Designs, Codes and Cryptography, 12, 39–48 (1997)
c 1997 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
°
On the Binary Self-Dual Codes with an
Automorphism of Order 2
STEFKA BUYUKLIEVA*
lpmivt@bgcict.acad.bg
Faculty of Mathematics and Informatics, University of Veliko Tarnovo, 5000 Veliko Tarnovo, Bulgaria
Communicated by: V. D. Tonchev
Received December 8, 1995; Revised August 5, 1996; Accepted September 9, 1996
Abstract. Methods to design binary self-dual codes with an automorphism of order two without fixed points are
presented. New extremal self-dual [40, 20, 8], [42, 21, 8], [44, 22, 8] and [64, 32, 12] codes with previously not
known weight enumerators are constructed.
Keywords: self-dual codes, automorphisms, weight enumerators
1. Introduction
A method for constructing binary self-dual codes via an automorphism of odd prime order is
given by Huffman and Yorgov [11], [20], [21]. Yorgov has found all inequivalent extremal
doubly-even codes of length n with an automorphism of odd prime order p for n = 40, p >
5 [20]; n = 56, p = 13 [21]; n = 64, p = 31 [22]. All inequivalent extremal singlyeven self-dual codes of length 40 with an automorphism of odd prime order are in [4].
Many extremal codes of lengths 42 and 44 are obtained using this technique [3], [16].
Huffman proved that any extremal self-dual doubly-even code of length 48 with a nontrivial
authomorphism of odd order is equivalent to the extended quadratic residue code [11]. There
are some previously known constructions that lead to codes with full automorphism group
of order a power of 2 and also to extremal self-dual codes with trivial full automorphism
group [2], [18].
The purpose of this paper is to examine some general properties of binary self-dual codes
with an automorphism of order 2 without fixed points and to apply these results to extremal
codes. We investigate two construction techniques for such codes.
In the next section we define our terminology and obtain general results considering selfdual codes with an automorphism of order 2 without fixed points. Theorems 2 and 3 are
for constructing such codes using self-orthogonal codes with a twice smaller length. In the
third section we apply theorem 2 for self-dual codes with minimal distances 2 and 4. In
the fourth section we construct some new extremal self-dual codes of lengths 40, 42 and 44
using self-orthogonal codes obtained from the extended Golay code. In the fifth section we
design extremal codes using quasi-cyclic self-orthogonal codes. The final section examines
extremal codes of length 64.
*
This work was partially supported by the Bulgarian National Science Fund under Contract No. MM-503/1995.
40
BUYUKLIEVA
All possible weight enumerators of putative or known extremal self-dual codes of these
lengths are given by Conway and Sloane [5]. These weight enumerators are:
1) for length 40:
W (y) = 1 + (125 + 16β)y 8 + (1664 − 64β)y 10 + · · ·
(1)
for singly-even codes.
2) for length 42:
W (y) = 1 + (84 + 8β)y 8 + (1449 − 24β)y 10 + · · ·
(2)
and
W (y) = 1 + 164y 8 + 697y 10 + · · · .
3) for length 44:
W (y) = 1 + (44 + 4β)y 8 + (976 − 8β)y 10 + · · ·
(3)
W (y) = 1 + (44 + 4β)y 8 + (1232 − 8β)y 10 + · · · .
(4)
and
4) for length 64:
W (y) = 1 + 2976y 12 + 45495y 16 + · · ·
for doubly-even codes and
W (y) = 1 + (1312 + 16β)y 12 + (22016 − 64β)y 14 + · · ·
(5)
W (y) = 1 + (1312 + 16β)y 12 + (23040 − 64β)y 14 + · · ·
(6)
and
for singly-even codes.
Extremal singly-even self-dual codes with length 40 are obtained in [4], [10], with length
42 are constructed in [17], [19], with length 44 in [1], [23], and also in papers [6], [7], [8].
Extremal self-dual [64, 32, 12] codes are obtained in [9], [12], [14], [15].
We construct two extremal self-dual codes of length 40 with weight enumerators (1)
for β = 4 and β = 10, four self-dual [42, 21, 8] codes with weight enumerators (2) for
β = 6, 14, 18 and 42, four extremal [44, 22, 8] codes with weight enumerators (3) for
β = 30 and 38, and (4) for β = 74 and 154, a doubly-even [64, 32, 12] code and an
extremal singly-even [64, 32, 12] code with weight enumerator (5) for β = 44. The codes
with weight enumerators (1) with β = 4, (2) with β = 6, 14 and 18, (3) with β = 30, (5)
with β = 44 are the first known self-dual codes with these weight enumerators.
41
ON THE BINARY SELF-DUAL CODES
2. Definitions and General Results
Let Fqn be the n-dimensional vector space over the Galois field Fq . Suppose (., .): Fqn ×
Fqn → Fq defines an inner product. Then if C is an [n, k] code over Fq , C ⊥ = {u ∈ Fqn :
(u, v) = 0 for all v ∈ C}. If C ⊆ C ⊥ , C is termed self-orthogonal and if C = C ⊥ , C is selfdual. In this work we will consider
Pn binary codes. In all such cases, if u = (u 1 , . . . , u n ), v =
u i vi ∈ F2 . A binary self-dual code in which all weights
(v1 , . . . , vn ) ∈ F2n , (u, v) = i=1
are divisible by four is termed doubly-even. If not all weights are divisible by four the code
is singly-even. Self-dual codes with the largest minimal weight for a given length are called
extremal.
For τ in Sn (the symmetric group of degree n) and v = (v1 , . . . , vn ) in F2n we define
vτ = (v1τ −1 , v2τ −1 , . . . , vnτ −1 ). The binary codes Cτ and C are called equivalent. If
Cτ = C the permu tation τ is an automorphism of the code C. Let C be a binary self-dual
[n, k, d] code and σ = (1, 2)(3, 4) . . . (n − 1, n) be an automorphism of C.
In this section we prove three theorems. The first one gives some important properties of
binary self-dual codes with an automorphism of order 2 without fixed points. Theorems 2
and 3 give methods for constructing such codes.
THEOREM 1 Let C be a binary self-dual [n, k = n2 ] code and σ = (1, 2)(3, 4) . . . (n −
1, n) be an automorphism of C. Let φ: C → F2k be the map defined by φ(v) = (α1 +
α2 , . . . , αn−1 + αn ) for v = (α1 , . . . , αn ) ∈ C. Then φ is a homomorphism, C ′ = ℑφ is
′′
a self-orthogonal [k, s] code and C = π(Ker φ) = (C ′ )⊥ , where π : Ker φ → F2k is the
map defined by π(v) = (α1 , . . . , αk ) for v = (α1 , α1 , . . . , αk , αk ) ∈ Ker φ.
Proof. Clearly φ is linear and hence φ is a homomorphism. Thus C ′ is a [k, s] code for some
(β1 , . . . , βn ) be codewords in
s. To show it is self-orthogonal,
Pk let v = (α1 , . . . , αn ) and w = P
k
C. Then (φ(v), φ(w)) = i=1
(α2i−1 + α2i )(β2i−1 + β2i ) = i=1
(α2i−1 β2i−1 + α2i β2i ) +
Pk
(α
β
+
α
β
)
=
(v,
w)
+
(v,
wσ
)
=
0
as
wσ
∈
C.
2i 2i−1
i=1 2i−1 2i
As (α1 , α2 , . . . , αn ) ∈ Ker φ if and only if α2i−1 = α2i for 1 ≤ i ≤ k, Ker φ = C1
where C1 = {(β1 , β1 , β2 , β2 , . . . , βk , βk ) ∈ C}. Let v1 , . . . , vt be a basis of C1 where
t = dim Ker φ and extend this to a basis v1 , . . . , vt , vt+1 , . . . , vk of C. Define C2 to be the
code with basis vt+1 , . . . , vk . Thus C = C1 ⊕ C2 . Since C1 = Ker φ, C ′ = ℑφ = φ(C2 ).
Furthermore the restriction of φ to C2 is one-to-one as Ker φ = C1 and C1 ∩ C2 = {0}.
Therefore s = dim ℑφ = dim C2 = k − t or s + t = k.
The map π : Ker φ → F2k is clearly one-to-one linear map, and thus dim C ′′ =
dim π(Ker φ) = t. As dim C ′ = s and s + t = k, to prove that C ′′ = (C ′ )⊥ , it suffices to
show that a vector in C ′ is orthogonal to a vector in C ′′ . Let v =P(α1 , . . . , αn ) ∈ C and
k
(α2i−1 + α2i )βi =
w = {(β1 , β1 , β2 , β2 , . . . , βk , βk ) ∈ Ker φ. Then (φ(v), π(w)) = i=1
(v, w) = 0, completing the proof.
′′
THEOREM 2 Let C ′ be a self-orthogonal [k, s, d ′ ] code and C be its dual code. Let τ : C ′ →
′′
F22k and ψ: C → F22k be the maps defined by τ (v) = (α1 , 0, α2 , 0, . . . , αk , 0) for v =
(α1 , α2 , . . . , αk ) ∈ C ′ and ψ(w) = (β1 , β1 , β2 , β2 , . . . , βk , βk ) for w = (β1 , β2 , . . . , βk ) ∈
′′
′′
C . Then C = τ (C ′ ) + ψ(C ) is a self-dual [2k, k, d] code with minimal distance d =
42
BUYUKLIEVA
′′
′′
′′
min{d ′ , 2d }, where d is the minimal distance of the code C , and σ = (1, 2)(3, 4) . . . (2k−
1, 2k) is an automorphism of C.
′′
′
Proof. Since τ and ψ are monomorphisms
T the ′′dimensions of codes τ (C ) and ψ(C ) are s
′
and k −s respectively. Obviously τ (C ) ψ(C ) = {0} and therefore the dimension of C is
′′
′′
s+k−s = k. The minimal distances of τ (C ′ ) and ψ(C ) are d ′ and 2d respectively. If v is a
′′
vector from C ′ and w 6= v is from C the weight of the vector τ (v)+ψ(w) is equal to the sum
′′
′′
of the weights of the vectors v +w and w from C and therefore is at least 2d . If v ∈ C ′ and
w = v the weight of the vector τ (v)+ψ(w) is equal to the weight of the vector v and so is at
′′
least d ′ .Thus the minimal distance of the code C is d = min{d ′ , 2d }. For v, v ′ ∈ C ′ we have
′′
(τ (v), τ (v ′ )) = (v, v ′ ) = 0, for w, w′ ∈ C (ψ(w), ψ(w′ )) = (w, w′ ) + (w, w′ ) = 0 and
′′
for v ∈ C ′ and w ∈ C (τ (v), ψ(w)) = (v, w) = 0. Hence all vectors in C are orthogonal
′′
to each other and thus C is a self-dual code. If w ∈ C , then ψ(w)σ = ψ(w) ∈ C.
If v ∈ C ′ , then τ (v)σ = τ (v) + ψ(v) ∈ C. Hence σ is an automorphism of C.
Remark 1. This method is similar to the |u|u + v| construct ion [13].
Remark 2. To obtain an extremal [2k, k, d] self-dual code we have to take a self-orthogonal
[k, s] code with minimal distance at least d and minimal distance for its dual code at least
d
.
2
′′
THEOREM 3 Let C ′ be a self-orthogonal [k, s, d ′ ] code and C be its dual code. Let
′′
ψ: C → F22k be the map defined in theorem 2 and µ: C ′ → F22k be the map defined by µ(v) = (α1 , 0, α2 , 0, . . . , αk−2 , 0, 0, 0, 0, 0) for v = (α1 , α2 , . . . , αk−2 , 0, 0) ∈
C ′ , µ(v) = (α1 , 0, α2 , 0, . . . , αk−2 , 0, 1, 0, 1, 1) for v = (α1 , α2 , . . . , αk−2 , 1, 0) ∈ C ′ ,
µ(v) = (α1 , 0, α2 , 0, . . . , αk−2 , 0, 1, 1, 1, 0) for v = (α1 , α2 , . . . , αk−2 , 0, 1) ∈ C ′ , µ(v) =
(α1 , 0, α2 , 0, . . . , αk−2 , 0, 0, 1, 0, 1) for v = (α1 , α2 , . . . , αk−2 , 1, 1) ∈ C ′ . Then C =
′′
µ(C ′ ) + ψ(C ) is a self-dual [2k, k, d] code and σ = (1, 2)(3, 4) . . . (2k − 1, 2k) is an
′′
′′
automorphism of C. If d ′ ≥ 2d then the minimal distance d of C is equal to 2d .
Proof. Similarly to theorem 2 the dimension of C is k. If v = (v1 , α1 , α2 ), v ′ =
′′
(v1′ , α1′ , α2′ ) ∈ C ′ , and w = (w1 , β1 , β2 ) ∈ C , where v1 , v1′ , w1 ∈ F2k−2 , and α1 , α2 ,
α1′ , α2′ , β1 , β2 ∈ F2 , then from the tables 1 and 2 we have (µ(v), µ(v ′ )) = (v, v ′ ) = 0
and (µ(v), ψ(w)) = (v, w) = 0 (the sums in both tables 1 and 2 are mod 2). Hence
all vectors in C are orthogonal to each other and thus C is a self-dual code. Furthermore
′′
wt (µ(v)) ≥ wt (v) ≥ d ′ ≥ 2d for all v ∈ C ′ . For the vectors v and w we have
′′
wt (µ(v) + ψ(w)) ≥ wt (v + w) + wt (w) ≥ 2d in all cases except when α1 = 1, α2 = 0
and β2 = 1 (see table 3). In this case wt (µ(v) + ψ(w)) = wt (v + w) + wt (w) − 2.
′′
′′
So we have wt (µ(v) + ψ(w)) ≤ 2d iff wt (v + w) = wt (w) = d . But wt (v + w) =
′′
wt (v) + wt (w) − 2wt (v ∗ w) and hence wt (v) = 2wt (v ∗ w) ≤ 2wt (w) = 2d ≤ d ′ . So
′
wt (v) = d and wt (v ∗w) = wt (w) and the support of w is a subset of the support of v. But
k lies in the support of w and does not lie in the support of v thus this case is imposible, hence
′′
′′
wt (µ(v)+ψ(w)) ≥ 2d in all cases. So we obtain that wt (u) ≥ 2d for all u ∈ C and hence
43
ON THE BINARY SELF-DUAL CODES
Table 1.
α1
α2
0
1
1
1
1
0
0
0
0
1
α1′
1
0
1
1
α2′
(v, v ′ )
(µ(v), µ(v ′ ))
0
1
1
1
(v1 , v1′ )
(v1 , v1′ ) + 1
(v1 , v1′ )
(v1 , v1′ ) + 1
(v1 , v1′ ) + 2
(v1 , v1′ )
(v1 , v1′ ) + 3
(v1 , v1′ ) + 2
(v1 , v1′ ) + 1
(v1 , v1′ ) + 2
Table 2.
′′
α1
α2
(v, w)
(µ(v), ψ(w))
0
1
1
0
0
1
(v1 , w1 )
(v1 , w1 ) + β1
(v1 , w1 ) + β1 + β2
(v1 , w1 )
(v1 , w1 ) + β1 + β2 + β2
(v1 , w1 ) + β1 + β2
′′
′′
′′
d = 2d (the minimal distance of ψ(C ) is 2d ). If w ∈ C , then ψ(w)σ = ψ(w) ∈ C.
If v ∈ C ′ , then µ(v)σ = µ(v) + ψ(v) ∈ C. Hence σ is an automorphism of C.
3. Some Self-Dual Codes with Minimal Distances 2 and 4
If we consider C ′ be the code {0} using the theorem 2 construction we obtain the self-dual
[2k, k, 2] code i 2k with a generator matrix
11
...
11 . . .
...
. . . 11
Let k be even and C ′ be the code {00 . . . 0, 11 . . . 1}. If we use theorem 2 we can construct
a self-dual [2k, k, 4] code with a generator matrix
1111 . . .
1111 . . .
.
.
.
. . . 1111
1010
. . . 1010
The weight enumerator of this code is
k/2 µ ¶
X
k 4i
y + 2k−1 y k
W (y) = 1 +
2i
i=1
For k = 4, 6, 8, 10 these codes are extremal.
44
BUYUKLIEVA
Table 3.
α1
α2
β1
β2
wt (w)
wt (v + w)
wt (µ(v) + ψ(w))
0
0
0
1
1
1
1
1
1
1
0
0
0
0
0
0
0
1
1
1
0
1
1
0
1
0
1
0
1
1
0
0
1
0
0
1
1
0
0
1
wt (w1 )
wt (w1 ) + 1
wt (w1 ) + 2
wt (w1 )
wt (w1 ) + 1
wt (w1 ) + 1
wt (w1 ) + 2
wt (w1 )
wt (w1 ) + 1
wt (w1 ) + 2
wt (v1 + w1 )
wt (v1 + w1 ) + 1
wt (v1 + w1 ) + 2
wt (v1 + w1 ) + 1
wt (v1 + w1 )
wt (v1 + w1 ) + 2
wt (v1 + w1 ) + 1
wt (v1 + w1 ) + 2
wt (v1 + w1 ) + 1
wt (v1 + w1 )
wt (v1 + w1 ) + wt (w1 )
wt (v1 + w1 ) + wt (w1 ) + 2
wt (v1 + w1 ) + wt (w1 ) + 4
wt (v1 + w1 ) + wt (w1 ) + 3
wt (v1 + w1 ) + wt (w1 ) + 3
wt (v1 + w1 ) + wt (w1 ) + 1
wt (v1 + w1 ) + wt (w1 ) + 1
wt (v1 + w1 ) + wt (w1 ) + 2
wt (v1 + w1 ) + wt (w1 ) + 2
wt (v1 + w1 ) + wt (w1 ) + 2
4. Some Extremal Self-Dual Codes Obtained from the Golay Code
THEOREM 4 There exist extremal binary self-dual [40, 20, 8] codes with weight enumerators (1) for β = 10 and β = 4 with an automorphism of order 2 without fixed points.
Proof. From the extended Golay code g24 by deleting the last 4 coordinates of the vectors
having 0 on them we obtain a self-orthogonal [20, 8, 8] code g20 . From this code using the
theorem 2 construction we obtain a self-dual [40, 20, 8] code with weight enumerator (1)
for β = 10. From the code g20 we construct a self-dual code with weight enumerator (1)
for β = 4 using theorem 3.
Remark. A self-dual code with weight enumerator (1) for β = 10 is found as a double
circulant code [5]. No other self-dual code with a weight enumerator (1) for β = 4 is
known.
THEOREM 5 There exist extremal binary self-dual [42, 21, 8] codes with weight enumerators (2) for β = 42 and β = 18 with an automorphism of order 2 without fixed points.
Proof. From the extended Golay code g24 by deleting the last 3 coordinates of the vectors
having 0 on them we obtain a self-orthogonal [21, 9, 8] code g21 . From this code using the
theorem 2 construction we obtain a self-dual [42, 21, 8] code with weight enumerator (2)
for β = 42. From the code g21 we construct a self-dual code with weight enumerator (1)
for β = 18 using theorem 3.
Remark. A self-dual code with a weight enumerator (2) for β = 42 is found with an
automorphism of order 11 [16]. Another self-dual [42, 21, 8] code with a weight enumerator
(2) with β = 18 is not known.
THEOREM 6 There exist extremal binary self-dual [44, 22, 8] codes with weight enumerators (4) for β = 154 and β = 74 with an automorphism of order 2 without fixed points.
Proof. From the extended Golay code g24 by deleting the last 2 coordinates of the vectors
ON THE BINARY SELF-DUAL CODES
45
having 0 on them we obtain a self-orthogonal [22, 10, 8] code g22 . From this code using
the theorem 2 construction we obtain a self-dual [44, 22, 8] code with weight enumerator
(4) for β = 154. From the code g22 we construct a self-dual code with weight enumerator
(4) for β = 74 using theorem 3.
Remark. Self-dual codes with a weight enumerator (4) with β = 154 are found with
automorphisms of orders 11 [16] and 5 [3]. An extremal code with a weight enumerator
(4) with β = 74 is obtained with an automorphism of order 5 i̧te Bu.
5. New Extremal Self-Dual Codes Obtained by Using Quasi-Cyclic Codes
THEOREM 7 There exist extremal self-dual codes of length 42 with a weight enumerators
(2) for β = 14 and for β = 6.
Proof. We construct a self-orthogonal [21, 7, 8] code as quasi-cyclic with a generator
matrix obtained from three circulant 7 × 7 matrices with first rows 0001101, 0000111 and
0001111. From this code we construct an extremal [42, 21, 8] code with weight enumerator
(2) for β = 14 using the construction from theorem 2. From theorem 3 construction we
obtain a self-dual code with weight enumerator (2) for β = 6.
THEOREM 8 There exist extremal self-dual codes of length 44 with weight enumerators (3)
for β = 38 and for β = 30.
Proof. From three circulant 7 × 7 matrices with first rows 0000001, 0010011 and 00111
11 and with parity check we obtain a self-orthogonal [22, 7, 8] code. This code by using
theorem 2 gives us an extremal self-dual [44, 22, 8] code with wei ght enumerator (3) for
β = 38. From this self-orthogonal code and theorem 3 we obtain a self-dual code with
weight enumerator (3) for β = 30.
Remark. A self-dual code with weight enumerator (3) for β = 38 is found as a double
circulant code [6], [7]. No other self-dual code with a weight enumerator (3) for β = 30 is
known.
6. Two Extremal Self-Dual Codes of Length 64
LEMMA 6 There exists a self-orthogonal [32, 11, 12] code with minimal distance 6 of its
dual code.
Proof. We will obtain such a code via an automorphism of order 5 with 6 independent
5-cycles. Let λ = (1, 2, 3, 4, 5) . . . (26, 27, 28, 29, 30) be an automorphism of the selforthogonal code C ′ of length 32. Denote the cycles of λ by Ä1 , . . . , Ä6 , and the fixed
points by Ä7 , Ä8 . Let Fλ (C ′ ) = {v ∈ C ′ : vλ = v} and E λ (C ′ ) = {v ∈ C ′ : wt (v|Äi ) ≡
0(mod 2), i = 1, . . . , 8}, where v|Äi is the restriction of v on Äi . Then the code C ′ is a
direct sum of the subcodes Fλ (C ′ ) and E λ (C ′ ). Clearly v ∈ Fλ (C ′ ) iff v ∈ C ′ and v is
46
BUYUKLIEVA
constant on each cycle. Let π: Fλ (C ′ ) → F28 be the projection map where if v ∈ Fλ (C ′ ),
(vπ )i = v j for some j ∈ Äi , i = 1, 2, . . . , 8. Denote by E λ (C ′ )∗ the code E λ (C ′ ) with the
last 2 coordinates deleted. For v in E λ (C ′ )∗ we let v|Äi = (v0 , v1 , . . . , v4 ) correspond to a
polynomial v0 + v1 x + · · · + v4 x 4 from P, where P is the set of even-weight polynomials in
F2 [x]/(x 5 + 1). Thus we obtain the map φ: E λ (C ′ )∗ → P 6 . P is a field with 16 elements.
From theorem 2 in [20] the code C ′ is a self-orthogonal iff the following two conditions
hold:
(i) π(Fλ (C ′ )) is a self-orthogonal binary code;
is a self-orthogonal code of length 6 over the field P under the inner
(ii) φ(E λ (C ′ )∗ )P
6
u i vi4 .
product (u, v) = i=1
The identity of P is the polynomial e = x + x 2 + x 3 + x 4 . The polynomial a = 1 + x
generates the cyclic group P ∗ , and β = x + x 4 is an element in this group of order 3 [4].
Let φ(E λ (C ′ )∗ ) be the self-orthogonal code over P with a generator matrix
¶
µ
e 0 e e β β2
0 e e a e a8
and π(Fλ (C ′ )) be the binary self-orthogonal [8, 3, 4] code with a generator matrix
11001010
11000101
11110000
From these two codes we obtain a self-orthogonal [32, 11, 12] code with a generat or matrix
01111000000111101111010010011000
10111000001011110111101000001100
11011000001101111011010101000100
11101000001110111101001011100000
00000011110111111000011111001000
G=
00000101111011101100101110100100
00000110111101100110110111010000
00000111011110100011111010101000
11111111110000000000111110000010
11111111110000000000000001111101
11111111111111111111000000000000
Its dual code is a [32, 21, 6] code.
THEOREM 9 There exists an extremal doubly-even self-dual code of length 64 with an
automorphism of order 2 without fixed points.
Proof. By applying theorem 2 to the code C ′ from lemma 6 we obtain a doubly-even
self-dual [64, 32, 12] code.
Remark. Since the self-orthogonal code has an automorphism of order 5 with 6 independent
5-cycles, this self-dual code has an automorphism of order 5 with 12 independent 5-cycles
ON THE BINARY SELF-DUAL CODES
47
and 4 fixed points. Extremal doubly-even self-dual [64, 32, 12] codes are obtained by
Pasquier [14], Kapralov and Tonchev [12], Yorgov [22], Harada and Kimura [9].
THEOREM 10 There exists a singly-even self-dual [64, 32, 12] code with weight enumerator
(5) with β = 44
W (y) = 1 + 2016y 12 + 19200y 14 + 237740y 16 + · · · .
Proof. Using theorem 3 and the self-orthogonal code from lemma 6 we construct a self-dual
[64, 32, 12] code with weight enumerator (5) for β = 44.
Remark. A code with weight enumerator (6) for β = 32 [5] and a code with weight
enumerator (5) with β = 18 [15] are known.
Acknowledgments
We thank the anonymous referees for their very helpful comments. We are grateful for
simplifying our proof of Theorem 1.
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