数理解析研究所講究録 1228 巻 2001 年 43-50

43Classification of Type II code over GF(4) of some small lengths
名古屋大学多元数理科学研究科 別宮耕一 (Koichi Betsumiya)
Graduate School of Mathematics, Nagoya University

1

Introductionof length
elements. Let $C$ be acode overbe the finite field withLet
. Let $B=\{b_{1}, b_{2}, \ldots, b_{r}\}$ be abasis of, which is asubspace of the vector space, )the representation. We denote by $\phi_{B}(a)=$ ( , a2,
over
. For
with respect to abasis $B$ , that is,overof, we also denote by
the vector in $GF(2)^{rn}$ obtained$u=$. We callthe binary image of with respect,,
by concatenating
for $1\leq i,j\leq r$ whereis called atrace-Orthogonal basis (TOB) if
to B.
.
over
function of
Tr denotes thewhose binary, Pasquier and Wolfmann studied self-dual codes overIn
images with respect to aTOB are binary TyPe II codes (that is, binary doubly-evenself-dual codes) including the extended Hamming code and the extended Golay code
are Type II codes with respect to a(cf. [8, 10]). We say that such codes over). Recently, it has been proved by the
TOB [4] (see [6] for TyPe II codes overfor self-dual codes overauthor that the TyPe II property with respect to ais independent of the choice of aTOB [1]. This allows us to call aType code if $C$
is aType II code with respect to aTOB, without areference to an explicit TOB. The
of length is defined by $rn$ , which is the length ofbinary length of acode overits binary image. The TyPe II codes with binary length uP to 24 have been classified(cf. [2, 4, 6]). We refer to [9] for the classification of binary TyPe II codes. The next
for any with binary length
problem would be to classify all TyPe II codes over32.
$2^{r}$$\mathrm{G}\mathrm{F}(2^{r})$$\mathrm{G}\mathrm{F}(2^{r})$$\mathrm{G}\mathrm{F}(2^{r})^{n}$$n$$\mathrm{G}\mathrm{F}(2^{r})$$\mathrm{G}\mathrm{F}(2)$$a\in \mathrm{G}\mathrm{F}(2^{r})$$a_{1}$$a_{r}$$\ldots$$\in \mathrm{G}\mathrm{F}(2)^{r}$$a= \sum_{i=1}^{r}a_{i}b_{i}$$\mathrm{G}\mathrm{F}(2)$$\phi_{B}(u)$$(u_{1}, u_{2}, \ldots,u_{n})\in \mathrm{G}\mathrm{F}(2^{r})^{n}$$\phi_{B}(u_{1})$$\ldots$$\phi_{B}(u_{n})$$C$$\phi_{B}(C)$$B$$\mathrm{T}\mathrm{r}(b_{ij}b)=\delta_{ij}$$\mathrm{G}\mathrm{F}(2^{r})$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\mathrm{G}\mathrm{F}(2)$$1980’ \mathrm{s}$$\mathrm{G}\mathrm{F}(2^{r})$$\mathrm{G}\mathrm{F}(2^{r})$$\mathrm{G}\mathrm{F}(4)$$\mathrm{T}\mathrm{O}\mathrm{B}$$\mathrm{G}\mathrm{F}(2^{r})$$C$$\mathrm{G}\mathrm{F}(2^{r})$$n$$\mathrm{G}\mathrm{F}(2^{r})$Theorem 1.1 (Munemasa [7]) The total number of Typeis given by$N_{II,r}(n)= \prod_{i=0}^{n/2-2}(2^{ri}+1)$

if$rn\equiv 0$$(\mathrm{m}\mathrm{o}\mathrm{d} 8)$

and$n\equiv 0$$(\mathrm{m}\mathrm{o}\mathrm{d} 4)$$\mathrm{I}\mathrm{I}$,, and 0othemise.

$r$$II\mathrm{G}\mathrm{F}(2^{r})$-codes of length

$n$

(1)

44
The formula (1) is called the mass formula for Type II codes over
. By The
orem 1.1, the possible cases for which there is aType II code overof lengthwith binary lengh 32 are $(n, r)=(32,1)$ , $(16, 2)$ , $(8, 4)$ and $(4, 8)$ . For the cases$(n, r)=(32,1)$ and $(n, r)=(8,4)$ , the complete classifications are given in [5] and [4],respectively. Furthermore, for the case $(n, r)=(4,8)$ , it is shown that there exists a
unique TyPe II code uP to permutation-equivalence [7].
In this paper, we give aclassification for the case $(n, r)=(16,2)$ , that is, Type II codes
overof length 16. It is the only remaining case to complete the classification ofType codes with binary length 32.
$\mathrm{G}\mathrm{F}(2^{r})$$\mathrm{G}\mathrm{F}(2^{r})$$n$$\mathrm{G}\mathrm{F}(4)$$\mathrm{I}\mathrm{I}$2Classification of Type II Codesover$\mathrm{G}\mathrm{F}(4)$of Length 16

In this section, we give aclassification of Type II codes of length 16 over
. A bisorted matrix is constructed by sorted vectors on alexic0graphical order with both directions North to South and West to East. We check all thebisorted
matrices ’s such that (I ) generates aType II code, whereis theidentity matrix. It is sufficient to consider such matrices to complete theclassification [3]. Indeed, we obtain 82588 distinct Type II codes by the method above.
Let $C$ be acode overof length 16, and let$\mathrm{G}\mathrm{F}(4)=$$\mathrm{G}\mathrm{F}(2)[\omega]/(\omega^{2}+\omega +1)$

$A$

$8\cross 8\mathrm{G}\mathrm{F}(4)$

$A$

$I$

$8\cross 8$$\mathrm{G}\mathrm{F}(4)$,$\tilde{C}=\{(\hat{c}_{1},\hat{c}_{2}, \ldots,\hat{c}_{16})|\mathrm{w}\mathrm{t}(c)=6, c=(c_{1}, c_{2}, \ldots, c_{16})\in C\}$

where
is the Hamming weight of and $6=0$ if . $=0$ and . $=1$ otherwise. Thenis anon-linear binary code of length 16. We now consider three invariants of codes
under the permutation-equivalence:$\mathrm{w}\mathrm{t}(c)$$c$$\hat{\mathrm{q}}$$\mathrm{q}$$\overline{C}$

1. the Hamming weight enumerator $W(C)$ ,2. the order

3. the order$|\mathrm{A}\mathrm{u}\mathrm{t}(C)|$$|\mathrm{A}\mathrm{u}\mathrm{t}(\tilde{C})|$of the permutation automorphism group of $C$ ,of the permutation automorphism group of$\tilde{C}$.

Calculating the invariants for every code using MAGMA, we find 48 codes,,with distinct sets of invariants. The minimum Hamming weightof , the Hamming weight enumerator $W(D:)$ of
and the values
axe listed in Table 1. The
weight enumerators axe listed in Table 2where only the coefficients of the monomials
for $i=3,4,6,7$ ,, 16 are given. For all weight enumerators, the coefficients of,the monomials,
andare 1, 0, 0and 0, respectively. We verified that
$D_{1}$$d(D_{i})$$D_{:}$$x^{16-i}y^{:}$$\ldots$$x^{16}$$x^{15}y$$x^{14}y^{2}$$x^{11}y^{5}$$|\mathrm{A}\mathrm{u}\mathrm{t}(D:)|$$D_{i}$$\ldots$$D_{48}$

45

Table 1: Properties of the Type II codes over$\mathrm{G}\mathrm{F}(4)$of lengh 16

46

Table 2: The weight enumerators

47
andare 7962624 and 36864, respectively. By Table 1,
are not permutation-equivalent. By Theorem 1.1, we have that$|\mathrm{A}\mathrm{u}\mathrm{t}(D_{32}\ovalbox{\tt\small REJECT}$$|\ovalbox{\tt\small REJECT} \mathrm{t}(D\mathrm{S}3)|$$\sum_{i=1}^{48}\frac{16!}{|\mathrm{A}\mathrm{u}\mathrm{t}(D_{i})|}$$D_{\mathrm{I}_{\rangle}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$\rangle$D_{48}$11925737086250$=$$=N_{II,2}(16)$.

Thus, we obtain the following:

Theorem 2.1 There exist 48 Type $II$ codes overequivalence.

of length 16, up to permutation-$\mathrm{G}\mathrm{F}(4)$Generator matrices of the codes are given in Section 3. The Probenius automorphismis the ffield automorphism on
defined by. Each of these 48 codes ispermutation-equivalent to its Frobenius image. Finally, we calculate the binary imagesof the codes using MAGMA. In Table 3, the binary codes
are given and the minimum
Hamming weights $d(\phi(C))$ of
are also given. We use the notation in [9] for thebinary Type II codes of length 32. There exist 7Type II codes whose binary images areof minimum Hamming weight 8uP to permutation-equivalence. Only 4codes among
the 5extremal binary Type codes of length 32 are the binary images of Type II codesover.
As aconsequence, we obtain the complete classification of Type II codes overwith binary length 32. The numbers of Type II codes overwith binary length
32 up to permutation-equivalence are in Table 4. In the table,,
anddenotethe number of codes up to permutation-equivalence, the number of codes whose binary
images are extremal, and the number of extremal binary codes obtained as the binary$a\mapsto a^{2}$$\mathrm{G}\mathrm{F}(4)$$\phi(C)$$\phi(C)$$\mathrm{I}\mathrm{I}$$\mathrm{G}\mathrm{F}(4)$$\mathrm{G}\mathrm{F}(2^{r})$$\mathrm{G}\mathrm{F}(2^{r})$$\# 1$$\# 2$$\# 3$

images, respectively.

3

Generator Matrices

In order to save space, we list generator matrices $(I\{a_{i,j}\})$ as$a_{1,1}a_{1,2}\cdots a_{1,8}$

where$\overline{\omega}$

denotes$\omega^{2}$,$a_{2,1}\cdots a_{2,8}$$000000\alpha\overline{w}$$D_{2}$:$\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}ax\overline{v}$,$000000\overline{\omega}(v$$D_{3}$:$000000a\overline{w}$,$000000_{\mathrm{t}}^{-}‘ w$,$D_{4}$:$000000_{A\lambda}‘\overline{v}$,$000000\overline{\omega}(v$, 00011100, 00101100, 0:$000000\{Ax\overline{v}$,$0000\mathrm{c}\mathrm{j}^{-}00$$D_{5}$$\ldots$,$a_{8,1}\cdots a_{8,8}$

,

.

:

$D_{1}$,, 000000uxD,$0000a\mathrm{x}\overline{v}00,0000‘\overline{\mathrm{a}}\mathrm{v}00$,$00(\mathit{1}x\overline{v}0000,$$\mathrm{O}\mathrm{O}\overline{\alpha}w\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O},$, 00011100, 00101100, 00110100, 00111000,,$\mathrm{O}\{\overline{u}v1‘\overline{a}v00$,$00,0\{\overline{u}uv1\overline{\omega}00$,$0000\omega\overline{\omega}00,00\omega(\overline{v}0000,0\{v\overline{\alpha}xv1\overline{\omega}00,$$000\alpha\overline{w}(v1\overline{\omega},$$00\omega 0$$1\overline{a}x_{k}^{-}\mathrm{x}v$, 0$‘ u\overline{d}\overline{u}v1$$‘\alpha\overline{v}100\overline{\alpha}w$,$0\overline{\omega}\alpha x\overline{v}00\mathrm{r}v1$,$a\mathrm{x}\overline{v}\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O},\overline{\iota\alpha}\mathrm{v}\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}$$‘\alpha\overline{v}000000,\overline{a}\mathrm{x}\mathrm{v}\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}$$\omega 01\overline{a}\lambda\overline{v}(v00,\overline{\omega}0\overline{\alpha}x_{k}xv100$$\omega \mathrm{o}_{(}\overline{u}\overline{v}1\omega 00,\overline{\omega}0_{\mathrm{A}}‘[](k\lambda\overline{v}100$$\omega 01‘\overline{u}_{\lambda}^{-}xv00,\overline{\omega}0‘\overline{v}\mathfrak{u}X\theta 100$

48

Table 3: Binary images of the Type II code over$\mathrm{G}\mathrm{F}(4)$of lengh 16

Table 4: The number of TyPe II codes with binary length 32
$r$

1
2
4
8

#1 #2
$\# 1$$\# 2$$\# 3$

85
48
6

5
7
1
0

5
4

1

1

0

reference[5]
Section 2
Sec
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$\{$$\mathrm{t}\mathrm{i}\mathrm{c}$$)\mathrm{n}$

[4]
[7]
$[4]$

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$[7]$

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$2$

49
$D_{6}$: 000000$‘ u\overline{v}$$D_{7}:000000‘ a\overline{v}$,, 0000uaD00, 000,$D_{8}$:$D_{9}$: 0000000xj,$\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}_{(u\overline{v}}$$0001‘\overline{u}u_{\mathrm{A}}^{-}w$$\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}\omega\zeta\overline{v}\mathrm{O}\mathrm{O},$$0000\alpha \mathrm{x}\overline{v}00,0011\omega 1$: OOOOOOci,$D_{11}$: 0000000xj, 000$D_{12}$:$D_{13}$:$\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}_{[\alpha\overline{v}}$$D_{14}:000000a\overline{w}$,,$\mathrm{o}\mathrm{o}\mathrm{o}_{(}\alpha_{\ell}x\overline{v}1\overline{\omega}$$‘\alpha u\overline{v}1\overline{\omega}$$1\overline{\omega}$$000\alpha \mathrm{x}_{4}^{-}xv1\overline{\omega}$$000(Ax\alpha\overline{v}1\overline{\omega},$$D_{10}$$\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{a}\mathrm{x}\mathrm{I}/$$‘\alpha_{l}^{-}\mathrm{x}v1\overline{\omega}$, 0010,, $0101\omega 1$,$1\overline{\omega}$$0\{\mathrm{t}w\mathrm{o}_{\{}u\alpha\overline{u}v$$00\omega 0\overline{\omega}1(\overline{a}v,$$111\overline{\omega}$$\mathrm{o}\mathrm{o}\mathrm{o}_{(}uv\overline{\omega}1\overline{\omega}$$\mathrm{O}\overline{\omega}\alpha xv\mathrm{O}\mathrm{O}1\overline{\omega}$$0\omega\overline{(\iota}\overline{w}00\omega 1,$$\mathrm{O}\alpha w1\omega 1\omega 1,$,,,, 00111000,$00\omega 1$,$‘\overline{u}4\lambda\overline{\alpha}d$$\mathrm{O}ax\overline{v}1\mathrm{O}\mathrm{O}\overline{\alpha}w$, $0110\omega 1$,$1\overline{\omega}$,, 0$0\omega 1\omega\overline{a}x_{l}^{-_{J}\mathrm{o}\mathrm{o}}$,$\omega 0‘\overline{\alpha}_{\iota}\overline{\mathrm{x}}v100,\overline{\omega}0\{uv1\overline{\omega}00$$\omega 1111\overline{\omega}00,--\ovalbox{\tt\small REJECT}^{---}00$$\mathrm{o}_{\{}\overline{\alpha}v10\circ[\overline{\alpha}v,$, 01110000,$0[a_{l}^{-}xv001\overline{\omega},$$0\omega 0\overline{\omega}\mathrm{O}1\overline{\alpha}w$,$\mathrm{O}\overline{a}\mathrm{x}uv\mathrm{O}\mathrm{O}1\overline{\omega}$$\omega 000\overline{\alpha}w1\overline{\omega},\overline{\omega}000\omega 1\overline{\alpha}w$,$‘\iota \mathrm{u}\overline{J}0\overline{\omega}0\omega 1,0\overline{\omega}1101\omega 1$$\omega 1$,, 00111000, 01011000,$01101\omega 1\overline{\omega}$, 00011100, 00101100, 01001100,$011101‘\overline{\mathrm{a}}\mathrm{v}$,, 01111$01110\omega 1\overline{\omega}$,$0\overline{\alpha}w$,, $000000(\overline{u}v$ , 00011100, 00101100, 01001100, 10001100,$D_{15}$:$D_{16}$: $000000a\overline{w}$ , 000000aw, 00011100,$\omega 1\overline{a}x^{-}\ovalbox{\tt\small REJECT} w1\overline{\omega},\overline{a}x\overline{\alpha}_{k}xv11(\overline{\alpha}v$$0\omega 1[\overline{u}\alpha\overline{v}00,0\overline{\omega}1‘\overline{\alpha}\overline{v}11\overline{\omega},$ $\omega 1$$0\omega 1\{\mathrm{A}\mathrm{x}_{\overline{\iota}}\overline{w}00,0\omega 1[\overline{\alpha}\alpha\overline{v}00,0\overline{\omega}0\overline{\mathrm{m}}^{-}x\overline{\alpha}\overline{u}v,$$\mathrm{o}\mathrm{o}\mathrm{o}_{[}ua\overline{v}1\overline{\omega}$$1\overline{\omega}1100,--\ovalbox{\tt\small REJECT}^{---}00$$\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}\mathrm{O}\alpha\overline{w}$$10011\overline{\omega}$,$\omega 1000[\overline{u}\overline{u}v,$$\omega 00$$‘\overline{\alpha}_{\overline{\iota}}\overline{w}00[\overline{u}\overline{a}v$$‘\overline{\alpha}\overline{v}000\alpha w1$$1111\overline{\omega},\overline{\omega}00\overline{a}\overline{\mathrm{u}}\overline{h}\overline{u}w$$\omega 000\overline{\omega}\mathrm{t}\overline{\alpha}v1,\overline{\omega}000\alpha xv1\overline{\omega}$$1111[\iota\overline{w}00,1111‘\overline{\mathrm{a}}\mathrm{v}00$$\mathrm{o}_{\{}\ovalbox{\tt\small REJECT}^{-}xu\overline{v}100,0\overline{\omega}_{\mathrm{A}}w\overline{\omega}100,1‘ u_{\lambda}\lambda\overline{v}0\overline{\omega}00,1\overline{\alpha}\overline{w}0\{uv00,111\overline{\alpha}w1\mathrm{O}\mathrm{O}$, 000$‘ uv\overline{\omega}1\overline{\omega}$,$D_{18}$: 000000aw, 000$‘ a\alpha\overline{v}1\overline{\omega}$$D_{19}$: 0000000xj,$D_{20}$:$D_{21}$: OOOOOOci,$D_{22}$: 000000uw, 000000uw,$D_{23}$:$D_{24}$: 00000111, 00001011, 00001101, 00001110, 01110000, 10110000, 11010000, 11100000$D_{25}$: 00000111, 00001011, 00010011, 00100011, 01000011, 10000011, 111111 1, 11111110$D_{26}$: 00000111, 00001011,$D_{27}$$D_{17}:000000_{[}4X\overline{\theta}$$\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}_{(v\overline{\omega}}$,,$\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}\mathrm{o}_{\omega(_{\lambda}J}^{-}$$000axv\overline{\omega}1\overline{\omega}$$\mathrm{o}\mathrm{o}\mathrm{o}_{\{}\mathrm{A}\kappa\alpha_{\mathrm{A}}^{-_{J}}1\overline{\omega}$$000\overline{\omega}[1w1\overline{\omega},$,$\mathrm{o}\mathrm{o}\mathrm{o}_{[}lx\overline{v}\omega 1\overline{\omega}$,$00\omega 1$,$00\omega 0\overline{\omega}1\overline{\alpha}w$,$0\omega 0\overline{\omega}01\overline{\alpha}\omega$,$\mathrm{O}\mathrm{O}\omega \mathrm{O}\overline{\omega}1\overline{a}w$,$0\omega 01111\overline{\omega}$$\mathrm{o}\mathrm{o}\mathrm{o}_{(\mathrm{A}}\mathrm{u}x\overline{v}1\overline{\omega}$$000\omega(u\overline{v}1\overline{\omega}$$000[\alpha\overline{\iota}w1\overline{\omega},$, 00111000,,$111\overline{\omega}$$011\iota\iota hX\iota w1,101‘\alpha Ahxv1$,$110M\lambda\alpha d1$,$01\overline{\omega}011\omega 1$,$10\overline{\omega}011\omega 1$,$\mathrm{o}_{[_{k}}^{-}x\overline{v}00‘ av1$,

$1\omega 1\overline{\omega}1100,1^{-}[\lambda\overline{w}$, 0$0\alpha\lambda v0[Ax\alpha_{\mathrm{A}}^{-}xv,$$\mathrm{o}_{\{}l\lambda lh^{-}x\overline{v}100$,$\omega 0\overline{a}\mathrm{x}v1\overline{\omega}00$,$0[Axv0‘ au_{A}^{-}x\theta$,$\omega 0\overline{\omega}0\omega 01\overline{\omega}$,, 1$1ax\overline{v}11\mathrm{O}\mathrm{O},$uvOOO,$‘\overline{u}_{L}^{-}w1\omega 00,1\omega 1\overline{\omega}1100,1\overline{\omega}\overline{a}x\overline{v}1\mathrm{O}1\overline{\omega}$,$0‘\overline{a}_{\mathrm{t}}w001\overline{\omega}$$\omega[_{\mathrm{J}}-\lambda\overline{v}0\omega 100,$, 11100000,[$\overline{u}v0\overline{\omega}1\omega 00,\overline{\omega}1\omega 1$$11\omega 1\overline{\omega}100$, $11\omega 00$$1\overline{\omega}0\overline{\omega}\overline{\omega}11\overline{\omega}$$1a\overline{w}\mathrm{O}\overline{a}xv\mathrm{O}\mathrm{O},$$111\overline{\omega}00\omega 1$

1100,$11^{-}\{\lambda x\overline{v}0\overline{\omega}1\overline{\omega}$$1\overline{\omega}1\omega 1$, 0:00000111, 00001011, 00010011,$001111\alpha\overline{\omega}$,$010111\{a\overline{\theta}$,$100111\{Ax\overline{v}$, lll[ xuv 0, 11$D_{28}$: 00000111, 00001011, 00010011,$0\mathrm{m}^{-}\mathrm{A}a\alpha_{\{}J0\omega$,$0\overline{m}x\alpha\alpha v\mathrm{O}\omega$,$1u\alpha\overline{\alpha}_{\overline{\iota}}\overline{w}1\omega$,$D_{29}$: 00000111, 00001011,$0\alpha M\overline{x}_{l}xv0\omega$,$D_{30}$: 00000111,$D_{31}$: 00000111, 00$D_{32}$: 00000111,$000a\mathrm{x}\overline{\iota}\mathrm{X}_{\lambda}\lambda\overline{v}1$$D_{33}$: 00000111,$\mathrm{o}\mathrm{o}_{(A}xv10_{A\lambda}^{-}‘\overline{v}$$D_{34}$: 00000111,$D_{35}$: 00000111,$D_{36}$: 00000111, 00001011,$\mathrm{O}u\ovalbox{\tt\small REJECT}^{-}x_{\mathrm{A}}u’ 0\omega$$D_{37}$: 00000111,$\mathrm{o}_{[}av01\overline{\mathfrak{a}}x\overline{v}0$$D_{38}$: 00000111,$D_{39}$: 00000111, 00(twl$D_{40}$:00000111,$D_{41}$: 00000111,$D_{42}$:$001111(v\overline{\omega},$$1\overline{\alpha}\overline{w}\mathrm{O}\mathrm{O}\alpha \mathrm{x}v$, 1$0\overline{\alpha}x\overline{v}\mathrm{O}\mathrm{O}\alpha w$, $1ax_{l}xuv01$$010111\omega[\overline{AJ},$,$\mathrm{o}\mathrm{o}\mathrm{o}_{(\overline{\iota}\mathrm{u}x\alpha_{kJ}^{-1}}$,,$0\omega 0\omega 1\overline{\omega}0\overline{\omega}$,$\mathrm{O}\mathrm{O}\mathrm{O}\omega\overline{a}_{\iota}\overline{w}1,00\omega 0\overline{\omega}1\{\overline{u}v$,,$\omega \mathrm{o}_{\overline{\alpha}um\iota J}\mathrm{o}$,,$\omega 0\overline{\omega}00\overline{\omega}1\omega$,$\omega 0\overline{\omega}a\iota uw0$,$0\omega 0\omega 1\overline{\omega}0\overline{\omega}$,,$0_{(uv}01\overline{\alpha}\overline{w}0$,$00aw10\overline{\omega}(\overline{v},$$0\omega 0\omega 1\overline{\omega}0\overline{\omega}$,,$\mathrm{O}\omega \mathrm{O}\mathrm{O}\overline{\omega}1ax\overline{v}$,,$0\omega 0\omega 1\overline{\omega}0\overline{\omega}$,$ooo_{[}\alpha\overline{u}_{A}x\overline{v}1$$\mathrm{O}\mathrm{O}\mathrm{O}\alpha w\overline{\omega}\overline{\omega}1,00\omega 0‘\overline{u}v1\overline{\omega}$$D_{43}:000ax\iota \mathrm{K}_{\mathrm{A}}^{-}\overline{w}1,00\omega \mathrm{o}_{(_{L}}^{-}\mathrm{u}J1\overline{\omega}$$0\omega 0\overline{\omega}0^{-}\{A\lambda v1,$,,$\mathrm{O}\overline{a}x\overline{v}$,,01110000,,$\mathrm{o}\mathrm{o}\mathrm{o}_{(}u_{l}^{-}xa\overline{\iota}\prime 1,0a\overline{w}00[\mathrm{A}\overline{w}1,$$011\mathrm{O}11a\lambda\overline{v}$$\omega 0\overline{\omega}axv0(4xd,$$\omega 0\overline{\omega}$

11110,$1\overline{\omega}\overline{\omega}000\omega\omega$, 1,$11\alpha x_{\lambda}xuv0$$1\overline{a}\lambda\overline{v}\overline{\omega}\mathrm{O}\overline{\omega}$$111a^{-}x\overline{u}_{\mathrm{A}}^{-}\mathrm{x}\overline{v}0$$\mathrm{O}\overline{a}\mathrm{x}\overline{v}\mathrm{O}1\mathrm{O}aw$,$111001\omega\overline{\omega}$$1^{-}\ovalbox{\tt\small REJECT}^{--}1,1\overline{a}\overline{m}\overline{\alpha}\overline{d}110,11001\alpha\overline{\omega}1$, $1axuuv011$ ,$\mathrm{O}\alpha xv01\overline{\omega}\overline{\omega}0$$11\omega 1\omega\omega 0\omega$$1,1[\overline{a}^{-}\mathrm{t}xA\lambda v1\overline{\omega}\omega,$$\mathrm{o}_{\iota_{\lambda}x\overline{v}}\mathrm{o}\mathrm{o}_{\{a\mathrm{t}J1}^{-},0\overline{a}\omega 0\mathrm{o}_{(}\mathrm{t}\overline{w}1$$0^{-}ww[[XtJ0\omega$$\mathrm{O}\mathrm{O}\mathrm{O}ax\overline{\alpha}_{\iota}\overline{w}1,001(\alpha v\overline{\omega}0\overline{\omega},$$\mathrm{O}\mathrm{O}\mathrm{O}\alpha x\overline{\alpha}u\overline{v}1$$01a\mathrm{x}v00^{-}(\mathrm{A}\overline{w},$,$0^{-}\ovalbox{\tt\small REJECT}^{----}0,1(Ax\overline{v}01101,1\overline{\omega}0\omega 1110,11\overline{\alpha}xv\mathrm{O}\mathrm{O}11$$000\alpha x\overline{\alpha}\mathrm{t}\lambda\overline{v}1,000\overline{a}hxv\overline{\omega}1,0^{-}\ovalbox{\tt\small REJECT} 0,0^{-}\ovalbox{\tt\small REJECT} 0$$‘\alpha v10(\overline{u}\overline{v},$$100$$‘\alpha_{\overline{\iota}}\overline{w}1\omega 1\overline{\alpha}\mathrm{x}v,\overline{\alpha}xv\mathrm{O}\overline{\omega}\mathrm{O}\mathrm{O}\omega 1,\overline{\omega}1\omega_{4}^{-}x_{\mathrm{A}}^{-}\mathrm{x}av1,1\overline{\omega}1m\lambda_{\mathrm{A}}^{-}x_{\mathrm{A}}^{-}xv$$\mathrm{O}\mathrm{O}\omega \mathrm{O}\overline{\alpha}\overline{w}1\omega$$00\mathrm{O}ah\overline{\mathrm{X}}_{l}-\lambda v1,$

$11\overline{\omega}$$10\overline{a}\mathrm{x}\overline{v}00(ud,$$1^{-}(\ovalbox{\tt\small REJECT}^{-}x_{A}^{-}x\overline{v}101$,11111110,$1\overline{\omega}\overline{\omega}00(Axv0$,$11\omega 1\omega \mathrm{O}\mathrm{a}\mathrm{x}\mathrm{v}$$1\overline{\omega}0\overline{\omega}0\omega 0\omega$$1\omega 1\overline{\omega}0110,1\mathrm{o}^{-}x\overline{\alpha}uv1\overline{a}w$$0^{-}\ovalbox{\tt\small REJECT}^{----}0,1^{-}\ovalbox{\tt\small REJECT}^{--}1,1\overline{\omega}00\overline{\omega}(a\mathrm{t}J0,11\overline{\omega}\omega 0101$$\mathfrak{u}x_{A}^{-}x\overline{v}\mathrm{o}_{(}\overline{\alpha}\overline{v}\overline{\omega}0,\overline{a}\omega \mathrm{o}_{\overline{\omega}[_{\mathrm{A}}}^{-}\overline{w}0\overline{\omega},\overline{\omega}1\omega 100$

$11,1\overline{\omega}1\omega 00$ $11$

$101$$\omega(\overline{\iota}\overline{w}1[\iota w1\overline{\omega},\overline{a}xv\mathrm{o}_{[}\overline{a}\overline{u}_{\lambda}^{-}\overline{w}0,\overline{\omega}1[u\overline{m}\alpha\overline{v}1,1\overline{\omega}1\omega 0$$\omega 1[\overline{u}\overline{a}\alpha\overline{d}1\omega,\overline{a}x\overline{\alpha}av11(\lambda x\overline{v},\overline{a}x\overline{\alpha}\overline{\alpha}v0^{-}(\mathrm{A}\overline{w}0,1^{-}\ovalbox{\tt\small REJECT}^{--}1$$\omega 1\overline{\omega}\alpha\overline{\iota}\mathrm{u}x\overline{v}1$,$‘\iota\overline{w}\overline{u}v1\overline{\omega}1\omega$,$(\overline{\alpha}\overline{v}001\omega 0\omega,\overline{a}x\overline{\alpha}x\overline{v}\mathrm{O}\overline{\omega}\mathrm{O}\overline{\omega},$$1\omega 1\omega 10\omega\omega$$‘\overline{u}\prime J\mathrm{O}\mathrm{O}\mathrm{O}\iota\alpha\overline{v}1,\overline{\omega}1a\mathrm{x}\overline{\alpha}au\overline{v}1,1\overline{\omega}1\overline{\omega}1\overline{\omega}0\overline{\omega}$$\omega 00\omega 1^{-}[kx\overline{v}0,$$‘ \mathrm{A}x_{\mathrm{A}}^{-}\overline{w}\mathrm{o}_{\overline{\omega}(\mathrm{t}}^{-}\overline{w}0,\overline{\alpha}w1\omega(\overline{\alpha}\overline{v}1\omega,\overline{\omega}1\{uv\overline{\omega}1\overline{a}xv,$$1‘\overline{\alpha}\overline{a}\overline{\alpha}\overline{v}110$$\omega 0001[\overline{\alpha}\overline{\alpha}v,$$\omega(\overline{u}\overline{v}1000\omega,\overline{\alpha}w1\overline{\omega}\mathrm{O}\mathrm{O}\omega \mathrm{O},\overline{\omega}1[kx\overline{v}0\omega 00,$

$1‘\overline{u}\overline{u}uv000$$\omega 0\omega 0\overline{\omega}1\overline{\omega}0$,$a\overline{w}1\overline{\omega}00\omega 0$,$‘\overline{\alpha}a\overline{\mathrm{t}J}1000\omega,\overline{\omega}10^{-}(Axv00\omega,$$1^{-}(\mathrm{t}\overline{\mathrm{u})}00a\omega 0$

50$D_{44}$:000\sim 工禾 1,,$\mathrm{O}\omega \mathrm{O}\mathrm{O}\overline{\omega}1\alpha\overline{w}$,$\omega 0001‘\overline{u}\overline{\alpha}v$, $ti1^{-}$ み -1,$D_{45}$: 000 は諷科 1, 00uOuwlw,$0\omega 0\omega 1\overline{\omega}0\overline{\omega}$,$\omega 0\omega 0\overline{\omega}1\overline{\omega}0$,$D_{46}$:$\mathrm{O}\omega \mathrm{O}\mathrm{O}\overline{\omega}1a\overline{w}$,$\omega 0\overline{\omega}10\omega 0\overline{\omega}$,$D_{47}:000‘ u\alpha\overline{\alpha}\overline{\theta}1,00\omega 0[\overline{\alpha}d1\overline{\omega},$$0\omega 0\omega 1\overline{\omega}0\overline{\omega}$,$\omega 0\omega 0\overline{\omega}1\overline{\omega}0$,$D_{48}:000[a\alpha\overline{a}^{-}‘ J1,00\omega 01\overline{\omega}(k-x\iota"$$0\omega 0\overline{\omega}\overline{\omega}0\omega 1$,$\omega 0\overline{\omega}01\omega 0\overline{\omega}$,$\mathrm{O}\mathrm{O}\omega \mathrm{O}\overline{\alpha}\mathrm{x}v1\overline{\omega}$$\mathrm{O}\mathrm{O}\mathrm{O}\omega u\kappa\overline{u}\overline{v}1,00\omega 0‘\overline{a}d1\overline{\omega}$,$(\lambda X\overline{d}1[\overline{\alpha}\alpha\overline{w}1,$$[\overline{u}\alpha\overline{\iota’}1\alpha w1\overline{\omega},\overline{\omega}1\overline{\alpha}w1\overline{a}\lambda\alpha v,$$\{\overline{u}_{\mathit{1}}x\overline{v}1‘\alpha v1\overline{\omega},\overline{\omega}10\{\overline{u}\overline{v}1\overline{\omega}1,1\overline{\omega}\overline{\omega}01\overline{\omega}1\overline{\omega}$$\omega 1\{\overline{\overline{m}}\alpha\overline{\alpha}d1,\overline{\alpha}\overline{w}\mathrm{O}\omega 1\omega 00,\overline{\alpha}x\overline{u}_{\lambda}x\overline{v}\mathrm{O}\overline{\omega}0\overline{\omega}$,$1\omega 1$,$1\iota\alpha\overline{M}\overline{w}1\overline{\omega}$$\omega 1‘\overline{\alpha}\iota\overline{m}\overline{w}1,\overline{\alpha}^{-}\lambda kX\mathrm{t}\lambda v1\overline{\omega}1,\overline{\alpha}\mathrm{x}^{-}\ovalbox{\tt\small REJECT}\lambda v1\omega 1\overline{\omega}$$\omega 1\overline{\omega}$$1\overline{a}\mathrm{x}u_{\mathrm{A}}^{-}x\overline{v}1\omega\omega$

10101, [$\overline{a}\overline{d}0\omega 1$

two,

$00$

Ll

$[\overline{u}\overline{v}[v00\omega 01,1\omega 1\overline{\omega}1010$

References[1] K. Betsumiya, The Type II property for self-dual codes over finite fields of characteristic two, (submitted).
[2] K. Betsumiya, T. A. Gulliver, M. Harada and A. Munemasa, On Type II codes over
, Lecture Notes in Computer Sci. (to appear).
$\mathrm{F}_{4}$[3] K. Betsumiya and M. Harada, Formally self-dual codes related to Type II codes,(submitted).
[4] K. Betsumiya, M. Harada and A. Munemasa, Type II codes overInform. Theory (to appear).

$\mathrm{F}_{2^{r}}$, IEEE Trans.

[5] J.H. Conway, V. Pless and N.J.A. Sloane, The binary self-dual codes of length up
to 32, J. Combin. Theory Ser. A60 (1992), 183-195.
[6] P. Gaborit, V. Pless, P. Sol\’e and 0. Atkin, Type II codes over F4, preprint.
[7] A. Munemasa, Type II codes and quadratic forms over a finite field of characteristic
two, The Ohio State University Mathematical Research Institute Monograph Series,(to appear).

[8] G. Pasquier, Binary self-dual codes construction from self-dual codes over aGalois, Combinatorial mathematics, North-Holland Math. Stud., 75, (1983), 519ffield$\mathrm{F}_{2^{m}}$526.
[9] E. M. Rains and N. J. A. Sloane, Self-dual codes, in “Handbook of Coding Theory,”edited by V. Pless and W. C. Huffman, North Holland, Amsterdam (1998).
[10] J. Wolfmann, Aclass of doubly even self dual binary codes, Discrete Math. 56(1985), 299-303