Eisenstein polynomials associated to binary codes (II)
Dedicated to the memory of Professor Hidehisa Naganuma
Manabu Oura
Abstract: In the first paper we determined the generators of the ringof Eisenstein polynomials in genus 2, those of which are described in terms ofcoding theory. In this second paper we go beyond coding theory and clarify
the ring of Eisenstein polynomials in arbitarary genus from invariant theory.

1. Introduction. There exist extensive, connected studies between
theory of modular forms and that of combinatorics, however, we miss thecorresponding theory of Eisenstein series from the combinatorial side. Our
approach of the series of papers will give a step to fill this gap. The main
theme of this second paper is to clarify the properties of the graded ring ofEisenstein polynomials in connection with the invariant theory which is freefrom coding theory.
To explain our approach, or to help for the reader to understand themeaning of our work, we shall recall the theory of Eisenstein series fromour standpoint. Let g be a positive integer. We denote by Γg the integral
symplectic group Sp(g, Z) and by Γg,0 the subgroup of Γg consisting of theelements of Γg with left bottom blocks equal to zero. Then Eisenstein series
could be defined as∑Γψk g (τ ) =det(cτ + d)−kfor even k > g + 1 in which the summation is extended over elements of
Γg composed of a, b, c, d modulo left multiplications of elements of Γg,0 .
This gives a Siegel modular forms of weight k for Γg . Here we take up two
properties of this series. First the Eisenstein series could be written as theweighted sum of the theta series of all classes of even unimodular lattices inrank 2k ≡ 0 (mod 8). This is a consequence of Siegel’s Hauptsatz. Secondly
key words: Eisenstein polynomial, normalization
Mathematics Subject Classification: Primary 11T71; Secondary 11F46
running title: Eisenstein polynomial

1the ring of Siegel modular forms of even weights for Γg is the normalizationof the ring of Eisenstein series in its field of fractions. This also comes fromthe work of Siegel. At any rate Eisenstein series are singled out by theirimportance in the theory of modular forms.
Throughout our investigations, we always have in mind the following
dictionary, which might be helpful for the reader. The notations will be
explained in the next section.
“infinite world” ←→ “finite world”
Γg ←→ Hg
Siegel modular form ←→ Hg -invariant polynomial
Eisenstein series ←→ Eisenstein polynomial
lattice ←→ code

2. Definition of Eisenstein polynomial and its first properties.
Let g be a positive integer. Here we assume that entries of each element ofGL(2g , C) are indexed by the elements of Fg2 . A subgroup Dg of GL(2g , C)
tis generated by diag(i aSa : a ∈ Fg2 ) for all S = t S ∈ Matg×g (Z). We denoteby Hg , which was studied in [8], a finite subgroup of GL(2g , C) generated byDg and()g)
1+i ((−1)a·b a,b∈Fg .
22The group Hg contains the subgroups
Fg2 , GL(g, F2 )under appropriate embeddings. As entries of each element of Hg are indexedby the ordered elements of Fg2 , such a matrix naturally acts on the polynomial
ring C[x] = C[xa : a ∈ Fg2 ]. An Eisenstein polynomial of weight ℓ for Hg is,by definition,1 ∑H
φℓ g (x) =(σx0 )ℓ .
|Hg | σ∈HgH

From the definition we see that φℓ g (x) is an element of the invariant ring
C[x]Hg . Let Kg be a stabilizer of x0 in Hg which is generated by GL(g, F2 )2and Dg . The group Kg corresponds to Γg,0 . It is easy to see that|Kg | ∑H
φℓ g (x) =(σx0 )ℓ|Hg |Kg \Hg ∋σand we putkg = |Kg \Hg | = 22+g (2g + 1)(2g−1 + 1) · · · (21 + 1).
Here we only mention that this is nothing else but the number of the minimal
vectors of Barnes-Wall lattice in dimension 2g+1 . We note
2|Hg | = 2g +2g+2 (4g − 1)(4g−1 − 1) · · · (41 − 1),|Kg | = |Dg | · |GL(n, F2 )|= 4g 2(g−1)g/2 · 2(g−1)g/2 (2g − 1)(2g−1 − 1) · · · (21 − 1)2

= 2g +g (2g − 1)(2g−1 − 1) · · · (21 − 1).
The explicit values of kg = |Hg |/|Kg | for g = 1, 2, 3, 4 are964680371589120
48514675507200, 240 =
, 4320 =, 146880 =.
419286016
330301440
We add one more finite group. Let Gg be a finite subgroup of GL(2g , C)
generated by Hg and the 8th root of unity. The index of Hg in Gg is 2.
Invariant theory of Gg works well in coding theory as we shall recall. Theweight enumerator WCg (x) of a binary code C in genus g is defined by∑ ∏WCg (x) =xana (v1 ,...,vg )24 =v1 ,...,vg ∈C a∈Fg2

where na (v1 , . . . , vg ) denotes the number of i such that a = (vi1 , . . . , vig ). It
is then known that the ring generated over C by the weight enumeratorsof Type II codes in genus g coincides with the invariant ring of Gg . This
theorem started with Gleason [1]. See [6]. Note that the invariant ring ofarbitrary finite group is integrally closed.
Till the end of this section we assume ℓ ≡ 0 (mod 8). Applying Theorem
6.3 in [5] to the doubly even code generated by all one vector, we have∏∑ g2g ∑(σx0 )ℓ =(2g + 2i )−1WC (x)|Gg | σ∈G
C0≤i<ℓ/2−1g=∏(2g + 2i )−10≤i<ℓ/2−13

∑
[C]ℓ!
Wg.
|AutC| CThe summation over C means that C runs through all Type II codes oflength ℓ, while that over [C] through all classes of Type II codes of length1 ∑Hgℓℓ. Since the sumσ∈Gg (σx0 ) is φℓ (x), we get Siegel’s Hauptsatz in|Gg |coding theory:∗Hφℓ g (x) =2∑ℓ!

∏g0≤i<ℓ/2−1(2g + 2i )[C]

1WCg (x).
|AutC|H

From this formula we see that φℓ g does not vanish for ℓ ≡ 0 (mod 8). Siegel’s
Φ-operator in number theory could be also considered in our context (cf. [8]).
Applying our “Φ-operator”{x(a1 a2 ...ag−1 ) ag = 0
Φ(x(a1 a2 ···ag ) ) =0ag = 1
to the formula, we getHΦ(φℓ g (x)) =1 2g−ℓ/2+1 + 1 Hg−1·φℓ (x).
22g + 1

3. Ring of Eisenstein polynomial. In this section the graded ring ofEisenstein polynomials is studied. We shall start with the following finitenesstheorem, which is a consequence of the fundamental theorem of symmetricpolynomials.
HTheorem 1. (1) The ring C[φℓ g (x)] of Eisenstein polynomials is finitelygenerated over C.
H(2) The ring C[φℓ g (x) : ℓ ≡ 0 (mod 8)] of Eisenstein polynomials ofweights divisible by 8 is finitely generated over C.
In order to obtain the normalization type theorem, we shall prepare twolemmata.
HLemma 2. C[x]Hg is integral over C[φℓ g (x)].
∗This formula seems to be known to some experts including Professor A.Munemasa [4].

4Proof. First we recall thatr∑zjk = 0, k = 1, 2, . . . , r

j=1

have no common zero other than z1 = z2 = · · · = zr = 0. We apply thisto our case. Suppose that φ8·1 (x) = φ8·2 (x) = · · · = φ8kg (x) = 0, where
kg = |Hg |/|Kg |. Since Hg contains Fg2 , σ(x0 ) = xa = 0 for σ = a ∈ Fg2 . This
completes the proof of Lemma 2.
In some parts of the following, we assume non-vanishingness of certain
Eisenstein polynomial. This seems very likely true, however, the author does
not have its proof in his hand. We give it as a conjecture.
Conjecture: there exists a non-vanishing Eisenstein polynomial of weight̸ 0 (mod 8).
≡Lamma 3. Assume that Conjecture is true. Then the field of fractions
Hof C[φℓ g ] coincides with the field of fractions of C[x]Hg .
Proof. This is a consequence of Corollary 4.4 in [5]. On the treatmentof the graded ring, we refer to [2]. For the graded ring S, we denote by
S (d) generated by elements of degrees multiple of d. A quotient field of S iswritten as F (S) and its degree 0 part as F0 (S).
The Galois group of the extension C(x)/C(x)Gg is Gg . If we take anHelement of σ from the automorphism group of C(x)/C(φℓ g ), then σ shouldpreserve φ8 . By the mentioned corollary, σ is an element of Gg . As a conHsequence, we have C(x)Gg = C(φℓ g : ℓ ≡ 0 (mod 8)). Write S for C[x]Hgand take a weight 4 element ξ from F (S).
C(x)Hg = F (C[x]Hg )= F (S)= F0 (S)(ξ)= F0 (S (8) )(ξ)
H= F0 (C(φℓ g (x) : ℓ ≡ 0
H= F0 (C(φℓ g ))(ξ)
H= C(φℓ g )5(mod 8)))(ξ)

This completes the proof of Lemma 3.
Combining the lemmata above, we get
Theorem 4. Assume that Conjecture is true. Then the invariant ringHC[x]Hg of the finite group Hg is the normalization of the ring C[φℓ g ] ofEisenstein polynomials in its field of fractions.
The following ”Theorem” could be obtained as ”Corollary” of Theorem
4 if we drop the assumption in Theorem 4. The proof is the same as that ofTheorem 4.
Theorem 5. The ring C[x]Gg of the weight enumerators of Type II codes
His the normalization of C[φℓ g : ℓ ≡ 0 (mod 8)] in its field of fractions.
4. Concluding remarks. (1) For g = 1, 2, the ring C[xa ]Hg could be
generated by Eisenstein polynomials. In our paper [7], we gave the generators
2of C[φHℓ : ℓ ≡ 0 (mod 8)], the weights ℓ of which are
8, 24, 32, 40, 48, 56, 64, 72, 80, 96.
This ring is strictly smaller than C[x]G2 and Theorem 5 says that if we takethe normalization, we get the whole ring C[x]G2 . In order to see this, takethe Golay code g24 of length 24. If we add the weight enumerator Wg224 tothe ring of Eisenstein polynomial, we get the whole ring. This means thatWg224 is integral over the ring of Eisenstein polynomial and is contained in thefield of Eisenstein polynomials. It is easy to find such isobaric polynomials
2P, Q, P ′ , Q′ as, omitting H2 in the notation of φHℓ ,(Wg224 )2 + P (φ8 , φ24 )Wg224 + Q(φ8 , φ24 , φ32 , φ40 , φ48 ) = 0,
Wg224 = P ′ (φ8 , φ24 , φ32 , φ40 , φ48 , φ56 )/Q′ (φ8 , φ24 , φ32 ).
(2) The approach of this paper suggests the theory of Eisenstein polynomials for arbitrary finite group. Along this line Yano [9] studied the rings ofEisenstein polynomials for finite unitary reflection groups.
(3) Suppose g = 1. Under the theta map, an Eisenstein polynomial istransformed into a modular form for SL(2, Z). Due to some computations,the derived modular forms seem to enjoy properties similar to Eisenstein
6series. For further investigations of Eisenstein polynomials, consult Miezaki[3].
Acknowledgment. The author would like to thank a referee for helpfulcomments. He was partially supported by JSPS KAKENHI (C) 25400014.

References[1] Gleason, A.M., Weight polynomials of self-dual codes and theMacWilliams identities. Actes du Congrs International des Mathmaticiens (Nice, 1970), Tome 3, pp. 211215. Gauthier-Villars, Paris, 1971.
[2] Igusa, J., Theta functions, Die Grundlehren der mathematischen Wissenschaften, Band 194. Springer-Verlag, New York-Heidelberg, 1972.
[3] Miezaki, T., in preparation.
[4] Munemasa, A., Codes, invariant polynomials and modular forms(injapanese), The first spring conference ”The rings of automorphic forms”(Organizers:T.Ibukiyama, et al.), Hamana lake, Japan, 2002, 135–161.
[5] Nebe, G., Rains, E., Sloane, N.J., The invariants of the Cliﬀord groups.
Des. Codes Cryptogr. 24 (2001), no. 1, 99–121.
[6] Nebe, G., Rains, E., Sloane, N.J., Self-dual codes and invariant theory. Algorithms and Computation in Mathematics, 17. Springer-Verlag,
Berlin, 2006.
[7] Oura, M., Eisenstein polynomials associated to binary codes, Int. J.
Number Theory 5(2009), no.4, 635–640.
[8] Runge, B., Codes and Siegel modular forms. Discrete Math. 148 (1996),
no. 1-3, 175–204.
[9] Yano, S., Master thesis, Kochi University (2011).
Department of Mathematics
Kochi University
Akebono-cho 2-5-1
Kochi, 780-8520
7

Japanemail: oura@kochi-u.ac.jp
Current address:
Division of Mathematical and Physical Sciences
Graduate School of Natural Science and Technology
Kanazawa University
Kakuma-machi, Kanazawa
Ishikawa 920-1192Japan
email: oura@se.kanazawa-u.ac.jp8