EISENSTEIN POLYNOMIALS ASSOCIATED TO BINARY CODES
Manabu Oura
Abstract. The Eisenstein polynomial is the weighted sum of the weight
enumerators of all classes of Type II codes of fixed length. In this note, weinvestigate the ring generated by Eisenstein polynomials in genus 2.
1. Introduction. Eisenstein series play important roles in the theory ofmodular forms. Here we would like to mention two points. One is that theEisenstein series is, possibly up to a constant factor, the weighted sum of thetheta series of all classes of even unimodular lattices of fixed dimension([20].
cf. [21], [19]). Another is that the ring A(Γg ) generated over the field Cof complex numbers by modular forms of even weights for the full modulargroup in genus g is the normalization of the graded ring Bg generated over
C by Eisenstein series. This might suggest that Bg is close to A(Γg ). In thetwo special cases g = 1, 2, we know that A(Γg ) coincides with Bg , however,this is no longer true for g > 2. See [7], [8].
The Eisenstein polynomial in the title is analogue to the Eisenstein series,that is, the weighted sum of the weight enumerators of all classes of TypeII codes of fixed length. By analogy with Eisenstein series, it is natural toinvestigate the ring E(g) generated over C by Eisenstein polynomials. It isa subring of the ring W(g) generated over C by weight enumerators of TypeII codes. In the first case g = 1, these two rings coincide but this does nothold if g ≥ 2. The objective of this note is to determine E(2) . We shallshow that E(2) is minimally generated by the ten Eisenstein polynomials ofdegrees
8, 24, 32, 40, 48, 56, 64, 72, 80, 96
and coincides with W(2) except for homogeneous parts of lower degrees.
We shall denote by Z, F2 the ring of rational integers, the field of twoelements, respectively. For a finite set M , we shall denote by |M | thenumber of elements of M .

2. Eisenstein polynomial. Let g be a positive integer. We understandgthat an element of Fg2 is a row vector. For A = (ea : a ∈ Fg2 ) ∈ Z2≥0 , we putdim A = dimF2 h(1a) ∈ Fg+1
|ea > 0i.
2
We introduce 2g variables xa of degree 1 for a ∈ Fg2 . For A = (ea : a ∈QgFg2 ) ∈ Z2≥0 , a monomial xA = a∈Fg xeaa is called admissible if the degree
2P
n = a∈Fg ea is a multiple of 8 and2
Xea aS t a ≡ 0 (mod 4)
a∈Fg
2

1

for all integral symmetric g × g matrices S. Here t a stands for the transposeof a. For n = 8, 16, 24, . . . , we define the Eisenstein polynomial of degree
n in genus g byn/2−dim A−1

Eg,n (xa : a ∈ Fg2 ) =XAY(2j + 1)

j=0Y

xA ,ea !

a∈Fg
2in which the summation is extended over the set of all admissible monomialsxA of degree n. The above definition of the Eisenstein polynomials is formal
and their meaning does not become clear. In order to obtain a better
understanding, we interpret the Eisenstein polynomial from coding theory
as stated in the introduction.
A linear code C of length n is a subspace of Fn2 . A linear code is called selfdual if it coincides with its dual with respect to the inner product x · y =Pxi yi . A linear code is called doubly even if the number of non-zerocoordinates for every element of the code is a multiple of 4. A self-dual anddoubly even code is simply called Type II. It is known that a Type II code
exists if and only if n is a multiple of 8. Two codes are called equivalent

if one coincides with the other after some coordinate permutation. Up tothis equivalence, the Type II codes are classified for n = 8, 16, 24, 32 ([13],[15], [3], cf. [4]). The class invariant polynomial is given byXY
Wg,C (xa : a ∈ Fg2 ) =x(v1i ,v2i ,...,vgi ) ,v1 ,v2 ,··· ,vg ∈C 1≤i≤nwhich is called the weight enumerator of the code C in genus g. The set ofcoordinate permutations that map a code C to itself forms a group, called
the automorphism group of C. We shall denote this group by Aut(C). Let
Mn denote the set of all Type II codes of length n. Then by [17] (cf. [1],[18]) we have
1 X
Wg,C (xa : a ∈ Fg2 )n!
C∈MnX1Wg,C (xa : a ∈ Fg2 ),=|Aut(C)|Eg,n (xa : a ∈ Fg2 ) =[C]in which the summation of the second line is extended over the set of allclasses [C] of Type II codes of length n. Hence the polynomial Eg,n (xa : a ∈Fg2 ) is called ‘Eisenstein polynomial’. We refer to [20] for the original caseof this identity (cf. [21], [19]). By [10], the cardinality of Mn is known toQn/2−2
be j=0 (2j + 1). Multiplying n!/|Mn |, we get the normalized Eisenstein2polynomialn!
Eg,n (xa : a ∈ Fg2 )|Mn |X=xna + · · · .

∗Eg,n(xa : a ∈ Fg2 ) =a∈Fg
2We refer to [9], [14], [6] for the general theory of codes. See also [11] inwhich the Eisenstein polynomial plays an important role.
3. Ring generated by Eisenstein polynomials. Before restricting
ourselves to the case g = 2, we observe that E(g) = W(g) holds if and only
if g = 1. In fact, the two algebraically independent Eisenstein polynomials
E1,8 and E1,24 generate W(1) . As we shall see later, the two rings do notcoincide for g = 2. In the case g ≥ 3, the dimension of the vector spaceof spanned by the weight enumerators of Type II codes of length 24 is atleast 5 by [16](see also [17], [11]), whereas there are only 3 products ofEisenstein polynomials of degree 24. Therefore the two rings in question donot coincide.
In the rest of this note, we assume that g = 2 (and may omit g = 2 forsimplicity). We refer to [5], [17], [12] for the invariant theory of this section.
We shall denote by E, W the ring generated over C by Eisenstein polynomials, the ring generated over C by weight enumerators of Type II codes,respectively. The ring E is a subring of W. We shall denote by Ew , Ww
the homogeneous part of degree w of E, W, respectively. The ring W canbe generated by the five elements of degrees 8, 24, 24, 32, 40 and has thedimension formulaX
1 + t32(dim Ww ) tw =(1 − t8 )(1 − t24 )2 (1 − t40 )w≥0= 1 + t8 + t16 + 3t24 + 4t32 + 5t40 + 8t48 + 10t56+ 12t64 + 17t72 + 21t80 + 24t88 + 31t96 + 37t104+ 42t112 + 52t120 + 60t128 + 67t136 + 80t144 + 91t152+ 101t160 + 117t168 + · · · .
Now, we shall start investigating the graded ring E of Eisenstein polynomials. By direct calculations with Magma [2], the dimensions of Ew for
w = 24, 32, . . . , 80 are
2, 3, 4, 6, 8, 11, 15, 20
and we have that Ew = Ww for w = 0, 8, 16, 88, 96, . . . , 168. In the course ofthis calculation, we know that none of the Eisenstein polynomials of degrees
8, 24, 32, 40, 48, 56, 64, 72, 80, 96
3

is redundant to generate the ring E. For w = 176, 184, . . ., we can foreseethat Ew = Ww . Actually this is the case. The proof is as follows.
e a subring of E generated by the above ten Eisenstein
We denote by E
polynomials. We observe that the ring W can be generated by the elements
E8 , E24 , Wg24 , E32 , E40 ,in which g24 denotes the extended Golay code of length 24. Because ofe we know that W is an E-moduleeWg424 ∈ E,generated by 1, Wg24 , Wg224 , Wg324 ,i.e.,e + EWe g + EW
e 2 + EWe 3 .
W=Eg2424

g24

e + EWe g + EW
e 2 + EWe 3 for degree
We shall show that every element of E
g24
g2424eeat least 88 is an element of E. As before, we shall denote by Ew the homogee Note that we have already that Ee w = Ew = Ww
neous part of degree w of E.
for w = 88, 96, . . . , 168.

It is enough to consider monomials
fgjbcdehi
ϕ = E8a E24
E32E40E48E56E64E72
E80E96
Wgk24 , for k = 1, 2, 3eand we shall show that each monomial of degree ≥ 88 is contained in E.

For each k = 1, 2, 3 we argue1 by induction and assume that ϕ is a minimal
e w = Ww for w =counterexample of degree n ≥ 88. Then n ≥ 176 since E88, . . . , 168 by calculation. Either ϕ = E` F for some Eisenstein polynomialjWgk24 . In the first case the degreeE` of degree ` = 8, 24, . . . , 80 or ϕ = E96of F isn − ` ≥ n − 80 ≥ 176 − 80 = 96 > 88e and then also ϕ ∈ E.
e In the second
hence by minimality of ϕ we have F ∈ E
j−1
case ϕ = E96
F has a factor F = E96 Wgk24 of degree

88 < 96 + 24k ≤ 96 + 72 = 168e hence also ϕ ∈ E.
ewhich lies in E,

fw = Ew = Ww for any w ≥ 88. We
By what we have proved, we get E

have thus obtained the following

Theorem. The graded ring generated over C by Eisenstein polynomials
in genus 2 is minimally generated over C by the ten Eisenstein polynomialsof degrees
8, 24, 32, 40, 48, 56, 64, 72, 80, 96.
1 The following argument was suggested by the referee. The author’s original contained

extra computations.

4For w = 24, 32, . . . , 80, the vector space Ew is strictly smaller than the vector
space Ww and the dimensions of these Ew ’s are
2, 3, 4, 6, 8, 11, 15, 20.
For w = 0, 8, 16 and w ≥ 88, the vector space Ew coincides with the vector
space Ww .
Acknowledgement. This note was written during the author’s stayin RWTH Aachen. He would like to thank Prof. Gabriele Nebe for thehospitality. He would also like to thank Prof. Nebe and the referee for thecomments on this note.

References[1] Broué, M., Codes correcteurs d’erreurs auto-orthogonaux sur le corpsà deux éléments et formes quadratiques entières définies positives àdiscriminant +1, Discrete Math. 17, 247-269 (1977).
[2] Cannon,J.,

etal.,

The

Magma

Computational

bra System for Algebra, Number Theory
http://magma.maths.usyd.edu.au/magma/andAlge-Geometry.

[3] Conway, J.H., Pless, V., On the enumeration of self-dual codes, J.
Comb. Theory, Ser. A 28, 26-53 (1980).
[4] Conway, J.H., Pless, V., Sloane, N.J.A., The binary self-dual codes oflength up to 32: A revised enumeration, J. Comb. Theory, Ser. A 60,
No.2, 183-195 (1992).
[5] Duke, W., On codes and Siegel modular forms, Int. Math. Res. Not.
1993, No.5, 125-136 (1993).
[6] Huffman, W.C., Pless, V., Fundamentals of error-correcting codes,Cambridge University Press (2003).
[7] Igusa, J., On Siegel modular forms of genus two, Am. J. Math. 84,
175-200 (1962).
[8] Igusa, J., On the graded ring of theta-constants, Am. J. Math. 86,219-246 (1964).
5

[9] MacWilliams, F.J., Sloane, N.J.A., The theory of error-correcting
codes, Part I and II, North-Holland Mathematical Library (1977).
[10] MacWilliams, F.J., Sloane, N.J.A., Thompson, J.G., Good code exist,
Good self dual codes exist, Discrete Math. 3, 153-162 (1972).
[11] Nebe, G., Kneser-Hecke-operators in coding theory, Abh. Math. Semin.
Univ. Hamb. 76, 79-90 (2006).
[12] Nebe, G., Rains, E.M., Sloane, N.J.A., Self-dual codes and invariant theory, Algorithms and Computation in Mathematics 17, Berlin:Springer (2006).
[13] Pless, V., A classification of self-orthogonal codes over GF(2), Discrete
Math. 3, 209-246 (1972).
[14] Pless, V., Introduction to the theory of error-correcting codes, 3rd ed.,Wiley-Interscience Series in Discrete Mathematics and Optimization,
Chichester: John Wiley & Sons (1998).
[15] Pless, V., Sloane, N.J.A., On the classification and enumeration of selfdual codes, J. Comb. Theory, Ser. A 18, 313-335 (1975).
[16] Runge, B., On Siegel modular forms II, Nagoya Math. J. 138, 179-197(1995).
[17] Runge, B., Codes and Siegel modular forms, Discrete Math. 148, No.13, 175-204 (1996).
[18] Rains, E.M., Sloane, N.J.A., Self-dual codes, Pless, V. S. (ed.) et al.,Handbook of coding theory, Amsterdam: Elsevier, 177-294 (1998).
[19] Serre, J.-P., Cours d’arithmétique, Presses Universitaires de France(1970).
[20] Siegel, C.L., Über die analytische Theorie der quadratischen Formen,Ann. Math. (2) 36, 527-606 (1935).
[21] Witt, E., Eine Identität zwischen Modulformen zweiten Grades, Abh.
math. Sem. Hansische Univ. 14, 323-337 (1941).

6