# Extremal Unimodular Lattices in Dimension 36

Authors: Masaaki Harada
Journal: International Journal of Combinatorics
 (2014)

---

## Abstract

New extremal odd unimodular lattices in dimension 36 are constructed. Some new odd unimodular lattices in dimension 36 with long shadows are also constructed.


---

## Body

## 1. Introduction

A (Euclidean) latticeL
⊂
R
n in dimension n isunimodular if L
=
L
*, where the dual lattice L
* of L is defined as {
x
∈
R
n
∣
(
x
,
y
)
∈
Z for all y
∈
L
} under the standard inner product (
x
,
y
). A unimodular lattice is calledeven if the norm (
x
,
x
) of every vector x is even. A unimodular lattice, which is not even, is calledodd. An even unimodular lattice in dimension n exists if and only if n
≡
0 (mod 8), while an odd unimodular lattice exists for every dimension. Two lattices L and L
′ areisomorphic, denoted by L
≅
L
′, if there exists an orthogonal matrix A with L
′
=
L
·
A, where L
·
A
=
{
x
A
∣
x
∈
L
}. The automorphism group Aut
(
L
) of L is the group of all orthogonal matrices A with L
=
L
·
A.Rains and Sloane [6] showed that the minimum norm min
⁡
(
L
) of a unimodular lattice L in dimension n is bounded by min
⁡
(
L
)
≤
2
n
/
24
+
2 unless n
=
23 when min
⁡
(
L
)
≤
3. We say that a unimodular lattice meeting the upper bound isextremal.The smallest dimension for which there is an odd unimodular lattice with minimum norm (at least)4 is 32 (see [1]). There are exactly five odd unimodular lattices in dimension 32 having minimum norm 4, up to isomorphism [7]. For dimensions 33, 34, and 35, the minimum norm of an odd unimodular lattice is at most 3 (see [1]). The next dimension for which there is an odd unimodular lattice with minimum norm (at least) 4 is 36. Four extremal odd unimodular lattices in dimension 36 are known, namely, Sp4(
4
)D8.4 in [1], G
36 in [2, Table 2], N
36 in [3, Section 3], and A
4
(
C
36
) in [4, Section 3]. Recently, one more lattice has been found, namely, A
6
(
C
36,6
(
D
18
)
) in [5, Table II]. This situation motivates us to improve the number of known nonisomorphic extremal odd unimodular lattices in dimension 36. The main aim of this paper is to prove the following.Proposition 1.
There are at least 26 nonisomorphic extremal odd unimodular lattices in dimension 36.The above proposition is established by constructing new extremal odd unimodular lattices in dimension36 from self-dual Z
k-codes, where Z
k is the ring of integers modulo k, by using two approaches. One approach is to consider self-dual Z
4-codes. Let B be a binary doubly even code of length 36 satisfying the following conditions:
(1)
the
minimum
weight
of
B
is
at
least
16
,
the
minimum
weight
of
its
dual
code
B
⊥
is
at
least
4
.
Then a self-dual Z
4-code with residue code B gives an extremal odd unimodular lattice in dimension 36 by Construction A. We show that a binary doubly even [36, 7] code satisfying conditions (1) has weight enumerator 1
+
63
y
16
+
63
y
20
+
y
36 (Lemma 2). It was shown in [8] that there are four codes having the weight enumerator, up to equivalence. We construct ten new extremal odd unimodular lattices in dimension 36 from self-dual Z
4-codes whose residue codes are doubly even [36, 7] codes satisfying conditions (1) (Lemma 4). New odd unimodular lattices in dimension 36 with minimum norm 3 having shadows of minimum norm 5 are constructed from some of the new lattices (Proposition 7). These are often called unimodular lattices with long shadows (see [9]). The other approach is to consider self-dual Z
k-codes (
k
=
5,6
,
7,9
,
19
), which have generator matrices of a special form given in (14). Eleven more new extremal odd unimodular lattices in dimension 36 are constructed by Construction A (Lemma 8). Finally, we give a certain short observation on ternary self-dual codes related to extremal odd unimodular lattices in dimension 36.All computer calculations in this paper were done by Magma [10].


## 2. Preliminaries

### 2.1. Unimodular Lattices

LetL be an odd unimodular lattice and let L
0 denote the even sublattice, that is, the sublattice of vectors of even norms. Then L
0 is a sublattice of L of index 2 [7]. Theshadow
S
(
L
) of L is defined to be L
0
*
∖
L. There are cosets L
1
,
L
2
,
L
3 of L
0 such that L
0
*
=
L
0
∪
L
1
∪
L
2
∪
L
3, where L
=
L
0
∪
L
2 and S
=
L
1
∪
L
3. Shadows for odd unimodular lattices appeared in [7] and also in [11, p. 440], in order to provide restrictions on the theta series of odd unimodular lattices. Two lattices L and L
′ areneighbors if both lattices contain a sublattice of index 2 in common. If L is an odd unimodular lattice in dimension divisible by 4, then there are two unimodular lattices containing L
0, which are rather L, namely, L
0
∪
L
1 and L
0
∪
L
3. Throughout this paper, we denote the two unimodular neighbors by
(2)
N
e
1
L
=
L
0
∪
L
1
,
N
e
2
(
L
)
=
L
0
∪
L
3
.The theta seriesθ
L
(
q
) of L is the formal power series θ
L
(
q
)
=
∑
x
∈
L
‍
q
(
x
,
x
). The kissing number of L is the second nonzero coefficient of the theta series of L, that is, the number of vectors of minimum norm in L. Conway and Sloane [7] gave some characterization of theta series of odd unimodular lattices and their shadows. Using [7, (
2
)
,
(
3
)], it is easy to determine the possible theta series θ
L
36
(
q
) and θ
S
(
L
36
)
(
q
) of an extremal odd unimodular lattice L
36 in dimension 36 and its shadow S
(
L
36
):
(3)
θ
L
36
q
=
1
+
(
42840
+
4096
α
)
q
4
+
(
1916928
-
98304
α
)
q
5
+
⋯
,
(4)
θ
S
L
36
q
=
α
q
+
(
960
-
60
α
)
q
3
+
(
3799296
+
1734
α
)
q
5
+
⋯
,
respectively, where α is a nonnegative integer. It follows from the coefficients of q and q
3 in θ
S
(
L
36
)
(
q
) that 0
≤
α
≤
16.


### 2.2. Self-DualZ
k-Codes and Construction A

LetZ
k be the ring of integers modulo k, where k is a positive integer greater than 1. A Z
k
-code
C of length n is a Z
k-submodule of Z
k
n. Two Z
k-codes areequivalent if one can be obtained from the other by permuting the coordinates and (if necessary) changing the signs of certain coordinates. A code C isself-dual if C
=
C
⊥, where the dual code C
⊥ of C is defined as {
x
∈
Z
k
n
∣
x
·
y
=
0 for all y
∈
C
}, under the standard inner product x
·
y.IfC is a self-dual Z
k-code of length n, then the lattice
(5)
A
k
(
C
)
=
1
k
x
1
,
…
,
x
n
∈
Z
n
x
1
mod
⁡
k
,
…
,
x
n
mod
⁡
k
∈
C
is a unimodular lattice in dimension n. This construction of lattices is called Construction A.


## 2.1. Unimodular Lattices

LetL be an odd unimodular lattice and let L
0 denote the even sublattice, that is, the sublattice of vectors of even norms. Then L
0 is a sublattice of L of index 2 [7]. Theshadow
S
(
L
) of L is defined to be L
0
*
∖
L. There are cosets L
1
,
L
2
,
L
3 of L
0 such that L
0
*
=
L
0
∪
L
1
∪
L
2
∪
L
3, where L
=
L
0
∪
L
2 and S
=
L
1
∪
L
3. Shadows for odd unimodular lattices appeared in [7] and also in [11, p. 440], in order to provide restrictions on the theta series of odd unimodular lattices. Two lattices L and L
′ areneighbors if both lattices contain a sublattice of index 2 in common. If L is an odd unimodular lattice in dimension divisible by 4, then there are two unimodular lattices containing L
0, which are rather L, namely, L
0
∪
L
1 and L
0
∪
L
3. Throughout this paper, we denote the two unimodular neighbors by
(2)
N
e
1
L
=
L
0
∪
L
1
,
N
e
2
(
L
)
=
L
0
∪
L
3
.The theta seriesθ
L
(
q
) of L is the formal power series θ
L
(
q
)
=
∑
x
∈
L
‍
q
(
x
,
x
). The kissing number of L is the second nonzero coefficient of the theta series of L, that is, the number of vectors of minimum norm in L. Conway and Sloane [7] gave some characterization of theta series of odd unimodular lattices and their shadows. Using [7, (
2
)
,
(
3
)], it is easy to determine the possible theta series θ
L
36
(
q
) and θ
S
(
L
36
)
(
q
) of an extremal odd unimodular lattice L
36 in dimension 36 and its shadow S
(
L
36
):
(3)
θ
L
36
q
=
1
+
(
42840
+
4096
α
)
q
4
+
(
1916928
-
98304
α
)
q
5
+
⋯
,
(4)
θ
S
L
36
q
=
α
q
+
(
960
-
60
α
)
q
3
+
(
3799296
+
1734
α
)
q
5
+
⋯
,
respectively, where α is a nonnegative integer. It follows from the coefficients of q and q
3 in θ
S
(
L
36
)
(
q
) that 0
≤
α
≤
16.


## 2.2. Self-DualZ
k-Codes and Construction A

LetZ
k be the ring of integers modulo k, where k is a positive integer greater than 1. A Z
k
-code
C of length n is a Z
k-submodule of Z
k
n. Two Z
k-codes areequivalent if one can be obtained from the other by permuting the coordinates and (if necessary) changing the signs of certain coordinates. A code C isself-dual if C
=
C
⊥, where the dual code C
⊥ of C is defined as {
x
∈
Z
k
n
∣
x
·
y
=
0 for all y
∈
C
}, under the standard inner product x
·
y.IfC is a self-dual Z
k-code of length n, then the lattice
(5)
A
k
(
C
)
=
1
k
x
1
,
…
,
x
n
∈
Z
n
x
1
mod
⁡
k
,
…
,
x
n
mod
⁡
k
∈
C
is a unimodular lattice in dimension n. This construction of lattices is called Construction A.


## 3. From Self-DualZ
4-Codes

From now on, we omit the term “odd” for odd unimodular lattices in dimension36, since all unimodular lattices in dimension 36 are odd. In this section, we construct ten new nonisomorphic extremal unimodular lattices in dimension 36 from self-dual Z
4-codes by Construction A. Five new nonisomorphic unimodular lattices in dimension 36 with minimum norm 3 having shadows of minimum norm 5 are also constructed.


### 3.1. Extremal Unimodular Lattices

EveryZ
4-code C of length n has two binary codes C
(
1
) and C
(
2
) associated with C:
(6)
C
(
1
)
=
c
mod
⁡
2
∣
c
∈
C
,
C
(
2
)
=
c
mod
⁡
2
∣
c
∈
Z
4
n
,
2
c
∈
C
.
The binary codes C
(
1
) and C
(
2
) are called the residue and torsion codes of C, respectively. If C is a self-dual Z
4-code, then C
(
1
) is a binary doubly even code with C
(
2
)
=
C
(
1
)
⊥ [12]. Conversely, starting from a given binary doubly even code B, a method for construction of all self-dual Z
4-codes C with C
(
1
)
=
B was given in [13, Section 3].The Euclidean weight of a codewordx
=
(
x
1
,
…
,
x
n
) of C is m
1
(
x
)
+
4
m
2
(
x
)
+
m
3
(
x
), where m
α
(
x
) denotes the number of components i with x
i
=
α
(
α
=
1,2
,
3
). The minimum Euclidean weight d
E
(
C
) of C is the smallest Euclidean weight among all nonzero codewords of C. It is easy to see that min
⁡
{
d
(
C
(
1
)
)
,
4
d
(
C
(
2
)
)
}
≤
d
E
(
C
), where d
(
C
(
i
)
) denotes the minimum weight of C
(
i
)
(
i
=
1,2
). In addition, d
E
(
C
)
≤
4
d
(
C
(
2
)
) and A
4
(
C
) has minimum norm min
⁡
{
4
,
d
E
(
C
)
/
4
} (see, e.g., [3]). Hence, if there is a binary doubly even code B of length 36 satisfying conditions (1), then an extremal unimodular lattice in dimension 36 is constructed as A
4
(
C
), through a self-dual Z
4-code C with C
(
1
)
=
B. If there is a binary [
36
,
k
] code B satisfying conditions (1), then k
=
7 or 8 (see [14]).Lemma 2.
LetB be a binary doubly even [36, 7] code satisfying conditions (1). Then the weight enumerator of B is 1
+
63
y
16
+
63
y
20
+
y
36.Proof.
The weight enumerator ofB is written as
(7)
W
B
y
=
1
+
a
y
16
+
b
y
20
+
c
y
24
+
d
y
28
+
e
y
32
+
(
2
7
-
1
-
a
-
b
-
c
-
d
-
e
)
y
36
,
where a, b, c, d, and e are nonnegative integers. By the MacWilliams identity, the weight enumerator of B
⊥ is given by
(8)
W
B
⊥
y
=
1
+
1
16
(
-
567
+
5
a
+
4
b
+
3
c
+
2
d
+
e
)
y
+
1
2
(
1260
-
10
a
-
10
b
-
9
c
-
7
d
-
4
e
)
y
2
+
1
16
-
112455
+
885
a
+
900
b
llllllllll
+
883
c
+
770
d
+
497
e
y
3
+
⋯
.
Since d
(
B
⊥
)
≥
4, the weight enumerator of B is written using a and b:
(9)
W
B
y
=
1
+
a
y
16
+
b
y
20
+
(
882
-
10
a
-
4
b
)
y
24
+
(
-
1638
+
20
a
+
6
b
)
y
28
+
(
1197
-
15
a
-
4
b
)
y
32
+
(
-
314
+
4
a
+
b
)
y
36
.
Suppose thatB does not contain the all-one vector 1. Then b
=
314
-
4
a. In this case, since the coefficients of y
24 and y
28 are -
374
+
6
a and 246
-
4
a, these yield that a
≥
62 and a
≤
61, respectively, which gives the contradiction. Hence, B contains 1. Then b
=
315
-
4
a. Since the coefficient a
-
63 of y
32 is 0, the weight enumerator of B is uniquely determined as 1
+
63
y
16
+
63
y
20
+
y
36.Remark 3.
A similar approach shows that the weight enumerator of a binary doubly even[
36,8
] code B satisfying conditions (1) is uniquely determined as 1
+
153
y
16
+
72
y
20
+
30
y
24.It was shown in [8] that there are four inequivalent binary [
36,7
,
16
] codes containing 1. The four codes are doubly even. Hence, there are exactly four binary doubly even [
36,7
] codes satisfying conditions (1), up to equivalence. The four codes are optimal in the sense that these codes achieve the Gray-Rankin bound, and the codewords of weight 16 are corresponding to quasi-symmetric SDP 2
-
(
36,16,12
) designs [15]. Let B
36
,
i be the binary doubly even [
36,7
,
16
] code corresponding to the quasi-symmetric SDP 2
-
(
36,16,12
) design, which is the residual design of the symmetric SDP 2
-
(
64,28,12
) design D
i in [8, Section 5] (
i
=
1,2
,
3,4
). As described above, all self-dual Z
4-codes C with C
(
1
)
=
B
36
,
i have d
E
(
C
)
=
16
(
i
=
1,2
,
3,4
). Hence, A
4
(
C
) are extremal.Using the method in [13, Section 3], self-dual Z
4-codes C are constructed from B
36
,
i. Then ten extremal unimodular lattices A
4
(
C
36
,
i
)
(
i
=
1,2
,
…
,
10
) are constructed, where C
36
,
i
(
1
)
=
B
36,2
(
i
=
1,2
,
3
), C
36
,
i
(
1
)
=
B
36,3
(
i
=
4,5
,
6,7
), and C
36
,
i
(
1
)
=
B
36,4
(
i
=
8,9
,
10
). To distinguish between the known lattices and our lattices, we give in Table 1 the kissing numbers τ
(
L
), {
n
1
(
L
)
,
n
2
(
L
)
}, and the orders #
Aut
(
L
) of the automorphism groups, where n
i
(
L
) denotes the number of vectors of norm 3 in N
e
i
(
L
) defined in (2)  (
i
=
1,2
). These have been calculated by Magma. Table 1 shows the following.Table 1
Extremal unimodular lattices in dimension36.
LatticesL
τ
(
L
)
{
n
1
(
L
)
,
n
2
(
L
)
}
# Aut(
L
)
Sp4(4)D8.4 in [1]
42840
{
480,480
}
31334400
G
36 in [2, Table  2]
42840
{
144,816
}
576
N
36 in [3]
42840
{
0,960
}
849346560
A
4
(
C
36
) in [4]
51032
{
0,840
}
660602880
A
6
(
C
36,6
(
D
18
)
) in [5]
42840
{
384,576
}
288
A
4
(
C
36,1
) in Section 3
51032
{
0,840
}
6291456
A
4
(
C
36,2
) in Section 3
42840
{
0,960
}
6291456
A
4
(
C
36,3
) in Section 3
51032
{
0,840
}
22020096
A
4
(
C
36,4
) in Section 3
51032
{
0,840
}
1966080
A
4
(
C
36,5
) in Section 3
51032
{
0,840
}
1572864
A
4
(
C
36,6
) in Section 3
42840
{
0,960
}
2621440
A
4
(
C
36,7
) in Section 3
42840
{
0,960
}
1966080
A
4
(
C
36,8
) in Section 3
42840
{
0,960
}
393216
A
4
(
C
36,9
) in Section 3
51032
{
0,840
}
1376256
A
4
(
C
36,10
) in Section 3
51032
{
0,840
}
393216
A
5
(
D
36,1
) in Section 4
42840
{
144,816
}
144
A
5
(
D
36,2
) in Section 4
42840
{
456,504
}
72
A
6
(
D
36,3
) in Section 4
42840
{
240,720
}
288
A
6
(
D
36,4
) in Section 4
42840
{
240,720
}
576
A
7
(
D
36,5
) in Section 4
42840
{
288,672
}
288
A
7
(
D
36,6
) in Section 4
42840
{
144,816
}
72
A
7
(
D
36,7
) in Section 4
42840
{
144,816
}
288
A
9
(
D
36,8
) in Section 4
42840
{
384,576
}
144
A
19
(
D
36,9
) in Section 4
42840
{
288,672
}
144
A
5
(
E
36,1
) in Section 4
42840
{
456,504
}
72
A
6
(
E
36,2
) in Section 4
42840
{
384,576
}
144Lemma 4.
The five known lattices and the ten extremal unimodular latticesA
4
(
C
36
,
i
)
(
i
=
1,2
,
…
,
10
) are nonisomorphic to each other.Remark 5.
In this way, we have found two more extremal unimodular latticesA
4
(
C
), where C are self-dual Z
4-codes with C
(
1
)
=
B
36,1. However, we have verified by Magma that the two lattices are isomorphic to N
36 in [3] and A
4
(
C
36
) in [4].Remark 6.
ForL
=
A
4
(
C
36
,
i
)
(
i
=
1,2
,
…
,
10
), it follows from τ
(
L
) and {
n
1
(
L
)
,
n
2
(
L
)
} that one of the two unimodular neighbors N
e
1
(
L
) and N
e
2
(
L
) defined in (2) is extremal. We have verified by Magma that the extremal one is isomorphic to A
4
(
C
36
,
i
).Fori
=
1,2
,
…
,
10, the code C
36
,
i is equivalent to some code C
36
,
i
¯ with generator matrix of the form
(10)
I
7
A
B
1
+
2
B
2
O
2
I
22
2
D
,
where A, B
1, B
2, and D are (
1,0
)-matrices, I
n denotes the identity matrix of order n, and O denotes the 22
×
7 zero matrix. We only list in (11) the 7
×
29 matrix M
i
=
A
B
1
+
2
B
2 to save space. Note that O
2
I
22
2
D in (10) can be obtained from I
7
A
B
1
+
2
B
2 since C
36
,
i
¯
(
2
)
=
C
36
,
i
¯
(
1
)
⊥. A generator matrix of A
4
(
C
36
,
i
) is obtained from that of C
36
,
i
(11)
M
1
=
01101111111000000110000333322
10101101001000111110011210223
11000011001110100111100010321
11010100101011010011012032011
00110101010101001101013312302
00000000000001111111112111333
00011111111110000001111213133
,
M
2
=
01101111111000000110002111100
10101101001000111110011210023
11000011001110100111100010123
11010100101011010011012032211
00110101010101001101013312300
00000000000001111111112113131
00011111111110000001111211331
,
M
3
=
01101111111000000110002131300
10101101001000111110011210223
11000011001110100111100010121
11010100101011010011012032011
00110101010101001101013310102
00000000000001111111112131131
00011111111110000001111233331
,
M
4
=
01101111111001111101100220313
10101101001001001110101231032
11000011001110110100102101101
00111001100011100101113010223
11011000011101100011100021110
00000000000001111111112111131
00011111111110000001113301311
,
M
5
=
01101111111001111101102202133
10101101001001001110101231032
11000011001110110100102101101
00111001100011100101113010223
11011000011101100011100221310
00000000000001111111112111131
00011111111110000001113301311
,
M
6
=
01101111111001111101102000313
10101101001001001110101231032
11000011001110110100102101103
00111001100011100101113210223
11011000011101100011100021110
00000000000001111111112313113
00011111111110000001113103333
,
M
7
=
01101111111001111101102202313
10101101001001001110101231030
11000011001110110100102101121
00111001100011100101113210201
11011000011101100011100021112
00000000000001111111112111113
00011111111110000001113301333
,
M
8
=
11111110000111111111000022313
11110001100111000000001130313
01000110111001110100011231021
10011100001011100001101233011
01010101010010101101003301321
00000000011111111111112331111
01111111100000000111111233313
,
M
9
=
11111110000111111111002200111
11110001100111000000001332311
01000110111001110100013211021
10011100001011100001103213011
01010101010010101101000331321
01111111100000000111111211113
00000000011111111111111303313
,
M
10
=
11111110000111111111002202313
11110001100111000000001130311
01000110111001110100011211021
10011100001011100001101213011
01010101010010101101003321123
00000000011111111111112313313
01111111100000000111111211111
.


### 3.2. Unimodular Lattices with Long Shadows

The possible theta series of a unimodular latticeL in dimension 36 having minimum norm 3 and its shadow are as follows:
(12)
1
+
(
960
-
α
)
q
3
+
(
42840
+
4096
β
)
q
4
+
⋯
,
β
q
+
(
α
-
60
β
)
q
3
+
(
3833856
-
36
α
+
1734
β
)
q
5
+
⋯
,
respectively, where α and β are integers with 0
≤
β
≤
α
/
60
<
16 [3]. Then the kissing number of L is at most 960 and min
⁡
(
S
(
L
)
)
≤
5. Unimodular lattices L with min
⁡
(
L
)
=
3 and min
⁡
(
S
(
L
)
)
=
5 are often called unimodular lattices with long shadows (see [9]). Only one unimodular lattice L in dimension 36 with min
⁡
(
L
)
=
3 and min
⁡
(
S
(
L
)
)
=
5 was known, namely, A
4
(
C
36
) in [3].LetL be one of A
4
(
C
36,2
), A
4
(
C
36,6
), A
4
(
C
36,7
), and A
4
(
C
36,8
). Since {
n
1
(
L
)
,
n
2
(
L
)
}
=
{
0,960
}, one of the two unimodular neighbors N
e
1
(
L
) and N
e
2
(
L
) in (2) is extremal and the other is a unimodular lattice L
′ with minimum norm 3 having shadow of minimum norm 5. We denote such lattices L
′ constructed from A
4
(
C
36,2
), A
4
(
C
36,6
), A
4
(
C
36,7
), and A
4
(
C
36,8
) by N
36,1, N
36,2, N
36,3, and N
36,4, respectively. We list in Table 2 the orders #
Aut
(
N
36
,
i
)
(
i
=
1,2
,
3,4
) of the automorphism groups, which have been calculated by Magma. Table 2 shows the following.Table 2
# Aut(N
36
,
i) (i = 1, 2, 3, 4).
LatticesL
# Aut(
L
)
A
4
(
C
36
) in [3]
1698693120
N
36,1
12582912
N
36,2
5242880
N
36,3
3932160
N
36,4
786432Proposition 7.
There are at least 5 nonisomorphic unimodular latticesL in dimension 36 with min
⁡
(
L
)
=
3 and min
⁡
(
S
(
L
)
)
=
5.


## 3.1. Extremal Unimodular Lattices

EveryZ
4-code C of length n has two binary codes C
(
1
) and C
(
2
) associated with C:
(6)
C
(
1
)
=
c
mod
⁡
2
∣
c
∈
C
,
C
(
2
)
=
c
mod
⁡
2
∣
c
∈
Z
4
n
,
2
c
∈
C
.
The binary codes C
(
1
) and C
(
2
) are called the residue and torsion codes of C, respectively. If C is a self-dual Z
4-code, then C
(
1
) is a binary doubly even code with C
(
2
)
=
C
(
1
)
⊥ [12]. Conversely, starting from a given binary doubly even code B, a method for construction of all self-dual Z
4-codes C with C
(
1
)
=
B was given in [13, Section 3].The Euclidean weight of a codewordx
=
(
x
1
,
…
,
x
n
) of C is m
1
(
x
)
+
4
m
2
(
x
)
+
m
3
(
x
), where m
α
(
x
) denotes the number of components i with x
i
=
α
(
α
=
1,2
,
3
). The minimum Euclidean weight d
E
(
C
) of C is the smallest Euclidean weight among all nonzero codewords of C. It is easy to see that min
⁡
{
d
(
C
(
1
)
)
,
4
d
(
C
(
2
)
)
}
≤
d
E
(
C
), where d
(
C
(
i
)
) denotes the minimum weight of C
(
i
)
(
i
=
1,2
). In addition, d
E
(
C
)
≤
4
d
(
C
(
2
)
) and A
4
(
C
) has minimum norm min
⁡
{
4
,
d
E
(
C
)
/
4
} (see, e.g., [3]). Hence, if there is a binary doubly even code B of length 36 satisfying conditions (1), then an extremal unimodular lattice in dimension 36 is constructed as A
4
(
C
), through a self-dual Z
4-code C with C
(
1
)
=
B. If there is a binary [
36
,
k
] code B satisfying conditions (1), then k
=
7 or 8 (see [14]).Lemma 2.
LetB be a binary doubly even [36, 7] code satisfying conditions (1). Then the weight enumerator of B is 1
+
63
y
16
+
63
y
20
+
y
36.Proof.
The weight enumerator ofB is written as
(7)
W
B
y
=
1
+
a
y
16
+
b
y
20
+
c
y
24
+
d
y
28
+
e
y
32
+
(
2
7
-
1
-
a
-
b
-
c
-
d
-
e
)
y
36
,
where a, b, c, d, and e are nonnegative integers. By the MacWilliams identity, the weight enumerator of B
⊥ is given by
(8)
W
B
⊥
y
=
1
+
1
16
(
-
567
+
5
a
+
4
b
+
3
c
+
2
d
+
e
)
y
+
1
2
(
1260
-
10
a
-
10
b
-
9
c
-
7
d
-
4
e
)
y
2
+
1
16
-
112455
+
885
a
+
900
b
llllllllll
+
883
c
+
770
d
+
497
e
y
3
+
⋯
.
Since d
(
B
⊥
)
≥
4, the weight enumerator of B is written using a and b:
(9)
W
B
y
=
1
+
a
y
16
+
b
y
20
+
(
882
-
10
a
-
4
b
)
y
24
+
(
-
1638
+
20
a
+
6
b
)
y
28
+
(
1197
-
15
a
-
4
b
)
y
32
+
(
-
314
+
4
a
+
b
)
y
36
.
Suppose thatB does not contain the all-one vector 1. Then b
=
314
-
4
a. In this case, since the coefficients of y
24 and y
28 are -
374
+
6
a and 246
-
4
a, these yield that a
≥
62 and a
≤
61, respectively, which gives the contradiction. Hence, B contains 1. Then b
=
315
-
4
a. Since the coefficient a
-
63 of y
32 is 0, the weight enumerator of B is uniquely determined as 1
+
63
y
16
+
63
y
20
+
y
36.Remark 3.
A similar approach shows that the weight enumerator of a binary doubly even[
36,8
] code B satisfying conditions (1) is uniquely determined as 1
+
153
y
16
+
72
y
20
+
30
y
24.It was shown in [8] that there are four inequivalent binary [
36,7
,
16
] codes containing 1. The four codes are doubly even. Hence, there are exactly four binary doubly even [
36,7
] codes satisfying conditions (1), up to equivalence. The four codes are optimal in the sense that these codes achieve the Gray-Rankin bound, and the codewords of weight 16 are corresponding to quasi-symmetric SDP 2
-
(
36,16,12
) designs [15]. Let B
36
,
i be the binary doubly even [
36,7
,
16
] code corresponding to the quasi-symmetric SDP 2
-
(
36,16,12
) design, which is the residual design of the symmetric SDP 2
-
(
64,28,12
) design D
i in [8, Section 5] (
i
=
1,2
,
3,4
). As described above, all self-dual Z
4-codes C with C
(
1
)
=
B
36
,
i have d
E
(
C
)
=
16
(
i
=
1,2
,
3,4
). Hence, A
4
(
C
) are extremal.Using the method in [13, Section 3], self-dual Z
4-codes C are constructed from B
36
,
i. Then ten extremal unimodular lattices A
4
(
C
36
,
i
)
(
i
=
1,2
,
…
,
10
) are constructed, where C
36
,
i
(
1
)
=
B
36,2
(
i
=
1,2
,
3
), C
36
,
i
(
1
)
=
B
36,3
(
i
=
4,5
,
6,7
), and C
36
,
i
(
1
)
=
B
36,4
(
i
=
8,9
,
10
). To distinguish between the known lattices and our lattices, we give in Table 1 the kissing numbers τ
(
L
), {
n
1
(
L
)
,
n
2
(
L
)
}, and the orders #
Aut
(
L
) of the automorphism groups, where n
i
(
L
) denotes the number of vectors of norm 3 in N
e
i
(
L
) defined in (2)  (
i
=
1,2
). These have been calculated by Magma. Table 1 shows the following.Table 1
Extremal unimodular lattices in dimension36.
LatticesL
τ
(
L
)
{
n
1
(
L
)
,
n
2
(
L
)
}
# Aut(
L
)
Sp4(4)D8.4 in [1]
42840
{
480,480
}
31334400
G
36 in [2, Table  2]
42840
{
144,816
}
576
N
36 in [3]
42840
{
0,960
}
849346560
A
4
(
C
36
) in [4]
51032
{
0,840
}
660602880
A
6
(
C
36,6
(
D
18
)
) in [5]
42840
{
384,576
}
288
A
4
(
C
36,1
) in Section 3
51032
{
0,840
}
6291456
A
4
(
C
36,2
) in Section 3
42840
{
0,960
}
6291456
A
4
(
C
36,3
) in Section 3
51032
{
0,840
}
22020096
A
4
(
C
36,4
) in Section 3
51032
{
0,840
}
1966080
A
4
(
C
36,5
) in Section 3
51032
{
0,840
}
1572864
A
4
(
C
36,6
) in Section 3
42840
{
0,960
}
2621440
A
4
(
C
36,7
) in Section 3
42840
{
0,960
}
1966080
A
4
(
C
36,8
) in Section 3
42840
{
0,960
}
393216
A
4
(
C
36,9
) in Section 3
51032
{
0,840
}
1376256
A
4
(
C
36,10
) in Section 3
51032
{
0,840
}
393216
A
5
(
D
36,1
) in Section 4
42840
{
144,816
}
144
A
5
(
D
36,2
) in Section 4
42840
{
456,504
}
72
A
6
(
D
36,3
) in Section 4
42840
{
240,720
}
288
A
6
(
D
36,4
) in Section 4
42840
{
240,720
}
576
A
7
(
D
36,5
) in Section 4
42840
{
288,672
}
288
A
7
(
D
36,6
) in Section 4
42840
{
144,816
}
72
A
7
(
D
36,7
) in Section 4
42840
{
144,816
}
288
A
9
(
D
36,8
) in Section 4
42840
{
384,576
}
144
A
19
(
D
36,9
) in Section 4
42840
{
288,672
}
144
A
5
(
E
36,1
) in Section 4
42840
{
456,504
}
72
A
6
(
E
36,2
) in Section 4
42840
{
384,576
}
144Lemma 4.
The five known lattices and the ten extremal unimodular latticesA
4
(
C
36
,
i
)
(
i
=
1,2
,
…
,
10
) are nonisomorphic to each other.Remark 5.
In this way, we have found two more extremal unimodular latticesA
4
(
C
), where C are self-dual Z
4-codes with C
(
1
)
=
B
36,1. However, we have verified by Magma that the two lattices are isomorphic to N
36 in [3] and A
4
(
C
36
) in [4].Remark 6.
ForL
=
A
4
(
C
36
,
i
)
(
i
=
1,2
,
…
,
10
), it follows from τ
(
L
) and {
n
1
(
L
)
,
n
2
(
L
)
} that one of the two unimodular neighbors N
e
1
(
L
) and N
e
2
(
L
) defined in (2) is extremal. We have verified by Magma that the extremal one is isomorphic to A
4
(
C
36
,
i
).Fori
=
1,2
,
…
,
10, the code C
36
,
i is equivalent to some code C
36
,
i
¯ with generator matrix of the form
(10)
I
7
A
B
1
+
2
B
2
O
2
I
22
2
D
,
where A, B
1, B
2, and D are (
1,0
)-matrices, I
n denotes the identity matrix of order n, and O denotes the 22
×
7 zero matrix. We only list in (11) the 7
×
29 matrix M
i
=
A
B
1
+
2
B
2 to save space. Note that O
2
I
22
2
D in (10) can be obtained from I
7
A
B
1
+
2
B
2 since C
36
,
i
¯
(
2
)
=
C
36
,
i
¯
(
1
)
⊥. A generator matrix of A
4
(
C
36
,
i
) is obtained from that of C
36
,
i
(11)
M
1
=
01101111111000000110000333322
10101101001000111110011210223
11000011001110100111100010321
11010100101011010011012032011
00110101010101001101013312302
00000000000001111111112111333
00011111111110000001111213133
,
M
2
=
01101111111000000110002111100
10101101001000111110011210023
11000011001110100111100010123
11010100101011010011012032211
00110101010101001101013312300
00000000000001111111112113131
00011111111110000001111211331
,
M
3
=
01101111111000000110002131300
10101101001000111110011210223
11000011001110100111100010121
11010100101011010011012032011
00110101010101001101013310102
00000000000001111111112131131
00011111111110000001111233331
,
M
4
=
01101111111001111101100220313
10101101001001001110101231032
11000011001110110100102101101
00111001100011100101113010223
11011000011101100011100021110
00000000000001111111112111131
00011111111110000001113301311
,
M
5
=
01101111111001111101102202133
10101101001001001110101231032
11000011001110110100102101101
00111001100011100101113010223
11011000011101100011100221310
00000000000001111111112111131
00011111111110000001113301311
,
M
6
=
01101111111001111101102000313
10101101001001001110101231032
11000011001110110100102101103
00111001100011100101113210223
11011000011101100011100021110
00000000000001111111112313113
00011111111110000001113103333
,
M
7
=
01101111111001111101102202313
10101101001001001110101231030
11000011001110110100102101121
00111001100011100101113210201
11011000011101100011100021112
00000000000001111111112111113
00011111111110000001113301333
,
M
8
=
11111110000111111111000022313
11110001100111000000001130313
01000110111001110100011231021
10011100001011100001101233011
01010101010010101101003301321
00000000011111111111112331111
01111111100000000111111233313
,
M
9
=
11111110000111111111002200111
11110001100111000000001332311
01000110111001110100013211021
10011100001011100001103213011
01010101010010101101000331321
01111111100000000111111211113
00000000011111111111111303313
,
M
10
=
11111110000111111111002202313
11110001100111000000001130311
01000110111001110100011211021
10011100001011100001101213011
01010101010010101101003321123
00000000011111111111112313313
01111111100000000111111211111
.


## 3.2. Unimodular Lattices with Long Shadows

The possible theta series of a unimodular latticeL in dimension 36 having minimum norm 3 and its shadow are as follows:
(12)
1
+
(
960
-
α
)
q
3
+
(
42840
+
4096
β
)
q
4
+
⋯
,
β
q
+
(
α
-
60
β
)
q
3
+
(
3833856
-
36
α
+
1734
β
)
q
5
+
⋯
,
respectively, where α and β are integers with 0
≤
β
≤
α
/
60
<
16 [3]. Then the kissing number of L is at most 960 and min
⁡
(
S
(
L
)
)
≤
5. Unimodular lattices L with min
⁡
(
L
)
=
3 and min
⁡
(
S
(
L
)
)
=
5 are often called unimodular lattices with long shadows (see [9]). Only one unimodular lattice L in dimension 36 with min
⁡
(
L
)
=
3 and min
⁡
(
S
(
L
)
)
=
5 was known, namely, A
4
(
C
36
) in [3].LetL be one of A
4
(
C
36,2
), A
4
(
C
36,6
), A
4
(
C
36,7
), and A
4
(
C
36,8
). Since {
n
1
(
L
)
,
n
2
(
L
)
}
=
{
0,960
}, one of the two unimodular neighbors N
e
1
(
L
) and N
e
2
(
L
) in (2) is extremal and the other is a unimodular lattice L
′ with minimum norm 3 having shadow of minimum norm 5. We denote such lattices L
′ constructed from A
4
(
C
36,2
), A
4
(
C
36,6
), A
4
(
C
36,7
), and A
4
(
C
36,8
) by N
36,1, N
36,2, N
36,3, and N
36,4, respectively. We list in Table 2 the orders #
Aut
(
N
36
,
i
)
(
i
=
1,2
,
3,4
) of the automorphism groups, which have been calculated by Magma. Table 2 shows the following.Table 2
# Aut(N
36
,
i) (i = 1, 2, 3, 4).
LatticesL
# Aut(
L
)
A
4
(
C
36
) in [3]
1698693120
N
36,1
12582912
N
36,2
5242880
N
36,3
3932160
N
36,4
786432Proposition 7.
There are at least 5 nonisomorphic unimodular latticesL in dimension 36 with min
⁡
(
L
)
=
3 and min
⁡
(
S
(
L
)
)
=
5.


## 4. From Self-DualZ
k-Codes (
k
≥
5
)

In this section, we construct more extremal unimodular lattices in dimension36 from self-dual Z
k-codes (
k
≥
5
).LetA
T denote the transpose of a matrix A. An n
×
n matrix is negacirculant if it has the following form:
(13)
r
0
r
1
⋯
r
n
-
1
-
r
n
-
1
r
0
⋯
r
n
-
2
-
r
n
-
2
-
r
n
-
1
⋯
r
n
-
3
⋮
⋮
⋮
-
r
1
-
r
2
⋯
r
0
.
Let D
36
,
i
(
i
=
1,2
,
…
,
9
) and E
36
,
i
(
i
=
1,2
) be Z
k-codes of length 36 with generator matrices of the following form:
(14)
I
18
A
B
-
B
T
A
T
,
where k are listed in Table 3, and A and B are 9
×
9 negacirculant matrices with first rows r
A and r
B listed in Table 3. It is easy to see that these codes are self-dual since A
A
T
+
B
B
T
=
-
I
9. Thus, A
k
(
D
36
,
i
)
(
i
=
1,2
,
…
,
9
) and A
k
(
E
36
,
i
)
(
i
=
1,2
) are unimodular lattices, for k given in Table 3. In addition, we have verified by Magma that these lattices are extremal.Table 3
Self-dualZ
k-codes of length 36.
Codes
k
r
A
r
B
D
36,1
5
(
0,0
,
0,1
,
2,2
,
0,4
,
2
)
(
0,0
,
0,0
,
4,3
,
3,0
,
1
)
D
36,2
5
(
0,0
,
0,1
,
3,0
,
2,0
,
4
)
(
3,0
,
4,1
,
4,0
,
0,4
,
4
)
D
36,3
6
(
0,1
,
5,3
,
2,0
,
3,5
,
1
)
(
3,1
,
0,0
,
5,1
,
1,1
,
1
)
D
36,4
6
(
0,1
,
3,5
,
1,5
,
5,4
,
4
)
(
4,0
,
3,2
,
4,5
,
5,2
,
4
)
D
36,5
7
(
0,1
,
6,3
,
5,0
,
4,5
,
4
)
(
0,1
,
6,3
,
5,2
,
1,5
,
1
)
D
36,6
7
(
0,1
,
1,3
,
2,6
,
1,4
,
6
)
(
0,1
,
4,0
,
5,3
,
6,2
,
0
)
D
36,7
7
(
0,0
,
0,1
,
5,5
,
5,1
,
1
)
(
0,5
,
4,2
,
5,1
,
1,3
,
6
)
D
36,8
9
(
0,0
,
0,1
,
0,4
,
3,0
,
0
)
(
0,4
,
1,5
,
3,5
,
1,7
,
0
)
D
36,9
19
(
0,0
,
0,1
,
15,15,9
,
6,5
)
(
14,16,0
,
14,15,8
,
8,3
,
12
)
E
36,1
5
(
0,1
,
0,2
,
1,3
,
2,2
,
0
)
(
2,0
,
1,0
,
1,1
,
2,3
,
1
)
E
36,2
6
(
0,1
,
5,3
,
4,4
,
1,1
,
0
)
(
4,0
,
1,3
,
4,2
,
3,0
,
1
)To distinguish between the above eleven lattices and the15 known lattices, in Table 1 we give τ
(
L
), {
n
1
(
L
)
,
n
2
(
L
)
}, and #
Aut
(
L
), which have been calculated by Magma. The two lattices have the identical (
τ
(
L
)
,
{
n
1
(
L
)
,
n
2
(
L
)
}, #
Aut
(
L
)
) for each of the pairs (
A
5
(
E
36,1
)
,
A
5
(
D
36,2
)
) and (
A
6
(
E
36,2
)
,
A
9
(
D
36,8
)
). However, we have verified by Magma that the two lattices are nonisomorphic for each pair. Therefore, we have the following.Lemma 8.
The 26 lattices in Table1 are nonisomorphic to each other.Lemma8 establishes Proposition 1.Remark 9.
Similar to Remark6, it is known [3] that the extremal neighbor is isomorphic to L for the case where L is N
36 in [3], and we have verified by Magma that the extremal neighbor is isomorphic to L for the case where L is A
4
(
C
36
) in [4].


## 5. Related Ternary Self-Dual Codes

In this section, we give a certain short observation on ternary self-dual codes related to some extremal odd unimodular lattices in dimension36.


### 5.1. Unimodular Lattices from Ternary Self-Dual Codes

LetT
36 be a ternary self-dual code of length 36. The two unimodular neighbors N
e
1
(
A
3
(
T
36
)
) and N
e
2
(
A
3
(
T
36
)
) given in (2) are described in [16] as L
S
(
T
36
) and L
T
(
T
36
). In this section, we use the notation L
S
(
T
36
) and L
T
(
T
36
), instead of N
e
1
(
A
3
(
T
36
)
) and N
e
2
(
A
3
(
T
36
)
), since the explicit constructions and some properties of L
S
(
T
36
) and L
T
(
T
36
) are given in [16]. By Theorem  6 in [16] (see also Theorem  3.1 in [2]), L
T
(
T
36
) is extremal when T
36 satisfies the following condition (a), and both L
S
(
T
36
) and L
T
(
T
36
) are extremal when T
36 satisfies the following condition (b):(a)
extremal (minimum weight12) and admissible (the number of 1’s in the components of every codeword of weight 36 is even),
(b)
minimum weight9 and maximum weight 33. For each of (a) and (b), no ternary self-dual code satisfying the condition is currently known.


### 5.2. Condition (a)

Suppose thatT
36 satisfies condition (a). Since T
36 has minimum weight 12, A
3
(
T
36
) has minimum norm 3 and kissing number 72. By Theorem  6 in [16], min
⁡
(
L
T
(
T
36
)
)
=
4 and min
⁡
(
L
S
(
T
36
)
)
=
3. Hence, since the shadow of L
T
(
T
36
) contains no vector of norm 1, by (3) and (4), L
T
(
T
36
) has theta series 1
+
42840
q
4
+
1916928
q
5
+
⋯. It follows that {
n
1
(
L
T
(
T
36
)
)
,
n
2
(
L
T
(
T
36
)
)
}
=
{
72,888
}.By Theorem  1 in [17], the possible complete weight enumerator W
C
(
x
,
y
,
z
) of a ternary extremal self-dual code C of length 36 containing 1 is written as
(15)
a
1
δ
36
+
a
2
α
12
3
+
a
3
α
12
2
β
6
2
+
a
4
α
12
β
6
2
2
+
a
5
β
6
2
3
+
a
6
β
6
γ
18
α
12
+
a
7
β
6
γ
18
β
6
2
,
using some a
i
∈
R
(
i
=
1,2
,
…
,
7
), where α
12
=
a
(
a
3
+
8
p
3
),  β
6
=
a
2
-
12
b,  γ
18
=
a
6
-
20
a
3
p
3
-
8
p
6,  and δ
36
=
p
3
(
a
3
-
p
3
)
3 and  a
=
x
3
+
y
3
+
z
3,  p
=
3
x
y
z,  and b
=
x
3
y
3
+
x
3
z
3
+
y
3
z
3. From the minimum weight, we have the following:
(16)
a
2
=
3281
13824
-
a
1
64
,
a
3
=
203
4608
-
9
a
1
256
,
a
4
=
1763
13824
+
3
a
1
128
,
a
5
=
-
277
13824
-
a
1
256
,
a
6
=
1133
1728
+
3
a
1
64
,
a
7
=
-
77
1728
-
a
1
64
.
Since W
C
(
x
,
y
,
z
) contains the term (
15180
+
2916
a
1
)
y
15
z
21, if C is admissible, then
(17)
a
1
=
-
15180
2916
.
Hence, the complete weight enumerator of a ternary admissible extremal self-dual code containing 1 is uniquely determined, which is listed in (18). 								(18)
x
36
+
y
36
+
z
36
+
78706260
x
12
y
12
z
12
+
682
(
x
18
y
18
+
x
18
z
18
+
y
18
z
18
)
+
7019232
(
x
15
y
15
z
6
+
x
15
y
6
z
15
+
x
6
y
15
z
15
)
+
29172
(
x
24
y
6
z
6
+
x
6
y
24
z
6
+
x
6
y
6
z
24
)
+
10260316
(
x
18
y
9
z
9
+
x
9
y
18
z
9
+
x
9
y
9
z
18
)
+
37995408
x
12
y
15
z
9
+
x
12
y
9
z
15
+
x
15
y
12
z
9
llllllllllllllllllllllllllll
+
x
15
y
9
z
12
+
x
9
y
12
z
15
+
x
9
y
15
z
12
+
3924756
x
12
y
18
z
6
+
x
12
y
6
z
18
+
x
18
y
12
z
6
lllllllllllllllllllllllllll
+
x
18
y
6
z
12
+
x
6
y
12
z
18
+
x
6
y
18
z
12
+
58344
x
12
y
21
z
3
+
x
12
y
3
z
21
+
x
21
y
12
z
3
llllllllllllllllllllll
+
x
21
y
3
z
12
+
x
3
y
12
z
21
+
x
3
y
21
z
12
+
102
x
12
y
24
+
x
12
z
24
+
x
24
y
12
llllllllllllllllll
+
x
24
z
12
+
y
12
z
24
+
y
24
z
12
+
170544
x
15
y
18
z
3
+
x
15
y
3
z
18
+
x
18
y
15
z
3
llllllllllllllllllllllll
+
x
18
y
3
z
15
+
x
3
y
15
z
18
+
x
3
y
18
z
15
+
641784
x
21
y
6
z
9
+
x
21
y
9
z
6
+
x
6
y
21
z
9
llllllllllllllllllllllll
+
x
6
y
9
z
21
+
x
9
y
21
z
6
+
x
9
y
6
z
21
+
6732
x
24
y
3
z
9
+
x
24
y
9
z
3
+
x
3
y
24
z
9
llllllllllllllllllll
+
x
3
y
9
z
24
+
x
9
y
24
z
3
+
x
9
y
3
z
24
.


### 5.3. Condition (b)

Suppose thatT
36 satisfies condition (b). By the Gleason theorem (see Corollary  5 in [17]), the weight enumerator of T
36 is uniquely determined as
(19)
1
+
888
y
9
+
34848
y
12
+
1432224
y
15
+
18377688
y
18
+
90482256
y
21
+
162551592
y
24
+
97883072
y
27
+
16178688
y
30
+
479232
y
33
.
By Theorem  6 in [16] (see also Theorem  3.1 in [2]), L
S
(
T
36
) and L
T
(
T
36
) are extremal. Hence, min
⁡
(
A
3
(
T
36
)
)
=
3 and min
⁡
(
S
(
A
3
(
T
36
)
)
)
=
5.Note that a unimodular latticeL contains a 3-frame if and only if L
≅
A
3
(
C
) for some ternary self-dual code C. Let L
36 be any of the five lattices given in Table 2. Let L
36
(
3
) be the set {
{
x
,
-
x
}
∣
(
x
,
x
)
=
3
,
x
∈
L
36
}. We define the simple undirected graph Γ
(
L
36
), whose set of vertices is the set of 480 pairs in L
36
(
3
) and two vertices {
x
,
-
x
}
,
{
y
,
-
y
}
∈
L
36
(
3
) are adjacent if (
x
,
y
)
=
0. It follows that the 3-frames in L
36 are precisely the 36-cliques in the graph Γ
(
L
36
). We have verified by Magma that Γ
(
L
36
) are regular graphs with valency 368, and the maximum sizes of cliques in Γ
(
L
36
) are 12. Hence, none of these lattices is constructed from some ternary self-dual code by Construction A.


## 5.1. Unimodular Lattices from Ternary Self-Dual Codes

LetT
36 be a ternary self-dual code of length 36. The two unimodular neighbors N
e
1
(
A
3
(
T
36
)
) and N
e
2
(
A
3
(
T
36
)
) given in (2) are described in [16] as L
S
(
T
36
) and L
T
(
T
36
). In this section, we use the notation L
S
(
T
36
) and L
T
(
T
36
), instead of N
e
1
(
A
3
(
T
36
)
) and N
e
2
(
A
3
(
T
36
)
), since the explicit constructions and some properties of L
S
(
T
36
) and L
T
(
T
36
) are given in [16]. By Theorem  6 in [16] (see also Theorem  3.1 in [2]), L
T
(
T
36
) is extremal when T
36 satisfies the following condition (a), and both L
S
(
T
36
) and L
T
(
T
36
) are extremal when T
36 satisfies the following condition (b):(a)
extremal (minimum weight12) and admissible (the number of 1’s in the components of every codeword of weight 36 is even),
(b)
minimum weight9 and maximum weight 33. For each of (a) and (b), no ternary self-dual code satisfying the condition is currently known.


## 5.2. Condition (a)

Suppose thatT
36 satisfies condition (a). Since T
36 has minimum weight 12, A
3
(
T
36
) has minimum norm 3 and kissing number 72. By Theorem  6 in [16], min
⁡
(
L
T
(
T
36
)
)
=
4 and min
⁡
(
L
S
(
T
36
)
)
=
3. Hence, since the shadow of L
T
(
T
36
) contains no vector of norm 1, by (3) and (4), L
T
(
T
36
) has theta series 1
+
42840
q
4
+
1916928
q
5
+
⋯. It follows that {
n
1
(
L
T
(
T
36
)
)
,
n
2
(
L
T
(
T
36
)
)
}
=
{
72,888
}.By Theorem  1 in [17], the possible complete weight enumerator W
C
(
x
,
y
,
z
) of a ternary extremal self-dual code C of length 36 containing 1 is written as
(15)
a
1
δ
36
+
a
2
α
12
3
+
a
3
α
12
2
β
6
2
+
a
4
α
12
β
6
2
2
+
a
5
β
6
2
3
+
a
6
β
6
γ
18
α
12
+
a
7
β
6
γ
18
β
6
2
,
using some a
i
∈
R
(
i
=
1,2
,
…
,
7
), where α
12
=
a
(
a
3
+
8
p
3
),  β
6
=
a
2
-
12
b,  γ
18
=
a
6
-
20
a
3
p
3
-
8
p
6,  and δ
36
=
p
3
(
a
3
-
p
3
)
3 and  a
=
x
3
+
y
3
+
z
3,  p
=
3
x
y
z,  and b
=
x
3
y
3
+
x
3
z
3
+
y
3
z
3. From the minimum weight, we have the following:
(16)
a
2
=
3281
13824
-
a
1
64
,
a
3
=
203
4608
-
9
a
1
256
,
a
4
=
1763
13824
+
3
a
1
128
,
a
5
=
-
277
13824
-
a
1
256
,
a
6
=
1133
1728
+
3
a
1
64
,
a
7
=
-
77
1728
-
a
1
64
.
Since W
C
(
x
,
y
,
z
) contains the term (
15180
+
2916
a
1
)
y
15
z
21, if C is admissible, then
(17)
a
1
=
-
15180
2916
.
Hence, the complete weight enumerator of a ternary admissible extremal self-dual code containing 1 is uniquely determined, which is listed in (18). 								(18)
x
36
+
y
36
+
z
36
+
78706260
x
12
y
12
z
12
+
682
(
x
18
y
18
+
x
18
z
18
+
y
18
z
18
)
+
7019232
(
x
15
y
15
z
6
+
x
15
y
6
z
15
+
x
6
y
15
z
15
)
+
29172
(
x
24
y
6
z
6
+
x
6
y
24
z
6
+
x
6
y
6
z
24
)
+
10260316
(
x
18
y
9
z
9
+
x
9
y
18
z
9
+
x
9
y
9
z
18
)
+
37995408
x
12
y
15
z
9
+
x
12
y
9
z
15
+
x
15
y
12
z
9
llllllllllllllllllllllllllll
+
x
15
y
9
z
12
+
x
9
y
12
z
15
+
x
9
y
15
z
12
+
3924756
x
12
y
18
z
6
+
x
12
y
6
z
18
+
x
18
y
12
z
6
lllllllllllllllllllllllllll
+
x
18
y
6
z
12
+
x
6
y
12
z
18
+
x
6
y
18
z
12
+
58344
x
12
y
21
z
3
+
x
12
y
3
z
21
+
x
21
y
12
z
3
llllllllllllllllllllll
+
x
21
y
3
z
12
+
x
3
y
12
z
21
+
x
3
y
21
z
12
+
102
x
12
y
24
+
x
12
z
24
+
x
24
y
12
llllllllllllllllll
+
x
24
z
12
+
y
12
z
24
+
y
24
z
12
+
170544
x
15
y
18
z
3
+
x
15
y
3
z
18
+
x
18
y
15
z
3
llllllllllllllllllllllll
+
x
18
y
3
z
15
+
x
3
y
15
z
18
+
x
3
y
18
z
15
+
641784
x
21
y
6
z
9
+
x
21
y
9
z
6
+
x
6
y
21
z
9
llllllllllllllllllllllll
+
x
6
y
9
z
21
+
x
9
y
21
z
6
+
x
9
y
6
z
21
+
6732
x
24
y
3
z
9
+
x
24
y
9
z
3
+
x
3
y
24
z
9
llllllllllllllllllll
+
x
3
y
9
z
24
+
x
9
y
24
z
3
+
x
9
y
3
z
24
.


## 5.3. Condition (b)

Suppose thatT
36 satisfies condition (b). By the Gleason theorem (see Corollary  5 in [17]), the weight enumerator of T
36 is uniquely determined as
(19)
1
+
888
y
9
+
34848
y
12
+
1432224
y
15
+
18377688
y
18
+
90482256
y
21
+
162551592
y
24
+
97883072
y
27
+
16178688
y
30
+
479232
y
33
.
By Theorem  6 in [16] (see also Theorem  3.1 in [2]), L
S
(
T
36
) and L
T
(
T
36
) are extremal. Hence, min
⁡
(
A
3
(
T
36
)
)
=
3 and min
⁡
(
S
(
A
3
(
T
36
)
)
)
=
5.Note that a unimodular latticeL contains a 3-frame if and only if L
≅
A
3
(
C
) for some ternary self-dual code C. Let L
36 be any of the five lattices given in Table 2. Let L
36
(
3
) be the set {
{
x
,
-
x
}
∣
(
x
,
x
)
=
3
,
x
∈
L
36
}. We define the simple undirected graph Γ
(
L
36
), whose set of vertices is the set of 480 pairs in L
36
(
3
) and two vertices {
x
,
-
x
}
,
{
y
,
-
y
}
∈
L
36
(
3
) are adjacent if (
x
,
y
)
=
0. It follows that the 3-frames in L
36 are precisely the 36-cliques in the graph Γ
(
L
36
). We have verified by Magma that Γ
(
L
36
) are regular graphs with valency 368, and the maximum sizes of cliques in Γ
(
L
36
) are 12. Hence, none of these lattices is constructed from some ternary self-dual code by Construction A.



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*Source: 792471-2014-11-30.xml*