The Invariants of the Clifford Groups.
The automorphism group of the Barnes-Wall lattice L_m in dimension 2^m(m \neq 3 ) is a subgroup of index 2 in a certain Clifford group \mathcal{C}_m of structure 2_+^{1+2m} . O^+(2m,2). This group and its complex analogue \mathcal{X}_m of structure (2_+^{1+2m}{\sf Y}Z_8) . Sp(2m, 2) have arisen in recent years in connection with the construction of orthogonal spreads, Kerdock sets, packings in Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs. In this paper we give a simpler proof of Runge@apos;s 1996 result that the space of invariants for \mathcal{C}_m of degree 2k is spanned by the complete weight enumerators of the codes C \otimes \Bbb{F}_{2^m}, where C ranges over all binary self-dual codes of length 2k; these are a basis if m \ge k-1. We also give new constructions for L_m and \mathcal{C}_m: let M be the \Bbb{Z}[\sqrt{2}]-lattice with Gram matrix \scriptsize\big[\begin{array}{@{}r@{\quad}r@{}} 2 & \sqrt{2} \\ \sqrt{2} & 2 \end{array} \big]. Then L_m is the rational part of M^{\otimes m}, and (M^{\otimes m} ). Also, if C is a binary self-dual code not generated by vectors of weight 2, then \mathcal{C}_m is precisely the automorphism group of the complete weight enumerator of C \otimes \Bbb{F}_{2^m}. There are analogues of all these results for the complex group \mathcal{X}_m, with doubly-even self-dual code instead of self-dual code.
Introduction
In 1959 Barnes and Wall [2] constructed a family of lattices in dimensions 2
They distinguished two geometrically similar lattices Lm #
. The automorphism
investigated in a series of papers by Bolt, Room and Wall [8], [9],
[50]. Gm is a subgroup of index 2 in a certain group Cm of structure 2 1+2m
We follow Bolt et al. in calling Cm a Cli#ord group. This group and its complex analogue
Xm are the subject of the present paper.
These groups have appeared in several di#erent contexts in recent years. In 1972 Brou-e
and Enguehard [12] rediscovered the Barnes-Wall lattices and also determined their automorphism
groups. In 1995, Calderbank, Cameron, Kantor and Seidel [13] used the Cli#ord
groups to construct orthogonal spreads and Kerdock sets, and asked "is it possible to say
something about [their] Molien series, such as the minimal degree of an invariant?".
Around the same time, Runge [39], [40], [41], [42] (see also [20], [36]) investigated these
groups in connection with Siegel modular forms. Among other things, he established the
remarkable result that the space of homogeneous invariants for Cm of degree 2k is spanned
by the complete weight enumerators of the codes
ranges over all binary
self-dual (or type I) codes of length 2k, and the space of homogeneous invariants for Xm of
degree 8k is spanned by the complete weight enumerators of the codes
ranges over all binary doubly-even self-dual (or type II) codes of length 8k. One of our goals
is to give a simpler proof of these two assertions, not involving Siegel modular forms (see
Theorems 4.9 and 6.2).
Around 1996, the Cli#ord groups also appeared in the study of fault-tolerant quantum
computation and the construction of quantum error-correcting codes [4], [15], [16], [29], and
in the construction of optimal packings in Grassmannian spaces [14], [17], [44]. The story of
the astonishing coincidence (involving the group C 3 ) that led to [14], [15] and [16] is told in
[16]. (Other recent references that mention these groups are [23], [30], [51].)
Independently, and slightly later, Sidelnikov [45], [46], [47], [48] (see also [28]) came across
the group Cm when studying spherical codes and designs. In particular, he showed that for
the lowest degree harmonic invariant of Cm has degree 8, and hence that the orbit
under Cm of any point on a sphere in R 2 m
is a spherical 7-design. (Venkov [49] had earlier
shown that for m # 3 the minimal vectors of the Barnes-Wall lattices form 7-designs.)
In fact it is an immediate consequence of Runge's results that for m # 3 Cm has a unique
harmonic invariant of degree 8 and no such invariant of degree 10 (see Corollary 4.13). The
space of homogeneous invariants of degree 8 is spanned by the fourth power of the quadratic
form and the complete weight enumerator of the code H
is the [8, 4, 4]
Hamming code. An explicit formula for this complete weight enumerator is given in Theorem
4.14.
Our proof of the real version of Runge's theorem is given in Section 4 (Theorem 4.9),
following two preliminary sections dealing with the group Cm and with generalized weight
enumerators.
In Section 5 we study the connection between the group Cm and the Barnes-Wall lat-
tices. We define the balanced Barnes-Wall lattice Mm to be the Z[ # 2]-lattice #
(Lemma 5.2), which leads to a simple construction: the Barnes-Wall
lattice is just the rational part of
1 . Furthermore
More precisely,
Also, if C is any binary self-dual code that is not generated by vectors of weight 2,
(Corollary 5.7). The proof of this makes use of the fact that Cm is a
maximal finite subgroup of GL(2 m , R) (Theorem 5.6). Although there are partial results
about the maximality of Cm in Kleidman and Liebeck [30], this result appears to be new.
The proof does not use the classification of finite simple groups.
The analogous results for the complex Cli#ord group Xm are given in Section 6. Theorem
6.2 is Runge's theorem. Extending scalars, let M m be the hermitian Z[# 8 ]-lattice Z[# 8
Mm . Then Xm is the subgroup of U(2 m , Q [# 8 ]) preserving M m (Proposition 6.4). Theorem
6.5 shows that, apart from the center, Xm is a maximal finite subgroup of U(2 m , C ), and
Corollary 6.6 is the analogue of Corollary 5.7.
Bolt et al. [8], [9], [10], [50] and Sidelnikov [45], [46], [47] also consider the group C (p)
obtained by replacing 2 in the definition of Cm by an odd prime p. In the final section we
give some analogous results for this group.
In recent years many other kinds of self-dual codes have been studied by a number of
authors. Nine such families were named and surveyed in [38]. In a sequel [35] to the present
paper we will give a general definition of the "type" of a self-dual code which includes all
these families as well as other self-dual codes over rings and modules. For each "type" we
investigate the structure of the associated "Cli#ord-Weil group" (analogous to Cm and Xm
for types I and II) and its ring of invariants.
The results in this paper and in Part II can be regarded as providing a general setting
for Gleason's theorems [24], [32], [38] about the weight enumerator of a binary self-dual code
(cf. the case Theorem 4.9), a doubly-even binary self-dual code (cf. the case
Theorem 6.2) and a self-dual code over F p (cf. the case Theorem 7.1).
They are also a kind of discrete analogue of a long series of theorems going back to Eichler
(see for example [7], [39], [40], [42]), stating that under certain conditions theta series of
quadratic forms are bases for spaces of modular forms: here complete weight enumerators of
generalized self-dual codes are bases for spaces of invariants of "Cli#ord-Weil groups".
2 The real Cli#ord group Cm
This initial section defines the real Cli#ord group Cm . The extraspecial 2-group E(m)
is a subgroup of the orthogonal group O(2 m , R). If
is the automorphism group of the 2-dimensional standard lattice. In general E(m) is the
m-fold tensor power of E(1):
E(m) :=
E(1)# -
and is generated by the tensor products of # 1 and # 2 with 2 - 2 identity matrices I 2 .
Definition 2.1 The real Cli#ord group Cm is the normalizer in O(2 m , R) of the extraspecial
2-group E(m).
The natural representation of E(m) is absolutely irreducible. So the centralizer of E(m)
in the full orthogonal group is equal to { - I 2 m}, which is the center of E(m). Then Cm/E(m)
embeds into the outer automorphism group of E(m). The quotient group E(m)/Z(E(m)) is
isomorphic to a 2m-dimensional vector space over F 2 . Since every outer automorphism has
to respect the {+1, -1}-valued quadratic form
it follows easily that the outer automorphism group of E(m) is isomorphic to O + (2m, 2), the
full orthogonal group of a quadratic form of Witt defect 0 over F 2 (see e.g. [51]).
Since the group 2 1+2m
is a subgroup of O(2 m , R) (cf. [10] or the explicit
construction below), we find that Cm
2). The order of Cm is
1).
To perform explicit calculations we need a convenient set of generators for Cm .
Theorem 2.2 Cm is generated by the following elements
(1) diag((-1) q(v)+a ), where q ranges over all {0, 1}-valued quadratic forms on F m
2 and
a # {0, 1},
(2) AGL(m, 2), acting on R 2 m
by permuting the basis vectors in F m
and
(3) the single matrix
h# I
# .
Proof. Let H be the group generated by the elements in (1) and (2). First, H contains the
extraspecial group E(m), since #
are in H and their images under
GL(m,
To see that H/E(m) is a maximal parabolic subgroup of O note that by [13]
the elements a # GL(m, 2) act on E(m)/Z(E(m))
2 as # a 0
0 a -tr # , and the elements
# , where b is the skew-symmetric matrix corresponding to the
bilinear form b q (x, y) := q(x
Since
is not in H, the group generated by H and this element is
Cm .
Corollary 2.3 Cm is generated by
1# I
2# I
h# I
where # is the particular quadratic form
3 Full weight enumerators and complete weight enumerator

We now introduce certain weight enumerators and show that they are invariant under the
real Cli#ord group. Let C # F N
2 be a linear code # of length N over the field F 2 . For m # N
let C(m) :=
be the extension of C to a code over the field with 2 m elements.
Let V be the group algebra V := Regarding
2 as an
m-dimensional vector space over F 2 , we have a tensor decomposition
(R 2 ).
In the same manner the group algebra embeds naturally into the
group algebra
(R 2 )).
Definition 3.1 The full weight enumerator of C(m) is the element
(This was called a generalized weight polynomial in [24] and an exact enumerator in [32,
Chapter 5].)
Fix a basis (a 1 , . , am ) of F m
. Then a codeword c # C(m) is just an m-tuple of
codewords in C. The element
corresponds to the m-tuple
which can also be regarded as an m-N-matrix M of which the rows are the elements
Lemma 3.2 Let
c1 ,. ,c m#C
e c
# m# N
(R
Then the
induced by identifying an m-tuple
with the codeword c :=
Proof. Let
C. The generator e c of R[C(m)] is
x
(R 2 )),
has a basis y 0 , y 1 . Under the identification above this element is mapped
onto
(y # (1)
(R 2 )),
which is the element e c #
# A binary linear code C of length N is a subspace of F N
2 . If C # C # , C is self-orthogonal; if
is self-dual [32], [38].
Definition 3.3 (Cf. [32, Chapter 5].) The complete weight enumerator of C(m) is the
following homogeneous polynomial of degree N in 2 m variables:
x a f (c)
where a f (c) is the number of components of c that are equal to f .
Remark 3.4 The complete weight enumerator of C(m) is the projection under # of the full
weight enumerator of C(m) to the symmetric power Sym N (V ), where #
is the R-linear mapping defined by x f
Theorem 3.5 Let C be a self-dual code over F 2 .
(i) The Cli#ord group Cm preserves the full weight enumerator fwe(C(m)).
(ii) The Cli#ord group Cm preserves the complete weight enumerator cwe(C(m)).
Proof. Let N be the length of C, which is necessarily even. Then Cm acts on R[F N
diagonally. This action commutes with the projection #
statement (ii) follows immediately from (i) by Remark 3.4. To prove (i) it is enough to
consider the generators of Cm .
The generators #
1# I
2# I
-# I 2 and
h# I
I 2 of Corollary 2.3
are tensor products of the form
x# I 2 m-1 . By Lemma 3.2 it is therefore enough to consider
the case generators. But then the matrix # 1 acts as #
mapping a codeword is the
all-ones vector. Since C is self-dual, 1 is in C and therefore # 1 only permutes the codewords
and hence fixes fwe(C). Analogously, the matrix # 2 changes signs of the components of the
codewords in the full weight enumerator: if is mapped to (-1) c i x c i .
Since the codewords in C have even weight, the tensor product x c
is fixed by
That h preserves the full weight enumerator follows from the MacWilliams
identity [32, Chapter 5, Theorem 14].
The generator d := diag((-1) #(v)
only occurs for m # 2. By
Lemma 3.2 it su#ces to consider the case 2. Again by Lemma 3.2, we regard d as acting
on pairs (c, c # ) of codewords in C. Then d fixes or negates (x c
and negates it if and only if c and c # intersect in an odd number of 1's. This is impossible
since C is self-dual, and so d also preserves fwe(C(m)).
The remaining generators in g # GL(m, 2) permute the elements of F 2 m . The codewords
c # C(m) are precisely the elements of the form
a fixed F 2 -basis for acts linearly on F m
mapping a i onto
, the word
c is mapped to
which again is in C(m). Hence these generators also fix
fwe(C(m)).
4 The ring of invariants of Cm
In this section we establish Runge's theorem that the complete weight enumerators of the
codes C(m) generate the space of invariants for Cm .
Definition 4.1 A polynomial p in 2 m variables is called a Cli#ord invariant of genus m if it
is an invariant for the real Cli#ord group Cm . Furthermore, p is called a parabolic invariant
if it is invariant under the parabolic subgroup P generated by the elements of type (1) and
(2) of Theorem 2.2, and a diagonal invariant if it is invariant under the group generated by
the elements of type (1).
The following is obvious:
Lemma 4.2 A polynomial p is a diagonal invariant if and only if all of its monomials are
diagonal invariants.
Let M be an m - N matrix over F 2 . We can associate a monic monomial -M # R[x f |
with such a matrix by taking the product of the variables associated with its
columns. Clearly all monic monomials are of this form, and two matrices correspond to the
same monic monomial if and only if there is a column permutation taking one to the other.
Lemma 4.3 A monic monomial -M is a diagonal invariant if and only if the rows of M
are orthogonal.
Proof. It su#ces to consider quadratic
m); we easily check that the action of diag((-1) q ij ) is to multiply -M by (-1) k , where k is
the inner product of rows i and j of M ; the lemma follows.
For
maps -M onto -M+b , where the matrix M + b has entries (M This implies
that -M is equivalent to -M # under the action of AGL(m, 2) if and only if the binary codes
#M, 1# and #M # , 1# are equivalent. We can thus define a parabolic invariant -m (C) for any
self-orthogonal code C containing 1 and of dimension at most m
-M .
We define -m (C) to be 0 if 1 # C or dim(C) > m+ 1. Since the invariants -m (C) are sums
over orbits, we have:
Lemma 4.4 A basis for the space of parabolic invariants of degree N is given by polynomials
of the form -m (C) where C ranges over the equivalence classes of binary self-orthogonal codes
of length N containing 1 and of dimension
Lemma 4.5 For any binary self-orthogonal code C containing 1,
-m (D).
Proof. From the definition,
where M ranges over m-N matrices with all rows in C. Let M be such a matrix. Then M
uniquely determines a subcode D := #M, 1# of C; we thus have
-m (D)
as required.
Theorem 4.6 A basis for the space of parabolic invariants is given by the polynomials
cwe(C(m)), where C ranges over equivalence classes of self-orthogonal codes containing 1
and of dimension
Proof. The equations in Lemma 4.5 form a triangular system which we can solve for the
polynomials -m (C). In particular, -m (C) is a linear combination of the cwe(D(m)) for
subcodes 1 # D # C.
denote the linear transformation
where P is the parabolic subgroup of Cm ; that is, X P is the operation of averaging over the
parabolic subgroup.
Lemma 4.7 For any binary self-orthogonal code C of even length N containing 1 and of
dimension N/2 - r,
The final sum is over all self-orthogonal codes C # containing C to index 2.
Proof. By the MacWilliams identity, we find that
where M ranges over m-N matrices such that the first row of M is in C # and the remaining
rows are in C. For each code 1 # D # C # , consider the partial sum over the terms with
D. If D # C, the partial sum is just -m (D), so in particular is a parabolic
invariant. The other possibility is that [D : D # 2. For a matrix M with #M,
define a vector v M # F m
2 such that (v M the ith row of M is in C, and (v M
otherwise. In particular, the partial sum we are considering is
-M .
If D is not self-orthogonal then this sum is annihilated by averaging over the diagonal
subgroup. Similarly, if we apply an element of AGLm (2) to this sum, this simply has the
e#ect of changing v M . Thus, when D # D # ,
-m (D).
Hence
-m (D),
where the sums are restricted to self-orthogonal codes D. Introducing a variable C #D, C#
into the second sum (note that since D # C # , C # C # precisely when D # D # ), this
becomes
-m (D).
Any given C # will, of course, contain each subcode of C exactly once, so we can remove the
condition
-m (D)
-m (D)
as required.
Lemma 4.8 Let V be a finite dimensional vector space, M a linear transformation on V ,
and P a partially ordered set. Suppose there exists a spanning set v p of V indexed by p # P
on which M acts triangularly; that is,
for suitable coe#cients c pq . Suppose furthermore that c only if p is maximal in
. Then the fixed subspace of M in V is spanned by the elements
Proof. Since the matrix is triangular, there exists another triangular matrix D
that conjugates C into Jordan canonical form. Setting
(d pp #= 0), we find
with c # su#ciently large n. In other words, each w p is
in the Jordan block of M with eigenvalue c pp . But the vectors w
the Jordan blocks of M on V are spanned by the corresponding Jordan blocks of C. In
particular, this is true for the block corresponding to 1.
Theorem 4.9 (Runge [42].) Fix integers k and m # 1. The space of homogeneous invariants
of degree 2k for the Cli#ord group Cm of genus m is spanned by cwe(C(m)), where C
ranges over all binary self-dual codes of length 2k; this is a basis if m # k - 1.
Proof. Let p be a parabolic invariant. If p is a Cli#ord invariant then
By Lemma 4.7, the operator X P
acts triangularly on the vectors cwem (C) (ordered
by inclusion); since
the hypotheses of Lemma 4.8 are satisfied. The first claim then follows by Lemma 4.8 and
Theorem 3.5. Linear independence for m # k - 1 follows from Lemma 4.4.
In fact a stronger result holds:
Theorem 4.10 For any binary self-orthogonal code C of even length N containing 1 and
of dimension N/2 - r,|C m | #
g#Cm
1#i#r
where the sum on the right is over all self-dual codes C # containing C.
To see that this is indeed stronger than Theorem 4.9, we observe that if p is an invariant
for
g#Cm
Since the space of parabolic invariants contains the space of invariants, the same is true of
the span of|C m | #
g#Cm
ranges over the parabolic invariants. By Theorem 4.10 each of these can be written
as a linear combination of complete weight enumerators of self-dual codes, and thus Theorem
4.9 follows.
Proof. For any self-orthogonal code C, let
g#Cm
Averaging both sides of equation (1) in Lemma 4.7 over Cm , we find
and solving for Em (C) gives
By induction on r (observing that the result follows from Theorem 3.5 when
1#i#r
cwem (C # ).
But each code C # is counted 2 r
times (corresponding to the 1-dimensional subspaces of
thus eliminating the sum over C # gives the desired result.
Note that
This gives a surjective map from the space of genus m complete weight enumerators to the
space of genus m- 1 complete weight enumerators. By Theorem 4.9 it follows that this also
gives a surjective map from the genus m invariants to the genus m- 1 invariants. (Runge's
proof of Theorem 4.9 proceeds by first showing this map is surjective, using Siegel modular
forms, and then arguing that this implies Theorem 4.9.) Since by Theorem 4.6 the parabolic
invariants of degree N become linearly independent when m # N
Corollary 4.11 Let #m (t) be the Molien series of the Cli#ord group of genus m. As m
tends to infinity, the series #m (t) tend monotonically to
where N 2k is the number of equivalence classes of self-dual codes of length 2k.
(For the definition of Molien series, see for example [5] or [32, Chapter 19].)
Explicit calculations for
Corollary 4.12 The initial terms of the Molien series of the Cli#ord group of genus m # 1
are given by
where the next term is 2t 12 for
showed that the lowest degree of a harmonic invariant of Cm is 8.
Inspection of the above Molien series gives the following stronger result.
Corollary 4.13 The smallest degree of a harmonic invariant of Cm is 8, and there is a
unique harmonic invariant of degree 8. There are no harmonic invariants of degree 10.
The two-dimensional space of homogeneous invariants for Cm of degree 8 is spanned by
the fourth power of the quadratic form and by hm := cwe(H
is the
[8, 4, 4] binary Hamming code. We can give hm explicitly.
Theorem 4.14 Let G(m, denote the set of k-dimensional subspaces of F m
2 . Then
v#d+U
v#d+U
v#d+U
x v . (2)
The second term on the right-hand side is equal to 14
runs
through unordered pairs of elements of F m
. The total number of terms is
Proof. We will compute cwe(H
(which is equal to cwe(H
defined by the generator matrix
A codeword corresponds to a choice of (a, b, c, d) # F m
, one for each row; from the columns
of the generator matrix we find that the corresponding term of the weight enumerator is
x d x c+d x b+d x b+c+d x a+d x a+c+d x a+b+d x a+b+c+d .
This depends only on the a#ne space #a, b, c# d. The four terms on the right-hand side
of Eq. (2) correspond to dim#a, b, the coe#cients are the number of ways of
choosing d) for a given a#ne space. If dim#a, b, for example, there are 7 - 6 - 4
ways to choose a, b, c and 8 ways to choose d, giving the coe#cient 8 - 7 - 6 -
Remarks
(1) The unique harmonic invariant of degree 8 integrates to zero over the sphere, and
so must have zeros on the sphere. The orbit of any such point under Cm therefore forms a
spherical 11-design, cf. [25]. This was already observed by Sidelnikov [48].
(2) The case dihedral group of order 16 with Molien series 1/(1-# 2 )(1-# 8 ),
as in Gleason's theorem on the weight enumerators of binary self-dual codes [24], [32, Problem
3, p. 602], [38].
(3) The case 2: C 2 has order 2304 and Molien series
(The reflection group [3, 4, 3], No. 28 on the Shephard-Todd list, cf. [5, p. 199], is a subgroup
of C 2 of index 2.) The unique harmonic invariants f 8 and f 12 (say) of degrees 8 and 12 are
easily computed, and then one can find real points
and f 12 vanish. Any orbit of such a point under C 2 forms a spherical 15-design of size 2304
(cf. [25]). We conjecture that such points exists for all m # 2.
(4) The group C 3 of order 5160960 has appeared in su#ciently many di#erent contexts
that it is worth placing its Molien series on record. It is p(#)/q(#), where p(#) is the
symmetric polynomial of degree 154 beginning
and
(5) For completeness, we mention that the Molien series for E(1) is 1
basic invariants
1 and x 2
1 . For arbitrary m the Molien series for E(m) is2n 2
5 Real Cli#ord groups and Barnes-Wall-lattices
In a series of papers [2], [8], [9], [10], [50], Barnes, Bolt, Room and Wall investigated a
family of lattices in
(cf. also [12], [18]). They distinguish two geometrically similar
lattices Lm # L # m in each dimension 2 m , for which if m #= 3 the automorphism groups
are subgroups Gm of index 2 in the real Cli#ord group Cm . When
are two versions of the root lattice
index 270 in Aut(L 3 ) and index 2 in C 3 .
The lattices Lm and L # m can be defined in terms of an orthonormal basis b 0 , . ,
of
as follows. Let V := F m
2 and index the basis elements
by the elements of
V . For each a#ne subspace U # V let # U #
correspond to the characteristic function
of
corresponds to an element of U and #
spanned by the set
{2 #(m-d+#)/2# U | 0 # d # m, U is a d-dimensional a#ne subspace of V } ,
Extending scalars, we define the Z[ # 2]-lattice
which we call the balanced Barnes-Wall lattice.
From the generating sets for Lm and L # m we have:
Remark 5.1 Mm is generated by the vectors # 2 m-d
runs
through the a#ne subspaces of V of dimension d.
Lemma 5.2 For all m > 1, the lattice Mm is a tensor product:
Proof. Write
as the direct sum of an (m - 1)-dimensional vector
space Vm-1 and a 1-dimensional space arrange the basis vectors so that
correspond to the elements in Vm-1 and b 2 m-1 , . ,
-1 to the elements in
# U be a generator for Mm , where d-dimensional linear
subspace U 0 of V and a
If U 0 # Vm-1 , then
m-d
m-1-d
Otherwise Um-1 := U 0 # Vm-1 has dimension d - 1 and U
some v m-1 # Vm-1 . If v m-1 # Um-1 , then
m-d
If v m-1 # Um-1 we have the identity
Hence Mm #
. The other inclusion follows more easily by similar arguments.

In view of Lemma 5.2, we have the following simple and apparently new construction for
the Barnes-Wall lattice Lm . Namely, Lm is the rational part of the Z[ # 2]-lattice
1 is the Z[ # 2]-lattice with Gram matrix # 2 # 2
# . For more about this construction
see [34].
Proposition 5.3 For all m # 1, the automorphism group Aut(Mm ) (the subgroup of the
orthogonal group O(2 m , R) that preserves Mm ) is isomorphic to Cm .
Proof. Let (v 1 , . , v 2 m) be a Z-basis for
is a Z-basis for Lm . Then m) is a Z[ # 2]-basis for
Hence Mm has a Z-basis ( # 2v 1 , . , #
Since the scalar products of the v i are integral, the Z-lattice Mm with respect to 1the
trace form of the Z[
# 2]-valued standard form on Mm is isometric to # 2L # m # Lm . In
particular, the automorphism group of the Z[ # 2]-lattice Mm is the subgroup of Aut( # 2Lm #
that commutes with the multiplication by # 2. Hence Aut(Mm ) contains
a subgroup of index at most two. Since
by Lemma 5.2, [Aut(Mm
Lemma 5.4 If m # 2, then the Z-span (denoted Z[C m ]) of the matrices in Cm acting on the
-dimensional Z[
2]-lattice Mm is Z[
Proof. We proceed by induction on m. Explicit calculations show that the lemma is true
by induction Z[C m-2
and
the automorphism group of Mm
contains C
2# Cm-2 . Hence

We now proceed to show that for m # 2 the real Cli#ord group Cm is a maximal finite
subgroup of GL(2 m , R). For the investigation of possible normal subgroups of finite groups
containing Cm , the notion of a primitive matrix group plays a central role. A matrix group
called imprimitive if there is a nontrivial decomposition
of V into subspaces which are permuted under the action of G. G is called primitive if it is
not imprimitive. If N is a normal subgroup of G then G permutes the isotypic components
of V |N . So if G is primitive, the restriction of V to N is isotypic, i.e. is a multiple of an
irreducible representation. In particular, since the image of an irreducible representation of
an abelian group N is cyclic, all abelian normal subgroups of G are cyclic.
Lemma 2. Let G be a finite group with Cm # G # GL(2 m , R) and let p be
a prime. If p is odd, the maximal normal p-subgroup of G is trivial. The maximal normal
2-subgroup of G is either E(m) if
Proof. We first observe that the only nontrivial normal subgroup of Cm that is properly
contained in E(m) is Therefore, if U is a normal subgroup of G,
U # E(m) is one of 1, Z(E(m)) or E(m).
The matrix group Cm and hence also G is primitive. In particular, all abelian normal
subgroups of G are cyclic. Let p > 2 be a rational prime and U E G a normal p-subgroup
of G. The degree of the absolutely irreducible representations of U that occur in R 2 m
|U is
a power of p and divides 2 m . So this degree is 1 and U is abelian, hence cyclic by the
primitivity of G. Therefore the automorphism group of U does not contain E(m)/Z(E(m)).
Since CG (U)#E(m) is a normal subgroup of Cm , it equals E(m) and hence E(m) centralizes
U . Since E(m) is already absolutely irreducible, U consists of scalar matrices in GL(2 m , R),
and therefore U = 1. If because Cm is the largest
finite subgroup of GL(2 m , R) that normalizes E(m). Since the normal 2-subgroups of G do
not contain an abelian noncyclic characteristic subgroup, the possible normal 2-subgroups
are classified in a theorem of P. Hall (cf. [27, p. 357]). In particular they do not contain
Cm /Z(E(m)) as a subgroup of their automorphism groups, so again U commutes with E(m),
and therefore consists only of scalar matrices.
Theorem 5.6 Let m # 2. Then the real Cli#ord group Cm is a maximal finite subgroup of
Proof. Let G be a finite subgroup of GL(2 m , R) that properly contains Cm . By Lemma
5.5, all normal p-subgroups of G are central. By a theorem of Brauer, every representation
of a finite group is realizable over a cyclotomic number field (cf. [43, -12.3]). In fact, since
the natural representation of G is real, it is even true that G is conjugate to a subgroup of
totally real abelian number field K containing
5.6]). Let K be a minimal such field and assume that G # GL(2 m , K). Let R be the
ring of integers of K. Then G fixes an RCm -lattice. By Lemma 5.4 all RCm -lattices are of
the form
I# Z[ # 2] Mm for some fractional ideal I of R, the group G fixes all RCm -lattices
and hence also
R# Z[ # 2] Mm . So any choice of an R-basis for Mm gives rise to an embedding
by which we may regard G as a group of matrices. Without loss of generality
we may assume that
Aut(R# Z[ # 2] Mm ). Then the Galois group # := Gal(K/Q [ # 2])
acts on G by acting componentwise on the matrices. Seeking a contradiction, we assume
It is enough to show that there is a nontrivial element # that acts trivially
on G, because then the matrices in G have their entries in the fixed field of #, contradicting
the minimality of K.
Assume first that there is an odd prime p ramified in K/Q , and let # be a prime ideal
of R that lies over p. Then p is also ramified in K/Q [ # 2] and therefore the action of the
ramification group, the stabilizer in # of #, on R/# is not faithful, hence the first inertia
group
is nontrivial (see e.g. [22, Corollary III.4.2]). Since G # := {g # G | g # I 2 m (mod #)} is
a normal p-subgroup of G, G Therefore all the elements in # act
trivially on G, which is what we were seeking to prove.
2 is the only ramified prime in K, which implies that
a ] for some a # 3,
and we are done. So
assume a > 3 and let # be the prime ideal of R over 2 (generated by (1-# 2 a)(1-1
2 a )) and let
# be the Galois automorphism defined by # 2 a+#
2 a +# -2 a-1
2 a ).
# and
Therefore # 2# . Since the subgroup G 2# := {g # G | g # I 2 m (mod 2#)} of G is trivial
(cf. [3, Hilfssatz 1]) one concludes that # acts trivially on G, and thus G is in fact defined
over
The theorem follows by induction.
Corollary 5.7 Let m # 1 and let C be a self-dual code over F 2 that is not generated by
vectors of weight 2. Then
Proof. The proof for the case will be postponed to Section 6. Assume m # 2.
We first show that the parabolic subgroup H # Cm acts irreducibly on the Lie algebra
R)), the set of real 2 m
. The group AGL(m, 2)
acts 2-transitively on our standard basis b 0 , . ,
. A basis for
is given by the matrices b ij := b
- 1. Since AGL(m, 2)
acts transitively on the b ij , a basis for the endomorphism ring End
is given by the orbits of the stabilizer of b 01 . Representatives for these orbits are b 01 , b 02 ,
b 23 and b 24 . But the generator corresponding to the quadratic form q(v 1 , . , therefore does not commute with the endomorphism
corresponding to b 02 or b 24 . Similarly the endomorphism corresponding to b 23 is ruled out
by q(v 1 , . ,
Let G := AutO(2 m ,R) (cwe(C(m)). Then G is a closed subgroup of O(2 m , R) and hence is a
Lie group (cf. [37, Theorem 3.4]). Since G contains Cm it acts irreducibly on Lie(O(2 m , R)).
Assume that G #= Cm . Then G is infinite by Theorem 5.6 and therefore G contains SO(2 m , R).
However, the ring of invariants of SO(2 m , R) is generated by the quadratic form
The only binary self-dual codes C that produce such complete weight enumerators are direct
sums of copies of the code {00, 11}.
6 The complex Cli#ord groups and doubly-even codes
There are analogues for the complex Cli#ord group Xm for most of the above results. (Z a
will denote a cyclic group of order a.)
Definition 6.1 The complex Cli#ord group Xm is the normalizer in U(2 m , Q [# 8 ]) of the central
product E(m)YZ 4 .
As in the real case, one concludes that
(cf. [33, Cor. 8.4]).
The analogue of Theorem 4.9 is the following, which can be proved in essentially the
same way.
Theorem 6.2 (Runge [42].) Fix integers N and m # 1. The space of homogeneous invariants
of degree N for the complex Cli#ord group Xm is spanned by cwe(C(m)), where C
ranges over all binary doubly-even self-dual codes of length N . (In particular, when N is not
a multiple of 8, the invariant space is empty.)
The analogues of Theorem 4.10 and Proposition 5.3 are:
Theorem 6.3 For any doubly-even binary code C of length N # 0(8) containing 1 and of
dimension
g#Xm
0#i<r
where the sum is over all doubly-even self-dual codes C # containing C.
Proposition 6.4 Let M m := Z[# 8
. Then the subgroup of U(2 m , Q [# 8 ]) preserving
M m is precisely Xm .
We omit the proofs.
For the analogue of Lemma 5.4, observe that the matrices in Xm generate a maximal order.
Even for the Z-span of the matrices in X 1 acting on M 1 is the maximal order Z[# 8 ] 2-2 .
Hence the induction argument used to prove Lemma 5.4 shows that Z[Xm
Therefore the analogue of Theorem 5.6 holds even for
Theorem 6.5 Let m # 1 and let G be a finite group such that Xm # G # U(2 m , C ). Then
there exists a root of unity # such that
Proof. As in the proof of Theorem 5.6, we may assume that G is contained in U(2 m , K)
for some abelian number field K containing # 8 . Let R be the ring of integers in K and T the
group of roots of unity in R. Then TXm is the normalizer in U(2 m , K) of TE(m) (cf. [33,
Cor. 8.4]). As before, the RXm -lattices in the natural module are of the form
where I is a fractional ideal of R. Since G fixes one of these lattices, it also fixes
As in the proof of Theorem 5.6, we write the elements of G as matrices with respect to a
basis for M m and assume that G is the full (unitary) automorphism group of
Then the Galois group # := Gal(K/Q [# 8 ]) acts on G. Assume that G #= TXm . Then TE(m)
is not normal in G. As in Lemma 5.5 one shows that the maximal normal p-subgroup of
G is central for all primes p. Let # be a prime ideal in R that ramifies in K/Q [# 8 ], and
let # be an element of the inertia group # . Then for all G, the image g # satisfies
is a normal p-subgroup,
where p is the rational prime divisible by #, it is central. Therefore the map g # a(g) is a
homomorphism of G into an abelian group, and hence the commutator subgroup G # is fixed
under #. Since any abelian extension K of Q that properly contains Q is ramified at some
finite prime of Q [# 8 ], we conclude that G # Aut(M m ). Since E(m)YZ 8
characteristic in Aut(M m ) and therefore also in G # YZ 8 , the group TE(m) is normal in G,
which is a contradiction.
Corollary 6.6 Assume m # 1 and let C be a binary self-dual doubly-even code of length N .
Then
Remarks
(1) The case unitary reflection group (No. 9 on the Shephard-Todd list)
of order 192 with Molien series 1/(1 - # 8 )(1 - # 24 ), as in Gleason's theorem on the weight
enumerators of doubly-even binary self-dual codes [24], [32, p. 602, Theorem 3c], [38].
(2) The case 2: X 2 has order 92160 and Molien series
This has a reflection subgroup of index 2, No. 31 on the Shephard-Todd list.
(3) The case 3: X 3 has order 743178240, and the Molien series can be written as
is the symmetric polynomial of degree 44 beginning
and
Runge [40] gives the Molien series for the commutator subgroup H
3 , of index 2 in
. The Molien series for X 3 consists of the terms in the series for H 3 that have exponents
divisible by 4. Oura [36] has computed the Molien series for H
4 , and that for X 4 can
be obtained from it in the same way. Other related Molien series can be found in [1].
Proof of Corollary 5.7, case
Let C be a self-dual binary code of length n with Hamming weight enumerator hwe C (x, y).
We will show that if C is not generated by vectors of weight 2 then AutO(2) (hwe C
Certainly G := AutO(2) (hwe C must show it is no larger. The only
closed subgroups of O(2) containing D 16 are the dihedral groups D 16k for k # 1 and O(2)
itself. So if the result is false then G contains a rotation
where # is not a multiple of #/4.
Consider the shadow S(C) of C [38]; that is, the set of vectors v # F n
2 such that
The weight enumerator of S(C) is given by S(x, y, i(x - y)). Then
G if and only if S(x, or in other words if and only if for all
is a multiple of 2#.
Now, pick a vector v 0 # S(C), and consider the polynomial W (x, y, z, w) given by
This has the following symmetries:
y, z, w),
y, z, w).
Furthermore, since only if
y, z, w).
To each of these symmetries we associate a 2 - 2 unitary matrix U such that (x, y) is
transformed according to U and (z, w) according to U . The first two symmetries generate
the complex group X 1 , which is maximally finite in PU(2) by Theorem 6.5. On the other
hand, we can check directly that
even up to scalar multiplication. Thus the three symmetries topologically generate PU(2);
and hence W is invariant under any unitary matrix of determinant -1. Since hwe C (x,
y, x, y), it follows that G = O(2). But then
hwe C (x,
implying that C is generated by vectors of weight 2.
This completes the proof of Corollary 5.7.
7 Cli#ord groups for p > 2
Given an odd prime p, there again is a natural representation of the extraspecial p-group
of exponent p, this time in U(p m , C ); to be precise, E p (1) is generated by
transforms
is the m-th tensor power of E p (1). The Cli#ord group C (p)
m is then defined to be
the normalizer in U(p m , Q [# ap ]) of E p (m), where a one finds that
(cf. e.g. [51]).
As before, the invariants of these Cli#ord groups are given by codes:
Theorem 7.1 Fix integers N and m # 1. The space of invariants of degree N for the
Cli#ord group C (p)
m is spanned by cwe(C(m)), where C ranges over all self-dual codes over F p
of length N containing 1.
Theorem 7.2 For any self-orthogonal code C over F p of length N containing 1 and of
dimension N/2 - r,|C (p)
0#i<r
where the sum is over all self-dual codes C # containing C (and in particular is 0 if no such
code exists).
Regarding maximal finiteness, the arguments we used for to prove Theorem 5.6 do
not carry over to odd primes, since the groups C (p)
m do not span a maximal order. Lindsey
[31] showed by group theoretic arguments that C (p)
1 is a maximal finite subgroup of SL(p, C )
(cf. [6] for the theorem below follows from [21] and [26].
Theorem 7.3 Let p > 2 be a prime and m # 1. If G is a finite group with C (p)
there exists a root of unity # such that
Proof. As before we may assume that G is contained in U(p m , K) for some abelian number
field K containing # p . Let L denote the set of rational primes l satisfying the following four
properties: (i) G is l-adically integral, (ii) l is unramified in K, (iii) |G| < |P GL(p m , l)|, (iv)
l splits completely in K. Since all but finitely many primes satisfy conditions (i)-(iii), and
infinitely many primes satisfy (iv) (by the -
Cebotarev Density Theorem), it follows that the
set L is infinite.
Fix a prime l over l # L. Since G is l-adically integral, we can reduce it mod l, obtaining
a representation of G in GL(p m , l). Since p is ramified in K, l #= p, so this representation
is faithful on the extraspecial group. Since the extraspecial group acts irreducibly, the
representation is in fact faithful on the entire Cli#ord group. Thus G mod l contains the
normalizer of an extraspecial group, but modulo scalars is strictly contained in PGL(p m , l)
(by condition (iii)). It follows from the main theorem of [30] that for p m
l and
coincide as subgroups of PGL(p m , l). For already follows from the
references in the paragraph preceding the theorem.
Fix a coset S of C (p)
m in G. For each prime l|l with l # L, the above argument implies
that we can choose an element g # S such that g # 1 (mod l). As there are infinitely many
such primes, at least one such g must get chosen infinitely often. But then we must actually
have in K, and since g has finite order, some root of unity # S .
Since this holds for all cosets S, G is generated by C (p)
together with the roots of unity
proving the theorem.
Remark 7.4 It is worth pointing out that the proof of the main theorem in [30] relies
heavily on the classification of finite simple groups, which is why we preferred to use our
alternative arguments when proving Theorem 5.6.



--R

Type II codes
Some extreme forms defined in terms of Abelian groups
Zur Galoiskohomologie definiter arithmetischer Gruppen
Mixed state entanglement and quantum error correction
Polynomial Invariants of Finite Groups
University of Chicago Press
in Theta functions (Bowdoin
The Cli



Une famille infinie de formes quadratiques enti-eres

A group-theoretic framework for the construction of packings in Grassmannian spaces
Quantum error correction orthogonal geometry
Quantum error correction via codes over GF (4)
packings in Grassmannian space
Lattices and Groups
Induction and structure theorems for orthogonal representations of finite groups
Notices 5
On finite linear groups in dimension at most 10
Algebraic Number Theory
On the faithful representations
Weight polynomials of self-dual codes and the MacWilliams identities
The football


On certain groups defined by Sidelnikov (in Russian)
algorithms and error correction (in Russian)
The Subgroup Structure of the Finite Classical Groups

The Theory of Error-Correcting Codes
Finite quaternionic matrix groups
A simple construction for the Barnes-Wall lattices
Generalized self-dual codes and Cli#ord- Weil groups
The dimension formula for the ring of code polynomials in genus 4
Algebraic Groups and Number Theory
in Handbook of Coding Theory
On
On
The Schottky ideal
Codes and
Linear Representations of Finite Groups
A family of optimal packings in Grassmannian manifolds
On a finite group of matrices and codes on the Euclidean sphere (in Russian)
On a finite group of matrices generating orbit codes on the Euclidean sphere
Spherical 7-designs in 2 n -dimensional Euclidean space
Orbital spherical 11-designs in which the initial point is a root of an invariant polynomial (in Russian)
"designs"

The automorphism group of an extraspecial p-group
--TR
Codes and Siegel modular forms
A Family of Optimal Packings in Grassmannian Manifolds
A Group-Theoretic Framework for the Construction of Packings in Grassmannian Spaces
Spherical 7-Designs in 2<math coding="latex" type="inline">^n</math>-Dimensional Euclidean Space

--CTR
F. L. Chiera, Type II Codes over$$\mathbb{Z}/2k\mathbb{Z}$$, Invariant Rings and Theta Series, Designs, Codes and Cryptography, v.36 n.2, p.147-158, August    2005
K. Betsumiya , Y. Choie, Jacobi forms over totally real fields and type II codes over Galois rings GR(2m, f), European Journal of Combinatorics, v.25 n.4, p.475-486, May 2004
invariants;spherical designs;self-dual codes;barnes-wall lattices;clifford groups