arXiv:1104.3727v4 [math.CO] 12 Nov 2012

A complete classification of doubly evenself-dual codes of length 40∗
Koichi Betsumiya
Graduate School of Science and Technology
Hirosaki University
Hirosaki 036–8561, Japanbetsumi@cc.hirosaki-u.ac.jp

Masaaki Harada
Department of Mathematical Sciences
Yamagata University
Yamagata 990–8560, Japan, andPRESTO, Japan Science and Technology Agency (JST)Saitama 332–0012, Japanmharada@sci.kj.yamagata-u.ac.jp

Akihiro Munemasa
Graduate School of Information Sciences
Tohoku University
Sendai 980–8579, Japan
munemasa@math.is.tohoku.ac.jp

November 13, 2012

Abstract
A complete classification of binary doubly even self-dual codes oflength 40 is given. As a consequence, a classification of binary extremal self-dual codes of length 38 is also given.
∗

This work was supported by JST PRESTO program.

1

1

Introduction

As described in [26], self-dual codes are an important class of linear codes
for both theoretical and practical reasons. It is a fundamental problem toclassify self-dual codes of modest lengths and much work has been done
towards classifying self-dual codes over Fq for q = 2 and 3, where Fq denotes
the finite field of order q and q is a prime power (see [26]).
Codes over F2 are called binary and all codes in this paper are binary.
The dual code C ⊥ of a code C of length n is defined as C ⊥ = {x ∈ Fn2 |x · y = 0 for all y ∈ C}, where x · y is the standard inner product. A code
C is called self-dual if C = C ⊥ . A self-dual code C is doubly even if allcodewords of C have weight divisible by four, and singly even if there is atleast one codeword of weight ≡ 2 (mod 4). It is known that a self-dual codeof length n exists if and only if n is even, and a doubly even self-dual code oflength n exists if and only if n is divisible by eight. The minimum weight d ofna self-dual code of length n is bounded by d ≤ 4⌊ 24⌋ + 6 if n ≡ 22 (mod 24),nd ≤ 4⌊ 24 ⌋ + 4 otherwise [20] and [25]. A self-dual code meeting the bound iscalled extremal.
Two codes C and C ′ are equivalent, denoted C ∼= C ′ , if one can beobtained from the other by permuting the coordinates. An automorphismof C is a permutation of the coordinates of C which preserves C. The set
consisting of all automorphisms of C is called the automorphism group of C
and it is denoted by Aut(C).
A classification of doubly even self-dual codes was done for lengths 8, 16
in [23], for length 24 in [24] and for length 32 in [9]. For length 40, only
some partial classifications have been done by various authors. Extremal
doubly even self-dual codes of length 40 with automorphism of a prime order
p having c cycles have been classified for (p, c) = (19, 2), (7, 5), (5, 4) in [28],(p, c) = (3, 6) in [6], (p, c) = (3, 8) in [16], and (p, c) = (5, 8) in [29]. The main
aim of this paper is to give a classification of doubly even self-dual codes oflength 40.
Theorem 1. There are 94343 inequivalent doubly even self-dual codes oflength 40, 16470 of which are extremal.
As a summary, we list in Table 1 the total number NT (n) of inequivalent
doubly even self-dual codes of length n and the number Nd (n) of inequivalent
doubly even self-dual codes of length n (n = 8, 16, . . . , 40) and minimum
weight d (d = 4, 8).
2

Table 1: Number of doubly even self-dual codes
Length n
8
16
24
32
40

NT (n)
1
2
9
85
94343

N4 (n)
1
2
8
80
77873

N8 (n)
1
5
16470

A classification of singly even self-dual codes of lengths up to 36 is known[3], [4], [9], [11], [23], [24]. As a consequence of Theorem 1, we give a classification of extremal singly even self-dual codes of length 38.
Generator matrices of all inequivalent doubly even self-dual codes oflength 40 and extremal self-dual codes of length 38 can be obtained electronically from [12]. All computer calculations in this paper were done byMagma [5].

2Classification method

In this section, we describe how to complete a classification of doubly evenself-dual codes of length 40.

2.1

Preliminaries

The weight enumerator of a doubly even self-dual code of length 40 can be
written as:
1 + A4 y 4 + (285 + 24A4 )y 8 + (21280 + 92A4 )y 12+ (239970 − 600A4 )y 16 + (525504 + 966A4 )y 20 + · · · + y 40 , (1)where Aw denotes the number of codewords of weight w (see e.g. [20]).
The number of distinct doubly even self-dual codes of length n is given[19] by the formula:n/2−2Y(2i + 1).
(2)
i=0

3

King [18] determined the number of distinct extremal doubly even self-dual
codes of length 40. Let N(40, d) denote the number of distinct doubly evenself-dual codes of length 40 and minimum weight d (d = 4, 8). Then we haveN(40, 4) = 4009357722800739726876619952910304312989584368968750,
N(40, 8) = 10263335567003567415076803513287627980544163840000000.

2.2

Minimum weight 4

Let C be a singly even self-dual code and let C0 denote the subcode ofcodewords having weight ≡ 0 (mod 4). Then C0 is a subcode of codimension
1. The shadow S of C is defined to be C0⊥ \C [10]. There are cosets C1 , C2 , C3
of C0 such that C0⊥ = C0 ∪ C1 ∪ C2 ∪ C3 , where C = C0 ∪ C2 and S = C1 ∪ C3 .
Proposition 2 (Brualdi and Pless [8]). Let C be a self-dual code of lengthn ≡ 6 (mod 8). Let C0 , C1 , C2 and C3 be as above. ThenC ∗ = {(v, 0, 0) | v ∈ C0 } ∪ {(v, 1, 1) | v ∈ C2 }∪ {(v, 1, 0) | v ∈ C1 } ∪ {(v, 0, 1) | v ∈ C3 }is a doubly even self-dual code of length n + 2.
There are 519492 inequivalent self-dual codes of length 36 [11]. By considering the direct sum of the unique self-dual code of length 2 and each ofthese codes, we have 519492 self-dual codes of length 38 and minimum weight
2. By Proposition 2, 519492 doubly even self-dual codes of length 40 andminimum weight 4 are constructed.
We examine the equivalence or inequivalence of codes as follows. Let C be
a doubly even self-dual code of length 40 and minimum weight d (d = 4, 8).
Let M(C) be the A8 × 40 matrix with rows composed of the codewords ofweight 8 in C, where the (1, 0)-matrix M(C) is regarded as a matrix over Z.
We define{nij | 1 ≤ i, j ≤ 40} \ {57} if C is extremal,
N(C) ={nij | 1 ≤ i, j ≤ 40}
otherwise,
where nij is the (i, j)-entry of M(C)T M(C), and M(C)T denotes the transposed matrix of M(C). The codewords of weight w in C are calculated by
the Magma function Words. Note that the codewords of weight 8 in C form4

a 1-(40, 8, 57) design when C is extremal. This means that nii = 57 for anyi (i = 1, 2, . . . , 40) and max{nij | 1 ≤ i, j ≤ 40} = 57 when C is extremal.
Then we consider the following:α(C) = (# Aut(C), A4 , max N(C), min N(C), #N(C)).
The automorphism group Aut(C) of the code C is calculated by the Magmafunction AutomorphismGroup. Of course, C and C ′ are inequivalent if α(C) 6=α(C ′ ). For a given set of codes, we divided into classes where each class contains codes C with identical α(C). Then we divided the codes in each classinto equivalence classes. This was done by the Magma function IsIsomorphic.
In this way, we checked equivalences among the above 519492 doubly
even self-dual codes of length 40 and minimum weight 4. Then we obtained
the set C40,4 of 77873 inequivalent doubly even self-dual codes with minimum
weight 4 satisfyingX
40!
= N(40, 4).
(3)#
Aut(C)C∈C
40,4This shows that there is no other doubly even self-dual code of length 40 andminimum weight 4. The numbers N(A4 ) of doubly even self-dual codes oflength 40 containing A4 codewords of weight 4 are listed in Table 2.

2.3

Minimum weight 8

For a set of coordinates I ⊂ {1, 2, . . . , n}, let π : Fn2 → Ft2 , π ′ : Fn2 → F2n−t
be the projection to the set of coordinates I, I ′ , respectively, where I ′ ={1, . . . , n} \ I and #I = t. For a code C of length n, the punctured code andthe shortened code of C on the set of coordinates I are the codes π ′ (C) and{π ′ (c) | c ∈ C, π(c) = 0}, respectively, where 0 denotes the zero vector.
If C is a doubly even code containing the all-one vector 1, then we denote
by qC : C ⊥ /C → F2 the map defined by qC (x+C) = wt(x)mod 2, where wt(x)2denotes the weight of x. It is easy to verify that the map qC is well-defined.
Let Ci be a doubly even code of length ni containing 1, for i = 1, 2. A
bijective linear map
f : C1⊥ /C1 → C2⊥ /C2(4)
is called an isometry if qC1 = qC2 ◦ f . The set of isometries (4) is denoted byΦ(C1 , C2 ). Note that an isometry exists only if n1 − n2 = 2(dim C1 − dim C2 )and n1 ≡ n2 (mod 8). For an isometry f ∈ Φ(C1 , C2 ), we define a code
D(C1 , C2 , f ) = {(x1 , x2 ) | x1 ∈ C1⊥ , x2 ∈ f (x1 + C1 )} ⊂ Fn2 1 +n2 ,5(5)

Table 2: Number of doubly even self-dual codes of length 40(0, 16470)(1, 20034)(2, 17276)(3, 12168)(4, 8471)(5, 5552)(6, 3916)(7, 2610)(8, 1932)(9, 1243)(10, 1093)(11, 669)(12, 605)(A4 , N (A4 ))(13, 382) (26, 47) (40, 12)(14, 374) (27, 16) (41, 1)(15, 231) (28, 38) (42, 9)(16, 236) (29, 13) (43, 3)(17, 143) (30, 29) (44, 7)(18, 160) (31, 7)(46, 7)(19, 100) (32, 22) (48, 4)(20, 104) (33, 3)(50, 4)(21, 54) (34, 25) (52, 6)(22, 90)(35, 3)(54, 2)(23, 37) (36, 11) (56, 1)(24, 59)(37, 4)(58, 4)(25, 26) (38, 11) (62, 2)(64, 3)(66, 1)(70, 3)(72, 1)(74, 1)(78, 1)(90, 1)(92, 1)(94, 2)(106, 1)
(134, 1)
(190, 1)

It is easy to see that D(C1 , C2 , f ) is a doubly even self-dual code. Conversely,
every doubly even self-dual code of length n1 + n2 containing a codeword ofweight n1 can be constructed by this method. Indeed, let x be a codeword ofweight n1 in a doubly even self-dual code C of length n1 + n2 . Let C2 (resp.
C1 ) be the shortened code of C on the support (resp. the complement of the
support) of x. Then C1 and C2 are doubly even codes. Moreover, C2⊥ (resp.
C1⊥ ) is the punctured code of C on the support (resp. the complement of the
support) of x. Let π and π ′ denote the projections onto the support of x andthe complement of the support of x, respectively. We define f : C1⊥ /C1 →C2⊥ /C2 by f (x1 + C1 ) = π ′ (x) + C2 , where x is a codeword of C satisfying
π(x) = x1 . Then D(C1 , C2 , f ) is equivalent to C.
For fixed codes C1 , C2 , the resulting code D(C1 , C2 , f ) depends on the
choice of an isometry f . However, some of these codes are equivalent toeach other. We will give a sufficient condition for two resulting codes to beequivalent. We need this criterion to reduce the amount of calculation to bereasonable.
First, we define some groups. For a doubly even code C containing 1,
we denote by G0 (C) the subgroup of GL(C ⊥ /C) induced by the action ofAut(C) on the linear space C ⊥ /C and denote by G1 (C) the subgroup Φ(C, C)
of GL(C ⊥ /C). By the definition, the group G0 (C) is a subgroup of G1 (C). If
6we replace f by σ2 ◦ f ◦ σ1 , where σi ∈ G0 (Ci ), then the resulting codes areequivalent, that is,D(C1, C2 , f ) ∼= D(C1, C2 , σ2 ◦ f ◦ σ1 ).
This means that, in order to enumerate the set of codes {D(C1, C2 , h) | h ∈Φ(C1 , C2 )} up to equivalence, we first fix f ∈ Φ(C1 , C2 ), and it suffices toenumerate the codes D(C1, C2 , f ◦ g), where g runs through a set of representatives for the double cosets(f −1 ◦ G0 (C2 ) ◦ f )\G1 (C1 )/G0 (C1 ).
We now apply this method with (n1 , n2 ) = (16, 24) in order to classify extremal doubly even self-dual codes of length 40. Note that from theweight enumerator (1), such a code has a codeword of weight 16. This meansthat every extremal doubly even self-dual code of length 40 is equivalent toD(C1 , C2 , f ) for some doubly even code C1 of length 16 containing 1, somedoubly even code C2 of length 24 containing 1, and f ∈ Φ(C1 , C2 ). All
doubly even codes of lengths 16 and 24 can be found in [21].
However, if dim C1 ≤ 2, then the degree of G1 (C1 ) ⊂ GL(C1⊥ /C1 ) as a
permutation group is too large to perform the double coset enumeration,so we only enumerated codes D(C1 , C2 , f ), where C1 is a doubly even codeof length 16 with dim C1 ≥ 3. Here, the group G1 (C1 ) was constructedby the Magma function GOPlus, and the double coset enumeration was
performed using the Magma function DoubleCosetRepresentatives. Then
we classified the resulting codes using the method described in the previous
subsection. In this way, we obtained a set of pairwise inequivalent 16468
extremal doubly even self-dual codes of length 40. It turns out that there are
two other codes. One is the code with automorphism group of order 6840
constructed in [28]. The other is the code H (1234) B6′ in the notation of [29]and this code has automorphism group of order 120.
In this way, we obtained the set C40,8 of 16470 inequivalent extremal
doubly even self-dual codes satisfyingX
40!
= N(40, 8).
(6)#Aut(C)C∈C
40,8

From (3) and (6), it follows that there is no other doubly even self-dualcode of length 40. This explains the number N(0) of extremal doubly evenself-dual codes of length 40 listed in Table 2. Therefore, we have Theorem 1.
7

3

Some properties

In this section, we give some properties of doubly even self-dual codes oflength 40.
The covering radius of a code C of length n is the smallest integer R
such that spheres of radius R around codewords of C cover the space Fn2 .
It is known that the covering radius is the same as the largest value among
weights of cosets. Here, the weight of a coset is the smallest weight of avector in the coset. The covering radius is a basic and important geometricparameter of a code. Assmus and Pless [2] began the study of the coveringradii of (extremal) doubly even self-dual codes.
Let R40,d be the covering radius of a doubly even self-dual code of length
40 and minimum weight d (d = 4, 8). Then, by the sphere-covering boundand the Delsarte bound (see [2]), 6 ≤ R40,8 ≤ 8 and 6 ≤ R40,4 ≤ 10. In Table
3, we list the numbers N(d, R) of doubly even self-dual codes with minimum
weight d and covering radius R. This was calculated by the Magma function
CoveringRadius. From the above calculation, we have the following:
Proposition 3. There is no doubly even self-dual code of length 40 withcovering radius 6.
Remark 4. In [14], based on a preprint by Michio Ozeki, the non-existence ofan extremal doubly even self-dual code with covering radius 6 was announced.
However, unfortunately, his preprint contained an error and, in his paper [22]he withdrew the above announcement. From the above calculation, the nonexistence of an extremal doubly even self-dual code with covering radius 6
was verified.
Remark 5. The two extremal doubly even self-dual codes with covering radius
7 can be found in [13] and [14].
Now we give some properties of extremal doubly even self-dual codes oflength 40. Let σ be an automorphism of odd prime order p. If σ has c
independent p-cycles and f fixed points, then σ is said to be of type p-(c, f ).
All extremal doubly even self-dual codes of length 40 with automorphismof type p-(c, f ) are known for p ≥ 5 (see [15, Table 3]). The cases with(p, c) = (3, 6) and (3, 8) were considered in [6] and [16], respectively. The
numbers N(p, c) of inequivalent extremal doubly even self-dual codes with
automorphism of type p-(c, f ) are listed in Table 4 for (p, c) = (3, 6), (3, 10)
and (3, 12). It is claimed in [6, Theorem 12] that N(3, 6) = 16. However, we
8Table 3: Covering radii of doubly even self-dual codes
R
6
7
8
9
10

N (4, R)
0
23
76768
954
128

N (8, R)
0
2
16468
-

verified that N(3, 6) = 17. Since the list of the 16 codes is not available, we
are unable to compare the result with ours.
Proposition 6. There are 17, 70 and 322 inequivalent extremal doubly evenself-dual codes of length 40 with automorphism of types 3-(6, 22), 3-(10, 10)
and 3-(12, 4), respectively.

Table 4: N(p, c) for (p, c) = (3, 6), (3, 10) and (3, 12)(p, c)N (p, c)(3, 6)
17(3, 10)70(3, 12)322

In Table 5, we list the numbers N(# Aut) of extremal doubly even selfdual codes with automorphism groups of order # Aut.
As we mentioned at the end of Subsection 2.3, we have the following:
Proposition 7. Let Cx denote the shortened code of C on the complementof the support of a codeword x. Then there are two inequivalent extremal
doubly even self-dual codes C of length 40 such that dim Cx ≤ 2 for all x ∈ Cwith wt(x) = 16.
Although the condition given in Proposition 7 can be characterized by
the vanishing of a coefficient in the weight enumerator of genus 3 (see [27]),
we have not been able to prove Proposition 7 directly, without classifying all
extremal doubly even self-dual codes of length 40.
In Table 6, we list the numbers N(dim) of extremal doubly even selfdual codes such that subcodes generated by codewords of weight 8 have9

Table 5: Orders of automorphism groups(1, 10400)(2, 3538)(3, 43)(4, 1189)(5, 2)(6, 68)(8, 459)(10, 8)(12, 80)(16, 233)(18, 1)(20, 4)(24, 41)(30, 2)(32, 70)(# Aut, N (# Aut))(36, 1)(256, 21)(3072, 3)(38, 1)(288, 4)(3840, 1)(40, 5)(320, 1)(4096, 1)(48, 34)(384, 12)(4608, 2)(60, 2)(512, 16)(5376, 1)(64, 75)(576, 3)(6144, 7)(72, 4)(720, 2)(6840, 1)(96, 12)(768, 7)(9216, 1)(114, 1)(1024, 3) (12288, 2)(120, 5)(1296, 1) (16384, 1)(128, 46) (1536, 10) (18432, 1)(144, 4)(1728, 1) (20480, 1)(160, 1)
(1920, 1) (20736, 1)(192, 12) (2048, 4) (32768, 1)(240, 2)(2688, 1) (49152, 3)(61440, 1)(65536, 1)(110592, 1)(147456, 1)(245760, 1)(737280, 1)(786432, 1)(983040, 1)(1474560, 1)(5505024, 1)(8257536, 1)(44236800, 1)(82575360, 1)

dimension dim. The dimension is the same as the 2-rank of the 1-(40, 8, 57)
design formed by the codewords of weight 8.
Table 6: Dimensions of subcodes generated by codewords of weight 8dim
N (dim)

4

17
5

18
1

19
14

20
16450

Extremal self-dual codes of length 38

Let D be a doubly even self-dual code of length 40. Let C be the code
obtained from D for which some particular pair of coordinates i, j are 00 and
11 and deleting these coordinates. Then C is a self-dual code of length 38.
Here, we say that C is obtained from D by subtracting coordinates i, j. In
addition, any self-dual code of length 38 is obtained from some doubly even10

self-dual code of length 40 by subtracting some two coordinates (see [9]).
Due to the computational complexity, we only completed a classification of
extremal self-dual codes of length 38. Note that there are at least 13644433
inequivalent self-dual codes of length 38 [11].
Any extremal self-dual code C of length 38 and its shadow S have one of
the following weight enumerators [10]: WC = 1 + 171y 8 + 1862y 10 + 10374y 12 + 36765y 14 + 84759y 16+128212y 18 + · · · ,
(7)

7
11
15
19
WS = 114y + 9044y + 118446y + 269080y + · · · , WC = 1 + 203y 8 + 1702y 10 + 10598y 12 + 36925y 14 + 84055y 16+128660y 18 + · · · ,
(8)

3
7
11
15
19
WS = y + 106y + 9072y + 118390y + 269150y + · · · .

Although the following two lemmas are somewhat trivial, it is useful infinding extremal self-dual codes of length 38.

Lemma 8. Any extremal self-dual code of length 38 with weight enumerator(7) (resp. (8)) is obtained from some extremal doubly even self-dual code oflength 40 (resp. some doubly even self-dual code of length 40 containing one
codeword of weight 4) by subtracting some two coordinates.
Proof. Let C be an extremal self-dual code of length 38 with weight enumerator (7) (resp. (8)). By Proposition 2, a doubly even self-dual code C ∗ oflength 40 is constructed. In addition, by (7) (resp. (8)), C ∗ is extremal (resp.
C ∗ contains one codeword of weight 4). The code C is obtained from C ∗ bysubtracting the last two coordinates. The result follows.
For the remainder of this section, we suppose that D is either an extremal
doubly even self-dual code of length 40 or a doubly even self-dual code oflength 40 containing one codeword of weight 4. Also, let Di,j denote theself-dual code of length 38 obtained from D by subtracting two coordinatesi, j.
Lemma 9. Let M(D) be the matrix with rows composed of the codewords ofweight 8 in D, where the (1, 0)-matrix M(D) is regarded as a matrix over Z.
(1) Suppose that D is extremal. Then the (i, j)-entry of M(D)T M(D) iszero if and only if Di,j is extremal.
11(2) Suppose that D contains one codeword x of weight 4. Then the (i, j)entry of M(D)T M(D) is zero and the pair of coordinates i, j in x are
10 or 01 if and only if Di,j is extremal.
Proof. There is a codeword of weight 8 in D for which the coordinates i, j
are 11 if and only if Di,j contains a codeword of weight 6. Suppose that D
contains one codeword x of weight 4. The coordinates i, j in x are 11 (resp.
00) if and only if Di,j contains a codeword of weight 2 (resp. 4).
By Lemma 9, from all inequivalent extremal doubly even self-dual codes
and all inequivalent doubly even self-dual codes containing one codeword ofweight 4, we constructed extremal self-dual codes of length 38 which need bechecked further for equivalences. Then we checked equivalences among thesecodes using the method similar to that given in Section 2. Finally, we havethe following:
Proposition 10. There are 2744 inequivalent extremal self-dual codes oflength 38. Of these 1730 have weight enumerator (7) and 1014 have weight
enumerator (8).
Remark 11. A classification of extremal self-dual codes of length 38 was very
recently obtained in [1] by somewhat different techniques. This was indicatedby Jon-Lark Kim in a private communication [17].
In Table 7, we list the numbers N(# Aut) of extremal self-dual codeswith automorphism groups of order # Aut for both weight enumerators (7)and (8).
Some historical comments (July 27, 2012). A classification of extremal
self-dual codes of length 38 was completed in [1], and a classification of all
self-dual codes of length 38 was completed in [7]. The paper [1] was submitted
before this paper was submitted, and the paper [7] was submitted after thispaper was submitted.

Acknowledgements
The authors would like to thank Jon-Lark Kim for providing information on
[1].

12

Table 7: Number of extremal self-dual codes of length 38(1, 1480)(2, 177)(3, 15)(1, 773)(2, 145)(3, 21)(# Aut, N (# Aut))
Weight enumerator (7)(4, 30) (9, 1)(24, 4)(6, 7) (12, 5) (36, 1)(8, 7) (18, 1) (168, 1)
Weight enumerator (8)(4, 38) (12, 3) (24, 10)(6, 10) (14, 1) (144, 1)(8, 8) (21, 1) (168, 1)(342, 1)

(216, 1)
(504, 1)

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