数理解析研究所講究録 1228 巻 2001 年 140-152

140

Combinatorial deformations oftheta series of various kinds
小関 道夫 (山形大 理学部)

Michio Ozeki
Department of Mathematics
Faculty of Science
Yamagata University
1-4-12 , Koshirakawa-chou, YamagataJapanemail address :ozeki@kszaoh3.kj.yamagata-u.ac.jp

1

Introduction

It is known that there are strong links between combinatorcs and number theory. From selfdual binary or ternary (or other kinds of) codes we obtain unimodular lattices. From various
kinds of weight enumerators we can define theta series of various kinds. However for the present
speaker there is one unsatisfactory point, which will be precisely desribed in the former part ofthe present talk.
In the latter part we propose some possible remedies.

2Some preliminaries
Binary self-dual codes

2.1

Let $\mathrm{F}_{2}=GF(2)$ be the field of 2elements. Let
be the vector space of dimension ,?
$[n,
k]$
over
. Alinearcode
is avector subspace of of dimension . In
, the innerproduct , which is denoted byforin , is defined as usual. The dual codeofis defined by
Vv{}.
$V=\mathrm{F}_{2}^{n}$$V$$\mathrm{C}$$\mathrm{F}_{2}$$(\mathrm{x},\mathrm{y})$$\mathrm{C}^{[perp]}=$Acode$\mathrm{C}$$\mathrm{x},\mathrm{y}$$V$$\mathrm{C}=\mathrm{C}^{[perp]}$$\mathrm{C}$$\in \mathrm{C}$.

An element in is called acodeword of C. Let
then the Hamming weightof the codeword$\mathrm{C}$$\mathrm{x}$$V$$\mathrm{C}^{[perp]}$$\mathrm{u}\in V|(\mathrm{u},\mathrm{v})=0$

is called self-dual if it satisfies$k$$\mathrm{x}$be acodeword of alinear $[n, k]$ code C.

$wt(\mathrm{x})$$\mathrm{x}=(x_{1}, x_{2}, \ldots, x_{n})$141
is defined to be the number ofsuch that $x_{i}\neq 0$ . The Hamming distance on
is alsodefined by.
code, then the weightLet
be aself-dual binaryof each codeword in is aneven number. Further, if the weight of each codeword inis divisible by 4, then the code iscalled adoubly even binary code. It is known that doubly even self-dual binary codes exist
only when the length of is amultiple of 8.
The minimum distancefor acode is defined by$d$$i’ \mathrm{s}$$\mathrm{C}$$d(\mathrm{x}, \mathrm{y})=wt(\mathrm{x}-\mathrm{y})$$\mathrm{C}$$wt(\mathrm{x})$$[n, \frac{n}{2}]$$\mathrm{C}$$\mathrm{x}$$\mathrm{C}$$\mathrm{x}$$\mathrm{C}$$\mathrm{C}$$n$$\mathrm{C}$$d(\mathrm{C})$$d( \mathrm{C})=\min_{\mathrm{u},\mathrm{v}\in \mathrm{C},\mathrm{u}\neq \mathrm{v}}d(\mathrm{u}, \mathrm{v})=\min_{\mathrm{u}\in \mathrm{C}-\{\mathrm{O}\}}wt(\mathrm{u})$.

There is awell known proposition:

Proposition 1Let$\mathrm{C}$be a doubly even self dual binary code of length$d( \mathrm{C})\leq 4[\frac{n}{24}]+4$$n_{f}$

then.

Definition :Adoubly even self dual binary code of length$n$satisfying$d( \mathrm{C})=4[\frac{n}{24}]+4$

is called an extremal code.
be aself-dual doubly even code of length . which is embedded in. Let
be any pair of vectors in,, then the number of common
l’s of the corresponding coordinates forandis denoted by. This is called theintersection number of and , andis nothing else.
Letbe adoubly even self-dual binary
code. The homogeneous weight enumeratorof the code is defined byLet$\mathrm{C}$$n$$(u_{1}, u_{2}, \cdots, u_{n})$$\mathrm{v}=(v_{1}, v_{2}, \cdots, v_{n})$$\mathrm{F}_{2}^{n}$$\mathrm{v}$$\mathrm{u}*\mathrm{v}$$\mathrm{v}$$\mathrm{u}$$\mathrm{u}$$wt(\mathrm{u})$$\mathrm{u}*\mathrm{u}$$\mathrm{C}$$[n, \frac{n}{2}]$$\mathrm{C}$$\mathrm{V}V_{\mathrm{C}}(x\backslash y)$$W_{\mathrm{C}}(x, y)= \sum_{\mathrm{v}\in \mathrm{C}}x^{n-wt(\mathrm{v})}y^{wt(\mathrm{v})}$.

Williams identity for binary self-dual code:

Abasic result is the
Theorem 1Let
following identity holds :$W_{\mathrm{C}}(x, y)$be the weight enumerator of a self-dual binary$\mathrm{V}V_{\mathrm{C}}(x, y)=W_{\mathrm{C}}(\frac{x+y}{\sqrt{2}}, \frac{x-y}{\sqrt{2}})$When$\mathrm{C}$$\mathrm{u}=$$\mathrm{F}_{2}^{n}$is adoubly even binary code then$W\mathrm{c}(x, y)$$[n, \frac{n}{2}]$code, then the.

(1)

has another inbariance property:(2)$W\mathrm{c}(x, iy)=W\mathrm{c}(x, y)$Definition of Jacobi polynomials for code
The homogeneous Jacobi polynomial
defined by$Jac(\mathrm{C}, \mathrm{v}:x, y, u, v)$

for$\mathrm{C}$with respect to$Jac(\mathrm{C},\mathrm{v};x, y, u, v)$$= \sum_{\mathrm{u}\in \mathrm{C}}x^{n-wt(\mathrm{v})-wt(\mathrm{u})+\mathrm{u}*\mathrm{v}wt(\mathrm{u})-\mathrm{u}*\mathrm{v}wt(\mathrm{v})-\mathrm{u}*\mathrm{v}\mathrm{u}*\mathrm{v}}y\cdot uv$$\mathrm{v}\in \mathrm{F}_{2}^{n}$

is

142
We will call the vector the reference vector of the Jacobi polynomial in some occasions.
in the polynomial
The weightis called the index of the polynomial.
$\mathrm{v}$$Jac(\mathrm{C},\mathrm{v};x, y, u, v)$$wt(\mathrm{v})$We get aWilliams type identity for$Jac(\mathrm{C}, \mathrm{v};x, y, u, v)$

be the weight enumerator of a doubly even self-dual binary
Theorem 2Let
code, then the following identity holds :$W_{\mathrm{C}}(x, y)$$Jac( \mathrm{C}, \mathrm{v};x, y, u, v)=Jac(\mathrm{C}, \mathrm{v};\frac{x+y}{\sqrt{2}}, \frac{x-y}{\sqrt{2}}, \frac{u+v}{\sqrt{2}}, \frac{u-v}{\sqrt{2}})$Since$\mathrm{C}$is doubly even, each codeword$\mathrm{u}$of$\mathrm{C}$.

(3)

has weight divisible by 4, and we get (C.f. [13])(4)$Jac(\mathrm{C}, \mathrm{v};x, iy, u, iv)=Jac(\mathrm{C},\mathrm{v};x, y, u, v)$

Certain finite groups and their invarint rings

2.2
Let

$[n, \frac{n}{2}]$$G_{1}$

be the group generated by$\sigma_{1}=\frac{1}{\sqrt{2}}$and$(\begin{array}{l}111-1\end{array})$.

$\sigma_{2}=(\begin{array}{ll}1 00 i\end{array})$This group is of order 192 and is known as N0.9 in Shephard-Todd’s list in [20]. The equations(1) and (2) show that the homogeneous weight enumerator
for adoubly even self-dual
binary code belongs to the ring of invariant polynomials
for the finite group of linear
be the subring oftransformations . Let
generated by the homogeneous weight
enumerators of doubly even self-dual binary codes.
Then Gleason’s theorem can be regarded that the ringcoincides with the ring.
We denote by $diag(G_{1}, G_{1})$ the group of linear transformations generated by$W\mathrm{c}(x,y)$$\mathbb{C}[x,y]^{G_{1}}$$\mathcal{W}[x,y]$$G_{1}$$\mathbb{C}[x,y]$$\mathcal{W}[x,y]$$\tilde{\sigma}_{1}=$$(\begin{array}{ll}\sigma_{1} OO \sigma_{1}\end{array})$and$\mathrm{C}[\mathrm{x}, y]^{G_{1}}$$\tilde{\sigma}_{2}=(\begin{array}{ll}\sigma_{2} OO \sigma_{2}\end{array})$.

Then the equations (3) and (4) show that the polynomial
for doubly evenself-dual binary code is apolynomial in the ringof simultaneous polynomial invariants for the groupin the sense of I. Schur [19] pages 9-14.
$Jac(\mathrm{C}, \mathrm{v};x, y, u, v)$$\mathbb{C}[x,y, u, v]^{diag(G_{1\prime}G_{1})}$$\mathrm{C}$$G_{1}$

Construction of lattices ffom binary codes

2.3

Construction A.
Let$\mathrm{C}$be adoubly even self-dual binary code of length .
be an element of , where is the ring of rational integers. Amap$(x_{1}, x_{2}, \cdots, x_{n})$$n$$\mathbb{Z}^{n}$$(n\equiv 0 (\mathrm{m}\mathrm{o}\mathrm{d} 8))$$\mathbb{Z}$by$\rho:\mathbb{Z}^{n}$$arrow$$\Downarrow\rfloor$$\mathrm{x}-$$(\mathbb{Z}/2\mathbb{Z})^{n}=\mathrm{F}_{2}^{n}$$1\Downarrow$$\mathrm{x}$m0d2$\rho$

Let
is defined$\mathrm{x}=$143
The set$L(\mathrm{C})$defined by$L( \mathrm{C})=\frac{1}{\sqrt{2}}\rho^{-1}(\mathrm{C})$is proved to be an even unimodular lattice of dimension
the Construction A.

$n$. This lattice construction is called

Construction B.
Let

be adoubly even self-dual binary code of length$\mathrm{C}$$\mathcal{L}(\mathrm{C})$$= \{\frac{1}{\sqrt{2}}\mathrm{x}|\mathrm{x}\in \mathbb{Z}^{n}$,$n$. The set defined by$\mathrm{x}mod2\in \mathrm{C},\sum_{i=1}^{n}x_{i}\equiv 0$$(\mathrm{m}\mathrm{o}\mathrm{d} 4)\}$is an even lattice. This lattice construction is called the Construction B.

2.4

Modular forms, Jacobi forms

2.4.1

Definition of modular forms

Let IH[ be the complex upper half plane. Let
on E[ is called amodular form of weight
following conditions (5) and (6) :$f(\tau)$$f( \frac{a\tau+b}{c\tau+d})=(c\tau+d)^{k}f(\tau)$

Aspecial case of (5) implies that
expansion :$f(\tau)$$\tau$$k$

be avariable on . Aholomorphic function( is even) with respect
if it satisfies$\mathfrak{H}$$k$$SL_{2}(\mathbb{Z})$hhoollddss ffoorr$(\begin{array}{ll}a bc d\end{array})\in SL_{2}(\mathbb{Z})$$\forall$satisfies $f(\tau+1)=f(\tau)$ , and$f( \tau)=\sum_{m\in \mathbb{Z}}a_{m}e^{2\pi im\tau}$$k$with respect to$f(\tau)$(5)

has aFourier.

$f( \tau)=\sum_{m\geq 0,m\in \mathbb{Z}}a_{m}e^{2\pi im\tau}$

The set of modular forms of weight

.

.

(6)

is denoted by $M(k)$ .

$SL_{2}(\mathbb{Z})$Definition of Jacobi forms

2.4.2

Let IH[ and be as above. Let be the complex plane and be avariable on C. Acomplex
is called aJacobi form of weight anddefined onvalued holomorphic function
if it satisfies the conditions (7), (8) and (9) :index with respect to the pair$\mathbb{C}$$z$$\tau$$\phi(\tau, z)$$h$$\mathbb{H}\cross \mathbb{C}$$(SL_{2}(\mathbb{Z}),\mathbb{Z})$$\forall$$(\begin{array}{ll}a bc d\end{array})\in SL_{2}(\mathbb{Z})$$\phi(\tau, z)=(c\tau+d)^{-k}e^{2\pi ih(\frac{-cz^{2}}{c\tau+d})_{\emptyset}}(\frac{a\tau+b}{c\tau+d},$ $\frac{z}{c\tau+d})$$\phi(\tau, z)=e^{2\pi ih(\lambda^{2}\tau+2\lambda z)}\phi(\tau, z+\lambda\tau+\mu)$$\phi(\tau, z)$

$k$

(7)
(8)

has aFourier expansion of the form$\phi(\tau, z)=\sum_{n\geq \mathrm{r}^{2}/4h}c(n, r)q^{n}\zeta^{\mathrm{r}}$

(9)

144
The set of Jacobi forms of weight k and index h with respect to the pair
by $J(k,$h).

is denot\^eTheta series, Jacobi theta series

2.5

Let be an even unimodular lattice of rank
the lattice is defined by
$L$

$n$

. ( is divisible by 8) Theta series attached$n$$\theta(\tau, L)=\sum_{\mathrm{x}\in L}e^{\pi i(\mathrm{x},\mathrm{x})\tau}$

It is known that
Let$(SL_{2}(\mathit{7}!\ovalbox{\tt\small REJECT},\mathit{7}!\ovalbox{\tt\small REJECT})$$\mathrm{y}\in L$$\theta(\tau, L)$satisfying$\mathrm{t}($.

is amodular form of weight $n/2$ .

$(\mathrm{y}, \mathrm{y})=2h$, then aJacobi theta series with respect to$\theta_{\mathrm{y}}(\tau, z, L)=\sum_{\mathrm{x}\in L}e^{\pi i(\mathrm{x},\mathrm{x})\tau+2\pi i(\mathrm{x},\mathrm{y})z}$$\mathrm{y}$is defined by.

This series is proved to be aJacobi form of weight $n/2$ and index .
$h$Jacobi’s theta functions

2.6
Let ,
$\epsilon$$\epsilon’$be numbers 0or 1. Then Jacobi’s theta functions are defined by$\theta$$\{\begin{array}{l}\epsilon\epsilon’\end{array}\}$$( \tau, z)=\sum_{h\in \mathbb{Z}}e^{\pi}:\{\tau(h+\epsilon/2)^{2}+2(h+\epsilon/2)(z+\epsilon’/2)\}$Amore popular notations are the right-hand sides of the followings$\theta$$\{\begin{array}{l}00\end{array}\}$$(\tau, z)$$=$$\theta_{3}(\tau, z)$$\{\begin{array}{l}01\end{array}\}$$(\tau, z)$$=$$\theta_{0}(\tau, z)$$\{\begin{array}{l}10\end{array}\}$$(\tau, z)$$=$$\theta_{2}(\tau, z)$$\{\begin{array}{l}\mathrm{l}1\end{array}\}$$(\tau, z)$$=$$\theta_{1}(\tau, z)$$\mathit{4}\mathit{1}$$\theta$$\theta$

These functions satisfy the following transformations :(Conf. [17],[23].)$\theta_{2}(-\frac{1}{\tau}, \frac{z}{\tau})=\sqrt{\frac{\tau}{i}}e^{\pi i\frac{z^{2}}{\tau}}\theta_{0}(\tau, z)$$\theta_{0}(-\frac{1}{\tau}, \frac{z}{\tau})=\sqrt{\frac{\tau}{i}}e^{\pi i\frac{z^{2}}{\tau}}\theta_{2}(\tau, z)$$\theta_{3}(-\frac{1}{\tau}, \frac{z}{\tau})=\sqrt{\frac{\tau}{i}}e^{\pi i\frac{z^{2}}{\tau}}\theta_{3}(\tau, z)$here we take the branch of the root sothat it takes the value 1at $\tau=i$ .

(10

(11
(12

145
lt we put$\varphi_{0}(\tau, z)=\theta_{0}(2\tau, 2z)$,$\varphi_{2}(\tau, z)=\theta_{2}(2\tau, 2z)$$\varphi_{0}(\tau)=\theta_{0}(2\tau, 0)$,$\varphi_{2}(\tau)=\theta_{2}(2\tau, 0)$,,$\varphi_{3}(\tau, z)=\theta_{3}(2\tau, 2z)$$\varphi_{3}(\tau)=\theta_{3}(2\tau, 0)$then we can show that$(- \frac{1}{\tau}$,$(- \frac{1}{\tau}$, )$\varphi_{2}$$\frac{z}{\tau})=\sqrt{\frac{\tau}{2i}}e^{\pi i\frac{z^{2}}{\tau}}\{\varphi_{3}(\tau,$$\frac{z}{\tau}$$\varphi_{3}$$=\sqrt{\frac{\tau}{2i}}e^{\pi i\frac{z^{2}}{\tau}}\{\varphi_{3}(\tau,$$z)$$-\varphi_{2}(\mathcal{T}_{)}Z)\}$$z)$$+\varphi_{2}(\mathcal{T}_{)}z)$ $\}$

holds.

3Statement of the Problem
Problenu 1.
In what way the extremal theta series (i.e. theta series for extremal lattices)of various
kinds (ordinary, Jacobi, Siegel) can be determined ?Is there aclear way that leads to direct
construction of extremal theta series ?At present before us there is avague and ugly indirect
wa.v that leads to such construction.

Here we give aprecise explanation of the problem. Throughout this section we let
doubly even self-dual binary code of length .

$\mathrm{C}$

be a

$n$

3.1

Ord inary theta series caseLetbe the weight enumerator of the code
correspondence$hV_{\mathrm{C}}(x, y)$$(\mathbb{C}[x, y]^{\mathrm{G}_{1}})_{h}$$arrow$$\Downarrow)$$W_{\mathrm{C}}(x, y)$$\mathrm{C}$, then the Brou\’e-Enguehard map is a$M_{n/2}$$\Downarrow)$–$W_{\mathrm{C}}(\varphi_{3}(\tau), \varphi_{2}(\tau))$Hereis the vector space of the -th homogeneous part of.
They proved this by showing thatis atheta seriesof the even
unimodular lattice.
The problem is that even if is an extremal code of length the resulting theta series
is not the extremal theta series. For instance whenis the binary Golay code of length 24,
thenis not the extremal theta series, but it is the theta series associated with theeven unimodular lattice of type(coming from the root lattice of). How the theta seriesof the Leech lattice can be derived ?
$h$$(\mathrm{C}[:\mathrm{c}, y]^{\mathrm{G}_{1}})_{h}$$\mathbb{C}[x., y]^{\mathrm{G}_{1}}$$\mathrm{L}V_{\mathrm{C}}(\varphi_{3}(\tau), \varphi_{2}(\tau))$$\theta(\tau, L(\mathrm{C}))$$L(\mathrm{C})$$\mathrm{C}$$n$$\theta(\tau, \mathrm{L}(\mathrm{C}))$$\mathcal{G}_{24}$$\theta(\tau, L(\mathcal{G}_{24}))$$A_{1}^{24}$$A_{1}^{24}$

Nu merical evidence 1.

$W_{\mathcal{G}_{24}}(x, y).=x^{24}+759x^{16}y^{8}+2576x^{12}y^{12}+759x^{8}y^{16}+y^{24}$146
is the weight enumerator of the binary Golay [24, 12, 8] code.
$W_{\mathcal{G}_{24}}(\varphi_{3}(\tau), \varphi_{2}(\tau))=$$\theta(\tau, L(\mathcal{G}_{24}))=$$1+48q+195408q^{2}+16785216q^{3}+397963344q^{4}+4629612960q^{5}+34417365696q^{6}+\cdots$here$q=e^{2\pi i\tau}$$\theta$. On the other hands theta series for Leech lattice is( , Leech)$\tau$.

$=$$1+196560q^{2}+16773120q^{3}+398034000q^{4}+4629381120q^{5}+34417656000q^{6}+\cdots$

This fact compels us to pose the following question :Can we modify the Broue-Enguehard map to the effect that the following diagram is commutative$\mathbb{C}[x, y]^{\mathrm{G}_{1}}$$arrow M_{n/2}\ni\theta(\tau, L(\mathrm{C}))$$\downarrow$$\downarrow$$\mathbb{C}[x, y]^{\mathrm{G}_{1}}$3.2$\mapsto M_{n/2}\ni \mathrm{e}\mathrm{x}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{l}$thetaJacobi theta series casewith areference vector
be aJacobi polynomial for the codeLetweight 1, the (so called and the simplest) Bannai-Ozeki map is acorrespondence$Jac(\mathrm{C}, \mathrm{v}, x, y, u, v)$$\mathrm{C}$$\mathrm{v}$of$arrow J_{1}(n/2)$$(\mathbb{C}[x, y, u, v]^{d\cdot ag(\mathrm{G}_{1\prime}\mathrm{G}_{1})}.)_{1}$$\square \cup$$\Downarrow$$-Jac(\mathrm{C}, \mathrm{v};\varphi_{3}(\tau),$$Jac(\mathrm{C},\mathrm{v}, x, y, u, v)$$\varphi_{2}(\tau),\varphi_{3}(\tau, z)$,$\varphi_{2}(\tau, z))$

Numerical evidence 2.
$Jac(\mathcal{G}_{24},\mathrm{v}, x, y, u, v)=$$x^{23}u+253x^{16}y^{7}v+506x^{15}y^{8}u+1288x^{12}y^{11}v+1288x^{11}y^{12}u+506x^{8}y^{15}v+$$253x^{7}y^{16}u+y^{23}v$

is the Jacobi polynomial of index 1for the Golay code. The image of Bannai-Ozeki map forthis polynomial is computed to be$Jac(\mathcal{G}_{24},\mathrm{v};\varphi_{3}(\tau)$,$\varphi_{2}(\tau)$,$\varphi_{3}(\tau, z),\varphi_{2}(\tau, z))=$147 147$1+(\zeta^{-2}+46+\zeta^{2})q+\{46(\zeta^{2}+\zeta^{-2})+32384(\zeta+\zeta^{-1})+130548\}q^{2}$$+\{130548(\zeta^{2}+\zeta^{-2})+9175896+3674112(\zeta+\zeta^{-1})\}q^{3}$$+\{\zeta^{4}+\zeta^{-4}+32384(\zeta^{3}+\zeta^{-3})+9175896(\zeta^{2}+\zeta^{-2})$$+95659392(\zeta+\zeta^{-1})+188227998\}q^{4}$$+(46(\zeta^{4}+\zeta^{-4})+3674112(\zeta^{-3}+\zeta^{3})+188227998(\zeta^{2}+\zeta^{-2})$$+1143025664(\zeta+\zeta^{-1})+1959757320)q^{5}$$+\{130548(\zeta^{4}+\zeta^{-4})+95659392(\zeta^{3}+\zeta^{-3})+1959757320(\zeta^{2}+\zeta^{-2})$$+8506630272(\zeta+\zeta^{-1})+13293010632\}q^{6}+\cdots$

,

.
where
This is verified to be the Jacobi theta series of index 1associated with the even unimodular
lattice of type.
Jacobi theta series of index 1associated with Leech lattice is 0by its meaning.
$\zeta=e^{2\pi iz}$

$A_{1}^{24}$

Numerical evidence 3.
There is aunique Jacobi polynomial$Jac(\mathcal{G}_{24}, \mathrm{v}, x, y, u, v)$

of index 2for the Golay code

$Jac(\mathcal{G}_{24}, \mathrm{v}, x, y, u, v)=$$x^{22}u^{2}+y^{22}v^{2}+352x^{15}y^{7}uv+1344x^{11}y^{11}uv+352x^{7}y^{15}uv+77x^{16}y^{6}v^{2}$$+330x^{14}y^{8}u^{2}+616x^{12}y^{10}v^{2}+616x^{10}y^{12}u^{2}+330x^{8}y^{14}v^{2}+77x^{6}y^{16}u^{2}$

From it we obtain$Jac(\mathcal{G}_{24}, \mathrm{v};\varphi_{3}(\tau)$,$\varphi_{2}(\tau)$,$\varphi_{3}(\tau, z)$,$\varphi_{2}(\tau, z)))=$$1+(2\zeta^{-2}+2\zeta^{2}+44)q+(\zeta^{4}+\zeta^{-4}+5016(\zeta^{2}+\zeta^{-2})+45056(\zeta+\zeta^{-1})+95262)q^{2}$$+\{44(\zeta^{4}+\zeta^{-4})+45056(\zeta^{3}+\zeta^{-3})+959288(\zeta^{2}+\zeta^{-2})+4149248(\zeta+\zeta^{-1})+6477944\}q^{3}$$+\{95262(\zeta^{4}+\zeta^{-4})+4149248(\zeta^{3}+\zeta^{-3})+95432704(\zeta+\zeta^{-1})+32752192(\zeta^{2}+\zeta^{-2})+13310453$$+\{2(\zeta^{6}+\zeta^{-6})+45056(\zeta^{5}+\zeta^{-5})+6477944(\zeta^{4}+\zeta^{-4})+95432704(\zeta^{3}+\zeta^{-3})$$+458048186(\zeta^{2}+\zeta^{-2})+1062150144(\zeta+\zeta^{-1})+1385304888\}q^{5}$$+\{5016(\zeta^{6}+\zeta^{-6})+4149248(\zeta^{5}+\zeta^{-5})+133104532(\zeta^{4}+\zeta^{-4})+1062150144(\zeta^{3}+\zeta^{-3})$$+3772360200(\zeta^{2}+\zeta^{-2})+7534989312(\zeta+\zeta^{-1})+9403848792\}q^{6}+\cdots$

This series is verified to be Jacobi theta series of index 2associated with the even unimodular
lattice of type
Jacobi theta series of index 2associated with the Leech lattice is computed to be$A_{1}^{24}$$\theta_{\mathrm{y}}$( ,$\tau$$Z_{\backslash }$Leech)$=$

148$1+\{\zeta^{4}+\zeta^{-4}+4600(\zeta^{2}+\zeta^{-2})+47104(\zeta+\zeta^{-1})+93150\}q^{2}$$+\{47104(\zeta^{3}+\zeta^{-3})+953856(\zeta^{2}+\zeta^{-2})+4147200((+\zeta^{-1})+6476800\}q^{3}$$+\{93150(\zeta^{4}+\zeta^{-4})+4147200(\zeta^{3}+\zeta^{-3})+32788800(\zeta^{2}+\zeta^{-2})$$+95385600(\zeta+\zeta^{-1})+133204500\}q^{4}$$+\{47104(\zeta^{5}+\zeta^{-5})+6476800(\zeta^{4}+\zeta^{-4})+95385600(\zeta^{3}+\zeta^{-3})$$+458086400(\zeta^{2}+\zeta^{-2})+1062195200(\zeta+\zeta^{-1})+1384998912\}q^{5}$$+\{4600(\zeta^{6}+\zeta^{-6})+4147200(\zeta^{5}+\zeta^{-5})+133204500(\zeta^{4}+\zeta^{-4})+1062195200(\zeta^{3}+\zeta^{-3})$$+3771829800(\zeta^{2}+\zeta^{-2})+7535462400(z+\zeta^{-1})+9403968600\}q^{6}+\cdots$

3.3

Siegel theta series caseWe do not give anumerical example, but it is evident that W. Duke’s correspondence fromthe multiple weight enumerator of aclass of doubly even self-dual binary codes of length
uses Construction A. And this should
to Sielgel modular forms via Siegel theta series for(in general case)arewithlattices of typeassociatedcause the obtained Siegel theta series
even if we use extremal codes.
$\mathrm{C}$$n$$L(\mathrm{C})$$A_{1}^{n}$

4First Approach
4.1

Ordinary theta series case

Our first approach for asolution to our problem is to find apolynomial which leads to theta
series of the Leech lattice via Broue- Enguehard map. It is easy to find such apolynomial. but
it may be difficult to give adeeper meaning to the polynomial. The polynomial in question is
given by$\overline{W_{Q_{24}}}(x, y)$$=$$x^{24}+759x^{16}y^{8}+2576x^{12}y^{12}+759x^{8}y^{16}+y^{24}$$-3(x^{20}y^{4}-4x^{16}y^{8}+6x^{12}y^{12}-4x^{8}y^{16}+x^{4}y^{20})$$=$$x^{24}-3x^{20}y^{4}+771x^{16}y^{8}+2558x^{12}y^{12}+771x^{8}y^{16}-3x^{4}y^{20}+y^{24}$We verify that$\overline{W_{\mathcal{G}_{24}}}(\varphi_{3}(\tau), \varphi_{2}(\tau))=\theta$

holds.
There are similar facts in dimensions 32 and 40( , Leech)$\tau$

149
4.2

Jacobi theta series caseIn this case we look for apolynomial in which satisfies$\overline{Ja}c(\mathcal{G}_{24\backslash }\mathrm{v};\varphi_{3}(\tau), \varphi_{2}(\tau),$$\varphi_{3}(\tau, z),\varphi_{2}(\tau, z)=\theta_{\mathrm{y}}$( , , LeecA) with$\tau$$z$$(\mathrm{y},\mathrm{y})=4$$\overline{Ja}c(\mathcal{G}_{24},\mathrm{v}, x, y,u, v)$$x^{22}u^{2}+y^{22}v^{2}+368x^{15}y^{7}uv+1312x^{11}y^{11}uv+368x^{7}y^{15}uv-1/2x^{20}y^{2}v^{2}$$=$$-5/2x^{18}u^{2}y^{4}+73x^{16}y^{6}v^{2}+330x^{14}y^{8}u^{2}+623x^{12}y^{1}0v^{2}+623x^{10}y^{12}u^{2}$$+330x^{8}y^{14}v^{2}+73x^{6}y^{16}u^{2}-5/2x^{4}y^{18}v^{2}-1/2y^{20}u^{2}x^{2}$

5Second Approach
In this section we discuss an arithmetical deformation of theta series of even unimodular lattice.
At present the research is not fully explored, but we expect this approach will be fruitful.
explain our idea by examining aspecial case.
$1\backslash \check{\mathrm{e}}$$24$$\mathrm{w}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{a}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}1\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}\mathcal{L}_{1}\mathrm{L}\mathrm{e}\mathrm{t}\mathcal{L}_{1}\mathrm{b}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{e}\mathrm{v}^{\nu}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}1\mathrm{a}\mathrm{r}1\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{o}.\mathrm{f}$$\mathrm{L}\mathrm{e}\mathrm{t}\mathrm{f}_{\mathrm{l}},\mathrm{f}_{2}\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o},\mathrm{n}$$\mathrm{f}_{24}\mathrm{b}\mathrm{e}\mathrm{m}\mathrm{u}\mathrm{t}\mathrm{u}\mathrm{a}111\mathrm{y}\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{g}\mathrm{o}\mathrm{n}\mathrm{a}1\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{n}\mathbb{R}^{24}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{s}\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{s}\mathrm{u}\mathrm{b}1\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}A_{1}^{24}.\mathrm{H}\mathrm{e}\mathrm{r}\mathrm{e}$satisfying
by the vectors$(\mathrm{f}_{i}, \mathrm{f}_{i})=2$, $i=1,2$ ,, 24 (the coupling denotes the inner product).
,
and the vecorts of the form$\cdots$$\mathrm{f}_{1}.\mathrm{f}_{2\backslash }\cdots$$\frac{1}{2}\sum_{i}^{24}\rho_{i}\mathrm{f}_{i}$here$\mathrm{V}1^{\vee}\mathrm{e}$usevectors$\mathcal{L}_{2}$\rho_{24})$is generated,

belongs to the Golay code.
to denote the Leech lattice, and give apresentation of it.
. $1\leq i\leq j\leq 24$ and the vectors of the form$(\rho_{1}, \rho_{2\backslash }\cdots.

$\mathcal{L}_{1}$$\mathrm{f}_{24}$$(\mathrm{m}\mathrm{o}\mathrm{d} 2)$$\mathcal{G}_{24}$$\mathcal{L}_{2}$is generated by the$\pm \mathrm{f}_{i}\pm \mathrm{f}_{J}\cdot$$\frac{1}{2}\sum_{i}^{24}\rho_{i}\mathrm{f}_{i_{\dot{J}}}$

where $(*)$$(\rho_{1}.\rho_{2}.

\cdots.\rho_{24})$(mod 2) belongs to the Golay code$\mathcal{G}_{24}$and$\sum_{i}^{24}\rho_{i}\equiv 0$$(\mathrm{m}\mathrm{o}\mathrm{d} 4)$

,

and

.

$\mathrm{x}_{0}=\frac{1}{4}(\mathrm{f}_{1}+\mathrm{f}_{2}+\cdots+\mathrm{f}_{23}-3\mathrm{f}_{24})$$\mathrm{L}\mathrm{L}^{\mathrm{v}}\mathrm{e}$observe that the lettice$\mathcal{L}_{0}=\mathcal{L}_{1}\cap \mathcal{L}_{2}$andis generated by the vectors :$\frac{1}{2}\sum_{i}^{24}\rho_{i}\mathrm{f}_{i}$with the condition $(*)$ .
It holds that$(\mathcal{L}_{0}+\mathbb{Z}\mathrm{f}_{1})\cup \mathcal{L}_{0}=\mathcal{L}_{1}$and$\pm \mathrm{f}_{i}\pm \mathrm{f}_{j}$, $1\leq i\leq j\leq 24$ ,,$(\mathcal{L}_{0}+\mathrm{Z}\mathrm{x}0)\cup \mathcal{L}_{0}=\mathcal{L}_{2}$. From these we get$\theta(\tau, \mathcal{L}_{1})=\theta(\tau.\mathcal{L}_{0})+\dot{\theta}(\tau, \mathcal{L}_{0}+\mathbb{Z}\mathrm{f}_{1})$,150$\theta(\tau,$

We look at$\theta(\tau, \mathcal{L}_{0})$,$\mathcal{L}_{2}=\theta(\tau,\mathcal{L}_{0})+\theta(\tau, \mathcal{L}_{0}+\mathbb{Z}\mathrm{x}_{0})$$\theta(\tau, \mathcal{L}_{0}+\mathbb{Z}\mathrm{f}_{1}),\theta(\tau, \mathcal{L}_{0}+\mathbb{Z}\mathrm{x}_{0})$.

more precisely.

$\theta(\tau, \mathcal{L}_{0}+\mathbb{Z}\mathrm{f}_{1})$$= \sum_{\mathrm{u}\in\dot{L}_{0}}e^{\pi i(\mathrm{u}+\mathrm{f}_{1},\mathrm{u}+\mathrm{f}_{1})\tau}$$= \sum_{\mathrm{u}\in\dot{L}_{0}}e^{\pi i\{(\mathrm{u},\mathrm{u})+2(\mathrm{f}_{1},\mathrm{u})+2\}\tau}$$=$$e^{2\pi i\tau} \sum_{\mathrm{u}\in \mathcal{L}_{0}}e^{\pi i\{(\mathrm{u},\mathrm{u})+2(\mathrm{f}_{1},\mathrm{u})\}\tau}$$=$$e^{2\pi i\tau}\theta_{\mathrm{f}_{1}}(\tau, z, \mathcal{L}_{0})|_{z=\tau}$

In the same way we see that$\theta(\tau, \mathcal{L}_{0}+\mathbb{Z}\mathrm{f}_{1})$$=$$e^{2\pi i\tau}\theta_{\mathrm{x}0}(\tau, z, \mathcal{L}_{0})|_{z=\tau}$by Construction B.
is seen to be obtained from the Golay code
The lattice
can be computed in a
In conclusion if theta series and Jacobi theta series for the lattice
We can expect thatlattice.
series
of
the
Leech
theta
computesystematic way, then we may,
will be controled by the information from the code (doubly,of index 2). Numerical computation shows thateven subcode of$\mathcal{G}_{24}$$\mathcal{L}_{0}$$\mathcal{L}_{0}$$\theta(\tau,\mathcal{L}_{0})$$\theta(\tau, \mathcal{L}_{0}+\mathbb{Z}\mathrm{f}_{1})$$\theta(\tau, \mathcal{L}_{0}+\mathbb{Z}\mathrm{x}_{0})$$g_{24}$$\theta(\tau, \mathcal{L}_{0})=1+98256q^{2}+5275648q^{3}+\cdots$,$\theta(\tau,\mathcal{L}_{0}+\mathbb{Z}\mathrm{f}_{1})=48q+97152q^{2}+11509568q^{3}+\cdots$$\theta(\tau,\mathcal{L}_{0}+\mathbb{Z}\mathrm{x}_{0})=98340q^{2}+11497472q^{3}+\cdots$,.

for the Leech lattice or extremal lattices in higher dimensions
Jacobi theta series of index
should be analyzed along this direction. At present the theory of Jacobi theta series for nonunimodular lattices is not at hand.
$\geq 2$

6Third Approachto the even unimodular lattices only use theSince there is amap from codes overto the class ofconstruction , there may arise anatural map from aclass of codes overthisdirection.
explored
yet
not
have
series.
We
extremal theta$\mathbb{Z}_{4}=\mathbb{Z}/4\mathbb{Z}$$\mathrm{A}$$\mathbb{Z}_{4}$References[1] E. Bannai and M. Ozeki, Construction of Jacobi forms from certain polynomials, Proc.
.Japan Academy Ser. A, Vo1.72 $(1996)12- 1$

151[2] E.R. Berlekamp, F.J. MacWilliams, and N.J.A. Sloane, Gleason’s Theorem on Self-Dual
Codes, IEEE Trans. Inform. Theory, IT-18 (1972), 409-414[3] M. Broue et M. Enguehard, Polynome des poids de certains codes et fonction theta decertains r\’eseaux, Ann. scien. Ec. Norm. Sup.
s\’erie, t.5 (1972) 157-181.
$4^{e}$[4] J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag

1988.
[5] W. Ebeling, Lattices and Codes, Vieweg 1994
[6] M. Eichler and D. Zagier, The Theory of Jacobi Forms, Birkhiuser, 1985
[7] F.J. MacWilliams, C.L. Mallows, and N.J.A. Sloane, Generalizations of Gleason‘s Theorem
on Weight Enumerators of Self-Dual Codes, IEEE Trans. Inf. Th. IT-18 (1972) 794-805[8] F.J. MacWilliams and N.J.A. Sloane, “The Theory of Error-Correcting Codes”,NorthHolland, Amsterdam, 1977.
[9] G.A. Miller, H.F. Blichfeldt, and L.E. Dickson, Theory and Applications of Finite Groups,
Dover , NY, 1961[10] M. Ozeki, Hadamard matrices and doubly even error correcting codes, J. Comb. Th. Ser.A
Vo1.44 (1987) 274-287[11] M. Ozeki, Examples of even unimodular extremal lattices of rank 40 and their Siegel theta
series of degree 2, J. Num. Th. Vo1.28 (1988) 119-131[12] M. Ozeki, On the notion of Jacobi polynomials for codes, , Math. Proc. Cambridge Philos.
Soc. Vol. 121 (1997) 15-30[13] M. Ozeki, Determination of covering radii of doubly even extremal binary self-dual codesof length 40, Theoretical Computer Science 235 (2000) 283-308[14] M. Peters, Jacobi theta series, Rocky Mt. Math. J. Vol. 19 (1989) 863-870[15] V. Pless, N.J.A. Sloane, On the classification and Enumeration of Self-Dual Codes, J.Comb.
Th. Ser. A18, (1975) 313-335[16] V. Pless, An Introduction to the Theory of Error-Correcting Codes, Wiley Interscience,
New York 1982.
[17] H.E. Rauch and A. Lebowitz, Elliptic Functions, Theta Functions, and Riemann Surfaces,
The Williams&Wilkins, 1973[18] B. Schoeneberg, Elliptic Modular Functions, Springer Verlag, 1974

[19] I. Schur, Vorlesungen liber Invariantentheorie, Springer (1968)[20] G.C. Shephard and J.A. Todd, Finite unitary reflection groups, Canad J. Math. 5(1953),

364-38

152[21] N.J.A. Sloane, Error-correcting codes and invariant theory: new applications of anineteenth-century technique, Amer. Math. Month. 84 (1977) 82-107.

[22] N.J.A. Sloane, Self-dual codes and lattices,in “ Relations between Combinatorics and Other
Parts of Mathematics”, Proc. Symp. in Pure Math., n0.34 (1979) 273-308.
[23] E.T. Whittakker, and G.N. Watson, Acourse of Modern Analysis, Cambridge Univ. Press,

1946