c54)| = 48
512
128128
128128
0
-1 -i
-1+t1 + ί1 -t
0
00
00
0
00
00
1
-1
-1
-1-1
0
00
00
0
2i
-2i
2i-2i
0
1 - i
1 + t
-i + i-1 - t
0
00
00
order = 12, 055 = xoooιn(-2, 1, -3, -4), |Cf
5p(8|2)(c55)| = 96
256128
25664
12864
64
64
00
02i
02
-2i
-2
0-2
0-2
22
-2
2
00
00
00
0
0
22
-2-2
2-2
-2
-2
00
0-2i
0-2
2t
2
00
00
00
0
0
00
0-2i
02
2ί
-2
3-1
33
-1
3
3
3
order = 12, c56 = #0001̂ (4, 1, -2,3), \C
Sp(
256
256
256
256
— t
-1
t
1
0
0
0
0
— t
1
t
-1
-1
-1
-1
-1
0
0
0
0
8,2)(C56)| = 24
0
0
0
0
0
0
0
0
0
0
0
0
order = 12, c57 = :r0oil^oo2in(-4, -1,3, -2), |CsP( 8 ,2)^57)1 = 48128
128128128512
1 - i
1 + i-1+t-1 - i
0
-2i
2i-2i
2i0
-2 - 2i
-2 + 2t2 + 2t
2-2ί0
-3-3-3
-3-1
-2 + 2t-2 - 2i
2-2i
2 + 2i0
4ί-4ι
4i
-4t0
3 + 3t3-3t
-3 - 3t'
-3 + 3ί0
4
44
40
order = 12, c58 = xoiii#oθ2ira(2,3, 1, -4), \CSp(8>2)(c5s)\ = 144128128128512128
-1 -i1 + i1 -i
0
2i2i
0-2t
00000
111
-11
-2 - 2i
2-2i
0-2 + 2i
0
00
00
1 - i
0-1 -i
4
4404
order = 12, 059 = xoooι^ooiizΌiιi27oi2in(l, —4,2,3), |C75p(8f2)(c59)| = 72
64384
64384
6464
-2i0
20
2i-2
-3-1
31
-33
2t020
-2i-2
11
11
11
0000
00
-2-2
22
-22
4t040
-4i-4
6262
66
CODE POLYNOMIALS IN GENUS 4 67
order = 12, c6o = zoooι*ooιixoi2i™(-l, -2,4, -3), iCs^β^ί^Go)! = 24
256
128256
128128
128
0
-1 - t0
1 - i
1 + i
-1+i
I
i— i
— i
i
— i
0
1 - i
0
-1 - t
-1 + i
1 + i
-1
-1-1
-1
-1
-1
0
00
00
0
-2i
02i
00
0
0
00
00
0
2
02
00
0
order = 12, c6ι = r0oil^olii^o02i^oi2in(l,3, -2, -4), |C5p(8)2)(c6ι )| = 57651232
19232
1923232
0-2 + 2t'
0-2-2i
02 + 2t2-2t '
0-6t-2t
6t2t6t
-6t
08 + 8t
08-8t
0-8 + 8t-8 - 8t'
-2-18-2
-18-2
-18-18
018 - 18t
018 + 18t
0-18- 18t-18 + 18t'
032t'
0-32i
0-32t
32t
0-26 - 26t
0-26 + 26t'
026 - 26t'26 + 26t
339-139-13939
order = 12, cβ2 = ^0111^0121 n(l, 4, 3, -2), \CSp
32192
32512
19232
32
-2 - 2i
0
2 + 2t'0
02 - 2t
-2 + 2t'
2t-2i
2i
0
2t-2i
-2t
00
00
00
0
-2-2
-22
-2-2
-2
2 + 2i
0
-2 - 2i0
0
-2 + 2i
2 - 2t
f8,2)(c62)| = 576
00
00
00
0
-2 + 2t*0
2-2t'0
0
2 + 2ϊ
-2 - 2i
7-1
7
3-1
7
7
order = 12, CGS = tfoooι#ooιια;oιiιsoi2in(4, -3, 1, -2), \CSp(
128
128512
128128
0
1 -i-1 - i
0
00
00
-2 - 2i
2-2i
02 + 2i
-2 + 2i
3
31
33
0
00
00
2i
0
8,2)(C63)| = 144
-1 - i
1 - i0
0
0000
order = 12, c64 = (#oooi^ooιι^oθ2in(-4, -1, -2, -3))2, |Cί5p(8ι2.)(c64)| = 152
64
76864
6464
2t0
-2
2-2i
000
00
4i04
-4
-4i
-2-2-2
-2-2
6t'0
-6
6
-6i
-808
8-8
2i02
-2
-2i
-9
3-9
-9-9
order = 12, c65 = (xoιιι*oθ2in(-4 f -2, -3, -I))2, |Cί
5p(8t2)(c65)| = 384
64
51264
6464
256
2t0
-2
-2i2
0
-4
04
-44
0
-4t0
-4
4i4
0
626
66
-2
6i0
-6
-6i6
0
-808
-88
0
-6t'0
-6
6i6
0
7
37
77
3
68 M. OURA
order = 12, c6e = ^ooιo^ooιιxoιιι^oo2ia7oi2in(~2, -4, 1,3), \CSp(&t2)(cββ)\ = 144256256256256
-1— i
i
1
0000
-1
i— t
1
11
11
0000
2-2-2
2
-22t
-2t2
0000
order = 14, c67 = xooιιn(-4, -2, l,-3), |CSp(βι2)(cβ7)| = 14256128128128256128
0
1+i-1 - t
-1 + i0
1 - t
— tt
t— i
i— i
000000
000000
000000
000000
01 - t
-1 - f t-1 - t
0
1 + i
02
22
02
order = 15, c6β = ^ooiι^(-4, 2, -3, 1), \CSj
64
64384
38464
64
2
-20
0
-2i
2i
1
1-1
1
-1
-1
-2
20
0-2t
2i
-4
-40
0-4
-4
-4
40
0
4i-4i
X8.2)(C68) | = 90
-1
-1
1
-1
1
1
4
-40
0
4i-4t
7
7
-1
-17
7
order = 15, c69 = (*oonzoo2in(2, -4, -1, -3))2, |C5p(8
256256
256256
-1
1t
— t
11
-1-I
-22
-2t
2i
22
22
1
1
t
— i
22
-2-2
2)(C69)|=90
-22
-2t
2i
1
1
11
order = 15, c70 = a7ooiι̂ (-4, -1, -2, -3), |C'
Sp(8f2)(c7o)| = 15
256
256
256
256
— i
i
I1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
order = 17, c7J = xoooι^ooιιn(-4, -1, -2,3), |C
f
5ί,(8|2)(c
7ι)| = 17
256
256
256
256
1
i
-1
— t
1
-1
1
-1
1
— i
-1
t
1
1
1
1
1
t
-1
— i
1
-1
1
-1
1
— i
-1
t
1
1
1
1
order = 17, c72 = (*oooι*ooιιπ(-4, -1, -2,3))3, |C5p(8
256256
256256
1
i
-1
— i
1
-1
1
-1
1
— t-1
i
11
1
1
1
i-1
— t
1-1
1
-1
2)(C72)| = 17
1
— i-1
t*
1
1
1
1
CODE POLYNOMIALS IN GENUS 4 69
order = 18, c73 = x
0ooιn(-4, -2, -1, -3), |C
Sp(8|2)(c
73)| = 18
256
256
128
128
128
128
0
0
1 - t
-1 -i
-1+i
1 + i
— ii
— i
i
— i
i
0
0
0
0
0
0
-1
-1
1
1
1
1
0
0
1 - t
-1 -i
-1+i
1 + i
i
— i
— i
i
— i
i
0
0
0
0
0
0
1
1
1
1
1
1
order = 20, en - *oooi*ooiin(-2, 1, -4,3), |Csp(8,2)(c74)| = 40
512128
128128128
0
-1 - t1 -i
0
2i
2i
0
1 - t-1 - t
-1-1
-1
-1-1
01 - i
-1 - t
0
2t
2i
0-2 - 2i
-2 + 2i2 + 2i
2 - 2i
13
333
order = 20, c75 = a?ιma?ιi2in(-4, -1,3, -2), |C5
128
128512
128128
1 + i-1 -i
0
-1+i1 - t
0
0000
1 - t
-1 + i0
-1 - t1 + i
1
1
11
1
-1 - t
1 + i0
1-t
-1 + i
pfβ.2)(c75)| =40
-2i-2i
0
2t2i
0
00
00
-1
-1
1-1
-1
order = 21, c?e = #oooι#ooιiπ(-4, -1, -3, -2), \C SP[B ,2)(C7*)\ = 21
256
256
256
256
1
-1
— t
i
I
1
-1
-1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
-1
t'
— t
1
1
1
1
order = 24, 077 = ίΌooi^oon^oθ2in(-4, -1, -2, -3), |Cf
5p(8t2)(c77)| = 48
128512
128128
128
-1 - i
0
-1 + i1 - t
1 + i
00
00
0
00
00
0
00
00
0
-1 -i
0
-1 + f1 -t
1+i
00
000
-1 + i0
-1 -i
1 + i1 - i
1-1
1
11
order = 24, c78 = a?om*oo2in(-4, -2, -3, -1), |CSp(8|2)(c78)| = 48
128
512128
128128
1 -i
0-1 - i
-1 + i1+i
-2t0
2i-2t
2t
-2 - 2i
02 - 2i
2 + 2i-2 + 2i
-4
0-4
-4-4
-3 + 3t
03 + 3i
3-3i-3 - 3i
4i
0-4i
4ί
-4i
3 + 3i0
-3 + 3i-3 - 3i
3-3i
5
-15
55
order = 30, c79 = *ooii*oo2in(2, -4, -1, -3), |C
5p(βi2)(c79)| = 30
256
256
256
256
1
t
— i
-1
1
-1
-1
1
0
0
0
0
0
0
0
0
-1
— t
t
1
0
0
0
0
0
0
0
0
1
1
1
1
70 M. OURA
order = 30, c80 = zooiin(-4, -2, -3, 1), |CsP(8,2)(c8θ)| = 30256
128128
128256
128
0
-1 - t
1 + t-1 + t
0
1 -i
-
-
-
0
1 - t-1 + i
1 + t'0
-1 -i
0
-2-2
-20
-2
0
2 + 2ϊ-2 - 2i
2 - 2t0
-2 + 2t
— t-3i
-3i
3ii
3i
0
-2 + 2x2-2ι
-2 - 2t0
2 + 2i
-1
33
3-1
3
REMARK. The determination of the conjugacy classes of Chevalley groupswere studied by several authors. For example, see [2], [6], [15].
4. Main result
We have obtained the surjective homomorphism
φ:H4-+Sp(*92)
with Kerφ = ΛΓ4 and a set {ci}0<i<8o of representatives of conjugacy classes of>$p(8,2) in the preceding sections. Our main result can be stated as follows:
Theorem 4.1. The Molien series of H4 is given by
+ 87/44 + 172f48 + 279ί52 + 550/56 + 960ί60 + 1782f64
where
D = (1 - - ί24)4(l - ί28χi - - r48)(l - r56χi - ί60χi - tβs)
x 1 - ί
4- 157f52 + 335ί56 + 549f60 + 1094ί64-h 186 U68 + 350H72 + 5965Γ6
4- 10728ί80 + 18041f84 + 31051/88 + 51025ί92 + 84427f96 + 134865ί100
1688292/124-f2447124^1284-3487706/132+4922301ί136
CODE POLYNOMIALS IN GENUS 4 71
+ 105549085/180+132161437f184 +
+ 247823660* 1 96 + 301 389903f 20° + 363960630* 204 + 4368 14071 f208
+ 1730214602ί244+1963201767?248 + 2216376776<2
+ 2785299743f260 + 31009837 10ί264 + 3436532034/268 + 3792023955*272
+ 7625054128ί308 + 8080605429?312 + 8530781308/316 + 8973489231ί320
+ 9404047193/324 + 9820361028ί328 + 10217690359ί332 + 10594101406ί336
+ 10944974102ί340 + 1 1268698558/344 + 1 1560953224ί348 + 1 1820605627/352
+ 12545212616/372 + 12566910422?376 + 12545212616ί380 + 12481889419ί384
+ 12375970644/388 + 12230003475ί392 + 12043796179/396 + 1 1820605627ί400
+ 11560953224ί404 + 11268698558/408 + 10944974102ί412 + 10594101406ί416
+ 10217690359ί420 + 9820361028ί424+9404047193ί428 + 8973489231/432
+ 853078 1308/436 + 8080605429/440 + 7625054128;444 + 7168581264?448
+ 6713128315ί452 + 6262771 875/456 + 5819175192ί460 + 53859171 15ί464
+ 496428443 1 ί468 + 4557273329ί472 + 41 6573 11 23ί476 + 3792023955/480
+ 3436532034ί484 + 3100983710ί488 + 2785299743ί492 + 2490597453<496
+ 2216376776/500+1963201767ί504+1730214602ί508
+ 1323941514/516+1149232929ί520
+ 363960630*548 + 301389903/552 +247823660ί556 + 202451 163ί560
+ 164140047ί564+132161437ί568 + 105549085ί572 + 83679250<576
+ 65756890ί580 + 51272203?584 + 39594318/588 + 30324507/592 + 22980000ί596
+ 17262549/600 + 12816307ί604 + 9427941ί608 + 6845055ί612+4922301ί616
+ 3487706ί620 + 2447124/624 + 1688292ί628 + 1 153627ί632 + 773052*636
72 M. OURA
12f 72° + f
724 + 3ί
728 + 2ί
736 - <
740 + ί
744 + ί
752. Π
References
[1] M. Broue and M. Enguehard: Polynόmes des poids de certains codes et functions theta de certainsreseaux, Ann. Scien. EC. Norm. Sup. 5 (1972), 157-181.
[2] B. Chang: The conjugate classes ofChevalley groups of type (G2\ J. Algebra 9 (1968), 190-211.[3] Conway, Curtis, Norton, Parker and Wilson: Atlas of finite groups, Oxford University
Press, 1985.[4] W. Duke: On codes and Siegel modular forms, Int. Math. Research Notices (1993), 125-136.[5] W. Ebeling: Lattices and Codes, A course partially based on lectures by F. Hirzeburch,
Vieweg, 1994.[6] H. Enomoto: The conjugacy classes of Chevalley groups of type (G2) over finite fields of
characteristic 2 or 3, J. Fac. Sci. Univ. Tokyo Sect. IA (1970), 497-512.[7] S.P. Glasby: On the faithful representations, of degree 2", of certain extensions of 2-groups by
orthogonal and symplectic groups, J. Austral. Math. Soc. (Series A) 58 (1995), 232-247.[8] W.C. Huffman: The biweight enumerator of selforthogonal binary codes, Disc. Math. 26
(1979), 129-143.[9] B. Runge: Codes and Siegel modular forms, Disc. Math. 148 (1996), 175-204.
[10] B. Runge: On Siegel modular forms, part I, J. Reine angew. Math. 436 (1993), 57-85.[11] B. Runge: On Siegel modular forms, part II, Nagoyya Math. J. 138 (1995), 179-197.[12] B. Runge: The Schottky ideal, in "Abelian Varieties" (Proceedings of the International Con-
ference held in Eggloffstein, Germany, October, 1993), Walter de Gruyter, Berlin-New York,1995, pp. 251-272.
[13] M. Schόnert, et. al: GAP: Groups, Algorithms and Programing, Lehrstuhl D fur Mathemtik,RWTH Aachen, 1994.
[14] G.C. Shephard and J.A. Todd: Finite unitary reflection groups, Canad. J. Math. 6 (1954),274-304.
[15] K. Shinoda: The conjugacy classes of Chevalley groups of type (F4) over finite fields ofcharacteristic 2, J. Fac. Sci. Univ. Tokyo 21 (1974), 133-159.
[16] N.J.A. Sloane: Error-correcting codes and invariant theory:New applications of a nineteenth-century technique, Amer. Math. Mathly 84 (1977), 82-107.
Graduate School of MathematicsKyushu UniversityHakozaki 6-10-1, Higashi-ku,Fukuoka, 812-81, Japane-mail address: [email protected]
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