Ring of the weight enumerators of d+nMakoto Fujii

Manabu Oura

Abstract
We show that the ring of the weight enumerators of a self-dual doubly even code d+nin arbitrary genus is finitely generated. Indeed enough elements to generate it are given.
The latter result is applied to obtain a minimal set of generators of the ring in genus
two.

1 Introduction
The weight enumerator plays an important role in coding theory and has connections withother branches in mathematics. We recall some of them to see the background of this paper.
Let C be a self-dual doubly even (Type II, for short) code of length n. The weight enumerator
WC (x, y) =∑xn−wt(v) y wt(v)v∈Chas invariant properties. The so-called MacWilliams identity is described asx+y x−yWC (x, y) = WC ( √ , √ )22
and the doubly evenness gives√
WC (x, y) = WC (x, −1y).
These being said, the weight enumerator of a Type II code is an element of the invariant ringC[x, y]G = {f (x, y) ∈ C[x, y] : σf = f ∀σ ∈ G}of the finite group G where G is of order 192 generated by() ()1 1 11 √0√,−10
2 1 −1
Keywords: code, weight enumerator
2010MSC: Primary 94B05, Secondary 05E15
Running Head: Ring of the weight enumerators of d+n1

andσf (x, y) = f (ax + by, cx + dy), σ =(ac)b.
dA remarkable fact is that the converse is also true, that is, Gleason [3] showed that the invariant
ring C[x, y]G is generated by the weight enumerators of Type II codes. Indeed we haveC[x, y]G = C[Wd+ (x, y), Wd+ (x, y)].
824

We shall describe some consequences of this equality. Since the degrees of the generators are
8 and 24, length of a Type II code is a multiple of 8. Non-existence of an extremal Type II
code for suﬃciently large n also follows from the above equality.
We observe that Wd+ (x, y) and Wd+ (x, y) are algebraically independent over C. A finite
824

group having such a property (i.e., whose invariant ring is generated by the algebraicallyindependent elements over C) is called a finite unitary reflection group. See [13].
The generalization of the above correspondence is investigated in, for example, [12, 7].
The invariance property of the weight enumerator gives rise to the relation with the modularforms. See [1, 2, 12]. In fact, the weight enumerator of a Type II code of length n is mapped
under the theta map to the Siegel modular form of weight n/2 in genus g. The modular form+of weight 8 which is obtained from the diﬀerence ψ (g) of the weight enumerators of d+
8 ⊕ d8and d+
16 is of great importance. We just mention two points in genus three. Witt [14] asked
if the modular form obtained from ψ (3) vanishes, and it was aﬃrmatively answered in [5, 6].
Runge [10] showed that the ring of Siegel modular forms for Γ3 is isomorphic to the quotientring of the invariant ring of some finite group divided by an ideal (ψ (3) ).
Let D(g) be the ring of the weight enumerator of d+n in genus g. This is a subring of the ringof the weight enumerators of Type II codes. As indicated above, D(1) coincides with the ringof the weight enumerators. In this paper, we show that D(g) is generated by the elements of
8 ≤ n ≤ 22g+3 .
Using this result, we show that D(2) is minimally generated by nine weight enumerators oflengths
8, 24, 32, 40, 48, 56, 64, 72, 80.

2 Preliminaries
Let F2 = {0, 1} be the field of two elements. Two vector spaces Fn2 and Fg2 appear in thefollowing. For technical reason, an element of Fn2 is regarded as a row vector, while that ofFg2 as a column vector. The space Fn2 is equipped with the inner productu · v = u1 v1 + · · · + un vn , u = (u1 , . . . , un ), v = (v1 , . . . , vn )2

and so is Fg2

 

α1β1
 ..  .. α · β = α1 β1 + · · · + αg βg , α =  .  , β =  .  .
αgβg

Since we deal with only binary linear codes in this paper, we call a subspace of Fn2 a codeof length n. The weight wt(u) of an element u ∈ Fn2 is the number of non-zero coordinates ofu. A code C is said to be self-dual if it coincides with its dualC ⊥ = {v ∈ Fn2 : u · v = 0, ∀u ∈ C},
doubly even ifwt(u) ≡ 0 (mod 4), ∀u ∈ C.
Codes with those two properties (self-duality and doubly evenness) are particularly interesting.
We use the term Type II instead of self-dual and doubly even. It is known that a Type II
code of length n exists if and only if n ≡ 0 (mod 8). The weight enumerator of a code C oflength n in genus g is∑(g)

WC (xα : α ∈ Fg2 ) = x
 · · · x.
uu11121n  .   . 
 .  .   . 
 .  .   . 
 .  

ugn
ug1
ug2xuu1 =(u11 ,...,u1n )∈C..
.

ug =(ug1 ,...,ugn )∈C

This definition is consistent with that in the previous section if we put x = x0 , y = x1 . Since(g)there does not occur any confusion, we shall use an abridged notation WC . It is clear that(g)WC is a homogeneous polynomial of total degree n in C[xα : α ∈ Fg2 ]. Let W(g) be the ringover C generated by the weight enumerators of all Type II codes in genus g. It is known thatW(g) is the invariant ring of the specified finite group (cf. [3, 12, 7]). In particular W(g) isfinitely generated. In Introduction we discussed this topic for g = 1.
Next we recall a Type II code d+n of length n for n ≡ 0 (mod 8) and its weight enumerator.
It is nice to start with a repetition code Rn of length n. The dual code of Rn can be describedasRn⊥ = {(u1 , . . . , un ) ∈ Fn2 : u1 + · · · + un = 0}

which has a generator matrix

······..
.

1 1 0
1 0 1
1 0 0···

00

.
1

The following n/2 × n matrix is a generator matrix of d+n , that is, the n/2 row vectors form a3

basis of d+n:



11
11

11



11
10

1100
00

00
1100

00
00
11···
······
..
.

00
1000
1000
10······

00 0000 00
00 00.

00 11
10 10The code d+n is then characterized asd+n = {(α1 + γ, α1 , α2 + γ, α2 , . . . , αn/2 + γ, αn/2 ) : α1 , · · · , αn/2 , γ ∈ F2 , α1 + · · · + αn/2 = 0}.
It is known to be Type II. The weight enumerator of d+n in genus g is expressed as(g)Wd + =n12g



∑



β,γ∈Fg2∑

n/2(−1)α·β xα+γ xα .

α∈Fg2

We can find this formula of genus two in [2]. For the completeness of this paper, we add aproof. We have n/2∏∑∑∑i1(−1)α ·β xαi +γ xαi  RHS = g2i=1 αi ∈Fgγ∈Fgβ∈Fg222
=1 ∑ 2gg  1γ∈F2∑1(−1)(α +···+αn/2α ,...,αn/2 ∈Fg2β∈Fg
2For a fixed γ, we divide the summation as∑∑=
α1 ,...,αn/2 ∈Fg2β∈Fg
2∑From the1n/2

α +···+αβ)·β

α1 +···+αn/2 =0β+
xα1 +γ xα1 · · · xαn/2 +γ xαn/2  .
∑.

α1 +···+αn/2 ̸=0β-part, we get=0∑2gxα1 +γ xα1 · · · xαn/2 +γ xαn/2

α1 ,...,αn/2 ∈Fg2
α1 +···+αn/2 =01

because of (−1)(α +···+αn/2)·β= 1 for any β ∈ Fg2 . Next fix α1 , . . . , αn/2 such that α1 + · · · +αn/2 ̸= 0. Then the number of β ∈ Fg2 which is orthogonal to α1 +· · ·+αn/2 (̸= 0) is 2g−1 . This
could be easily understood if you consider the dual code of a code generated by α1 +· · ·+αn/2 .

4n81624324048566472808896104112120128(2)dim Wn(2)dim Dn11345810121721243137425260

1123468111520243036425159(2)(2)

Table: Dimensions of Wn and Dn∑

At any rate, from the1

α +···+αβ-part, we get 0. Finally we have thatn/2∑

̸=0∑xα1 +γ xα1 · · · xαn/2 +γ xαn/2 .

1n/2γ∈Fg∈Fg
2 α ,...,α2

α1 +···+αn/2 =0In view of the characterization of d+n this is nothing else but the definition of the weight
enumerator of d+n in genus g. This proves the formula.
We denote by D(g) the ring generated over C by the weight enumerators of d+n (n =8, 16, 24, . . .) in genus g. The ring D(g) is a subring of W(g) . These rings are graded as⊕

W(g) =n≡0

D(g)

=W(g)
n ,(mod 8)

⊕n≡0

D(g)
n .

(mod 8)In [2], W(2) is determined. Let g24 be the Golay code of length 24. It is then(2)

(2)

(2)

(2)

(2)

(2)

(2)

82440824

4032, Wd+ ]Wd+, Wd+ ] ⊕ C[Wd+ , Wd+ , Wg(2)W(2) = C[Wd+ , Wd+ , Wg(2)2424
and the dimension formula is given as follows:∑dim W(2)n =n

1 + t32.
(1 − t8 )(1 − t24 )2 (1 − t40 )The dimensions of small n in genus 2 are given in Table.
Finally we recall the following Φ-operator
Φ(xα ) = xα and Φ(xα ) = 0,01α ∈ Fg−1,2( )α∈ Fg2 .
∗It is known that the Φ-operator maps the weight enumerator of a code in genus g to that ingenus g − 1.

3 Results
Our first objective is to prove
5Theorem 1. (1) D(g) is finitely generated over C.
(g)(2) A set of generators of D(g) can be obtained from Wd+ for n ≤ 22g+3 .
nProof. Since (1) follows from (2), we shall show (2). If we ignore the coeﬃcient
enumerator of d+n has the form
n/2

X1

n/2+ X2

1, the weight2g

n/2+ · · · + X22g

for fixed Xi ∈ C[xα : α ∈ Fg2 ]. For a better understanding, we put Zi = Xi4 . Here we remindthat n ≡ 0 (mod 8). Then we can say that D(g) is generated by the forms
Z1 + Z2 + · · · + Z22g ,Z12 + Z22 + · · · + Z222g ,..
.
If we apply the fundamental theorem of symmetric polynomials, we can conclude that D(g)can be generated by
Z1i + Z2i + · · · + Z2i 2g , 1 ≤ i ≤ 22g .
Translating this into the condition of the length n, we have1≤n 1· ≤ 22g .
2 4Hence, in order to generate D(2) , it is enough for n to range from 8 through 22g+3 mod 8.
This completes the proof.
We shall examine the case g = 1. From (2) of Theorem 1, we possess four elements oflengths 8, 16, 24, 32 to generated D(1) . Because of((1)Wd+8)2(1)

(1)

16

32

= Wd+ and Wd+ = −

5 ( (1) )4 8 (1) (1)Wd ++ Wd + Wd + ,33 8824

we get
(1)

(1)

824

D(1) = C[Wd+ , Wd+ ].
Notice that our argument in this section does not give guarantee as to the fact W(1) = D(1) .
We proceed to the higher genus. Table shows that D(2) is strictly smaller than W(2) . In
fact, we shall show
Proposition 2. We have thatW(g) = D(g) if and only if g = 1.

6(2)Proof. We have only to prove D(g) ⊊ W(g) for all g ≥ 2. We know that Wg24 ∈/ D(2) . Now
suppose that()3(g)
(g)
(g)
(g)
+ bWd+ Wd+ + cWd+ .
Wg(g)
=aW
24
d+881624

for some g ≥ 3. If we successively apply the Φ-operator to both sides, we get()3(2)
(2)
(2)
(2)
Wg(2)
=aW+ bWd+ Wd+ + cWd+ .
+24d881624(2)We have thus a contradiction to the fact Wg24 ∈/ D(2) . This completes the proof.
We turn our attention to the case g = 2.
(2)

Theorem 3. The ring D(2) is minimally generated by nine elements Wd+ of lengths
n8, 24, 32, 40, 48, 56, 64, 72, 80.
(2)

Proof. By (2) of Theorem 1, D(2) is generated by the weight enumerators Wd+ (x) of d+n ofnlengths 8, 16, . . . , 22·2+3 = 128. By calculating the dimension of the homogeneous part foreach n, we get the result. This completes the proof.
Theorem 4. The ring W(2) is the normalization of D(2) in its field of fractions.
(2)

(2)

(2)

(2)

Proof. The ring W(2) is generated by the Wd+ ’s and Wg24 . Because of dim W88 = dim D88 (=n)8(
(2)
(2)
(2)
24), Wd+ Wg24 should be written as a linear combination of the Wd+ ’s. We can say more.
n8)7((2)(2)(2)
The product Wd+ Wg24 is indeed in D80 by calculation. At any rate, we see that W(2)8(2)and D(2) have the same field of fractions. Since it can be shown that Wg24 is a root of a monic
quadratic equation over D(2) by explicit calculation, W(2) is integral over D(2) . We give thementioned forms above explicitly in Appendix. Since the invariant ring of a finite group isnormal, so is W(2) . This completes the proof.
We conclude this paper with some comments.
As a finite analogue of Eisenstein series, we studied E-polynomials (cf. [8, 9]). Since d+
8 isa unique Type II code of length 8, we obtain the identity between an E-polynomial of weight(g)

8 and Wd+ . The resulting identity seems to be non-trivial.
8Let τ be an element of the Siegel upper-half space of genus g. For α, β ∈ Fg2 , we define aThetanullwert{ () ()()}
[ ]
∑√
1t1
111αp+ α τ p+ α +t p+ αβ .
θ(τ ) =
exp 2π −1β22222gp∈Z7We put fα (τ ) = θ[ ]α0(2τ ). It is known (cf. [4]) that
( [ ] )2∑γθ(τ ) =(−1)α·β fα+γ (τ )fα (τ ).
βgα∈F2(g)Under the theta map xα → fα (τ ), we derive the theta series ϑD+ (τ ) of an even unimodular
nlattice Dn+ from the weight enumerator of d+n in genus g. Therefore we have that(g)(g)ϑD+ (τ ) = Wd+ (fα (τ ) : α ∈ Fg2 )nn
n/21 ∑ ∑(−1)α·β fα+γ (τ )fα (τ )
= g2β,γ∈Fgα∈Fg22([ ] )n∑1γθ(τ )= gβ
2gβ,γ∈F2

which was given in [5] without coding theory.
(2)

Appendix: Expressions of Wg24

(2)

(7We shall denote by dn instead of d+n and by C instead of WC . For example, d8 means)7(2)Wd+ . In the first formula, if we divide both sides by d78 , we get a rational expression of
8g24 by the dn ’s. In the second formula, we can see that g24 is a root of a monic quadratic
equation over D(2) .

d78 g24 = 60068993523/2765440 · d80 − 180183157847/10370400 · d240
− 20022997841/553088 · d32 d48 − 2860428263/69136 · d24 d56+ 20022997841/414816 · d224 d32 − 20022997841/207408 · d8 d72+ 240240240009/1382720 · d8 d32 d40 + 20022997841/103704 · d8 d24 d48− 20022997841/233334 · d8 d324 + 361030987317/2212352 · d28 d64− 721615331745/4424704 · d28 d232 − 180492013471/518520 · d28 d24 d40− 11605081037/138272 · d38 d56 + 162089538457/829632 · d38 d24 d32− 6162271423/51852 · d48 d48 + 98965418167/622224 · d48 d224+ 1437603895651/6913600 · d58 d40 − 1819759052111/33185280 · d68 d32− 1943814249461/12444480 · d78 d24 + 119236217012539/2986675200 · d10
8 .

8

2g24
= −53361/9728 · d48 + 41699/4864 · d224 + 2863707/124640 · d8 d40− 55228635/1595392 · d28 d32 + 200123/199424 · d38 d24 + 61863307/7976960 · d68+ (161/152 · d24 − 3289/12464 · d38 )g24 .

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Graduate School of Natural Science and Technology, Kanazawa University,Kakuma-machi, Kanazawa, Ishikawa 920-1192, Japan
E-mail address: makotofkanazawa@gmail.com
E-mail address: oura@se.kanazawa-u.ac.jp
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