Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-conjugate weight enumerators and invariant theory

Let K be a field, Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} a finite group of field automorphisms of K, F the Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-fixed field in K, and G≤GLv(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G\le {{\,\textrm{GL}\,}}_v(K)$$\end{document} a finite matrix group. Then the action of Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} defines a grading on the symmetric algebra of the F-space Kv\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K^v$$\end{document} which we use to introduce the notion of homogeneous Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-conjugate invariants of G. We apply this new grading in invariant theory to broaden the connection between codes and invariant theory by introducing Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-conjugate complete weight enumerators of codes. The main result of this paper applies the theory from Nebe, Rains, Sloane to show that under certain extra conditions these new weight enumerators generate the ring of Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-conjugate invariants of the associated Clifford–Weil groups. As an immediate consequence, we obtain a result by Bannai et al. that the complex conjugate weight enumerators generate the ring of complex conjugate invariants of the complex Clifford group. Also the Schur–Weyl duality conjectured and partly shown by Gross et al. can be derived from our main result.

of all harmonic G-invariant polynomials up to degree (t, t), then the G-orbit xG gives rise to a projective t-design.
The textbook [9] gives a very general notion of a Type of a code.Associated to a Type ρ and an integer m ≥ 1, there is a finite complex matrix group C m (ρ), the associated Clifford-Weil group of genus m, such that the genus-m complete weight enumerator of any code of Type ρ and length N is an invariant polynomial of C m (ρ), homogeneous of degree N .The Weight Enumerator Conjecture states that the space of homogeneous degree N invariants of C m (ρ) is spanned by these weight enumerators.If v is the size of the alphabet of the codes of Type ρ, then C m (ρ) consists of matrices of size v m .Despite of this exponentially growing dimension, the space of invariants of a given degree N can be obtained by enumerating all codes of length N of Type ρ.
For the Type of doubly even binary codes, where v = 2, the associated Clifford-Weil groups are the complex Clifford groups X m ≤ GL 2 m (C) which have a tight connection to quantum information theory (see [11]).The main result of [1] shows that the ring of complex conjugate invariants of X m is spanned by the genus-m complex conjugate weight enumerators of self-dual doubly even binary codes.The paper also enumerates all such codes up to length (5,5) therewith proving a conjecture from [11] that whenever an X morbit forms a projective 4-design, then it is automatically a projective 5-design.
We extend the approach in [1] to the more general set-up in [9].Any Type ρ also determines an abelian number field K such that the associated Clifford-Weil groups consist of matrices over K. Given a subgroup Γ of the automorphism group of K, we introduce the concept of Γ-conjugate weight enumerators of codes of Type ρ.These are elements of the ring of Γ-conjugate invariants (Definition 2.1) of C m (ρ).Our main result, Theorem 6.3, shows that the space of Γ-conjugate homogeneous invariants of C m (ρ) is spanned by these weight enumerators of codes of a given length.Therefore the enumeration of such codes of small length determines the Γ-conjugate invariants of small degree of the genus-m Clifford-Weil group associated to ρ for all m ∈ N.
With a view to possible applications to measurement schemes for low rank matrix recovery from complex projective t-designs as in [6], Section 7 gives a few examples of self-dual codes of Type (N, N ) for N ≤ 4 for several small representations ρ.Section 8 contains, as another application of Theorem 6.3, a short proof of the Schur-Weyl duality conjectured and partly established in [4].

Conjugate polynomials.
Let K be a field and let Γ := {α 1 , . . ., α n } ≤ Aut(K) be a group of field automorphisms of K that is finite of order n.Put to denote the fixed field of Γ.Then K/F is a Galois extension with Galois group Γ.
For a vector space U over K of finite dimension v, we denote by K[U ] the ring of polynomials on U .We now restrict scalars and write U F for U regarded as an nv-dimensional space over F .For a K-basis (u 1 , . . ., u v ) of U and an where the functions y ji : U F → F form the dual basis to the previously chosen basis B. The latter is a polynomial ring in nv variables over F with well studied gradings given by multi-degrees.As we are mostly interested in invariant theory of finite groups over fields of characteristic 0, we assume that K is an infinite field and see elements of Definition 2.1.For v ∈ N, we denote by v := {1, . . ., v} and by and defined as the K-subspace spanned by the monomial functions M (( The degree of such a monomial is The K-algebra structure of K[x • Γ] is inherited from the K-algebra structure of K K v .In particular, the multiplication of two monomials is given by and as usual the degree of the product is just the sum of the degrees of the two factors.
The Γ-action provides a finer notion of degree of a polynomial in ( This shows that the linear functions (x i • α j : (i, j) ∈ v × n) in K K v are algebraically independent over K.

Corollary 2.2. The space of homogeneous polynomials of degree
As these monomials form a basis, its dimension is 3. Invariant theory.We keep the assumptions of the previous section, in particular, K is an infinite field and Γ = {α 1 , . . ., α n } is a group of automorphisms of K of finite order n.Let G ≤ GL v (K) be a group.Then the right action of G on K v defines a right action of G by K-algebra automorphisms on the K-algebra of K-valued functions on K v by This action preserves the subalgebra K[x • Γ] as well as its subspaces of homogeneous polynomials of a given degree.
Via the isomorphism ϕ from Equation (1), the In particular, classical invariant theory gives us Molien's formula for the Hilbert series of this invariant ring.
The grading from Definition 2.1 refines the classical degree function thus giving a notion of Γ-conjugate Hilbert series of the invariant ring K[x • Γ] G , generalising the Forger series (where K = C and F = R) [3] to our situation.

Definition 3.2. For any
denote the dimension of the space of G-invariant conjugate polynomials that are homogeneous of degree d.Putting the Γ-conjugate Hilbert series of the ring K[x • Γ] G .

Molien's theorem.
If G is finite and K has characteristic 0, then Molien's theorem [2, Theorem 2.5.2]gives a useful expression of the Hilbert series of the classical invariant ring of G.With a completely analogous proof (see also [3] for K = C and F = R), we obtain the following theorem.Theorem 3.3.Let K be a field of characteristic 0 and let G be a finite subgroup of GL v (K).Then .
Proof.For a given degree d ∈ N n 0 , the Reynolds operator Since char(K) = 0, the dimension of this fixed subspace is equal to the trace of P d .Therefore

As trace(P
To do so, we may and will assume that K contains a |G|th primitive root of unity.Then each g ∈ G is diagonalizable over K so after a suitable choice of basis, we assume that g −1 = diag(λ 1 , . . ., λ v ).Then for α j ∈ Γ, The monomials from Definition 2.1 form an eigenvector basis for the action of .
4. The Type of a self-dual code.This section briefly recalls the relevant notions from [9].Classically a self-dual code C of length N over a finite field F q is a linear subspace C ≤ F N q that is self-dual (i.e., C = C ⊥ ) with respect to the standard inner product.Loosely speaking, to define self-dual codes in a more general sense, we need a ring R, a left R-module V , and a non-singular form β on V such that β can be used to define the orthogonal code C ⊥ of an R-submodule C ≤ V N .Extra conditions (such as being doubly even) can be imposed by means of isotropy conditions with respect to some set Q of quadratic maps on V .We call such an admissible quadruple ρ := (R, V, Q, β) a Type, and the self-dual isotropic codes that arise from ρ are called codes of Type ρ.More precisely, a Type is a representation of an abstract form ring.

Finite representations of form rings. A finite representation of a form ring
For a code C in ρ, the orthogonal module is MacWilliams transformations ([9, Section 2.2]) map the weight enumerator of a code to the one of its dual and fix weight enumerators of self-dual codes.To obtain a generating set of the associated Clifford-Weil group, we need to include MacWilliams transformations for representatives of the conjugacy classes of primitive symmetric idempotents.

Definition 4.2. An idempotent
as right R-modules.Such an isomorphism is given by left multiplication with some v ι ∈ ι J Rι with inverse u ι ∈ ιRι J such that u ι v ι = ι and v ι u ι = ι J (see [9,Section 3.5.3]).
As ν(ρ) is a finitely generated (and hence finite) Remark 4.5.Put f := f (ρ) and let a 1 , . . ., a N ∈ Z be prime to f and put a := (a 1 , . . ., a N ).Then the orthogonal sum is a finite representation of the form ring (R, Φ).In fact, we do not know a counterexample and [9, Theorem 5.5.5 and Theorem 5.5.7]assert the truth of the Weight Enumerator Conjecture for fairly large classes of finite form rings including matrix rings over finite fields.
We recall the action of the associated Clifford-Weil group for the representation ρ a from Remark 4.5 with respect to the C-basis (b w : w = (w 1 , . . ., w N ) ∈ V N ) of the group algebra of V N : where and Here the ι runs through the set of all R × -conjugacy classes of symmetric idempotents in R and u ι , v ι ∈ R are as in Definition 4.2.
So the transformations m r are represented as permutation matrices on the chosen basis and the transformations d φ as diagonal matrices.It is clear that fwe(C) is invariant under all d φ and all m r if C is an isotropic code in ρ a .For self-dual codes, the invariance under h ι,uι,vι follows from a general MacWilliams theorem, see [ put γ a to denote the Galois automorphism of K(ρ) that is the identity on F and raises ζ f to the ath power.Then γ : (Z/f Z) × → Gal(K(ρ)/F ), a → γ a , is a group isomorphism.So for all a ∈ Z prime to f , the groups C(ρ (a) ) and γ a (C(ρ)) are equal.

Conjugate weight enumerators and the main theorem.
We keep the notation of the previous section, in particular, ρ is a finite representation of a form ring (R, Φ) and K := K(ρ) is an abelian extension of the field F from Definition 5.2.Put f := f (ρ) and Γ := Gal(K/F ) =: {α 1 , . . ., α n }, a finite group of order n isomorphic to (Z/f Z) × .We always assume that there is some length N 0 such that there is a self-dual isotropic code in ρ N0 .Then C(ρ) is a finite subgroup of GL v (K), where v = |V |.We also assume that the Weight Enumerator Conjecture 5.1 holds for the form ring (R, Φ).For a = (a 1 , . . ., a N ) as in Remark 4.5, we consider the Clifford-Weil groups C(ρ a ) introduced in Section 5. Definition 6.1.We say that a satisfies the sign condition if for all symmetric idempotents ι ∈ R.

Higher genus Clifford-Weil groups.
The Clifford-Weil groups associated to a representation ρ form an infinite sequence of matrix groups C m (ρ) in exponentially growing dimension v m for m ∈ N that comes with ring epimorphisms Φ m : Inv(C m (ρ)) → Inv(C m−1 (ρ)) that are isomorphisms on the spaces of invariants of small degree.A similar phenomenon occurs for the space of Γ-conjugate invariants.
The full genus-m weight enumerator fwe m (C) of C in ρ is the sum over all m-tuples of code words of C. For C ≤ V N , we can think of such an m-tuple c ∈ C m as a matrix c ∈ V m×N , where the rows correspond to code words.For In the notation of Definition 6. x ci • α j the Γ-conjugate complete weight enumerator of genus m of the code C in ρ a .Now the alphabet of C m is V m and hence a module over the ring Mat m (R) of m × m-matrices over R.This gives rise to a notion of a matrix ring of a form ring (see [9,Section 1.10]) such that R m ⊗ ρ is a representation of this matrix ring.The genus-m Clifford-Weil group is Denote by K[x (m) ] the symmetric algebra of V m .Theorem 6.3 has the following immediate consequence.Corollary 6.5.Assume that ρ satisfies the Weight Enumerator Conjecture 5.1 and that a satisfies the sign condition.Then the space of invariants of C m (ρ) in K[x (m) • Γ] d is spanned by the Γ-conjugate complete weight enumerators of genus m of codes of Type ρ a .Remark 6.6.The map Φ m mapping x (v1,...,vm) tr to x (v1,...,vm−1) tr if v m = 0 and to 0 if v m = 0 defines a ring epimorphism that preserves the grading from Section 2 and maps ccwe m (C) to ccwe m−1 (C) for any code C in ρ a .In particular, it induces an epimorphism from the space of invariants of If all self-dual isotropic codes in ρ a are generated by m − 1 elements, then this epimorphism is in fact an isomorphism.In this case, two codes are equivalent (see Definition 6.2) if and only if they have the same genus-m weight enumerator and these weight enumerators of the equivalence classes of codes of Type ρ a form a basis of the space of invariants of degree d.

Complex conjugate invariants.
In this section, we give the most important example where all a i from Remark 4.5 are ±1.As γ −1 is the complex conjugation − and the field F 1 from Definition 5.2 is totally real, we always have (ι, −1) = 1 for all symmetric idempotents ι ∈ R and hence the sign condition is fulfilled.
In particular, the main result of [1] is an immediate consequence of Theorem 6.3.Whereas [1] only considers doubly even, self-dual binary codes, Theorem 6.3 allows to consider complex conjugate invariants for more general Clifford-Weil groups.
In view of possible applications to complex projective designs, we are mainly interested in codes of Type ρ (1 N ,(−1) N ) which we call Type (N, N ) for short.As we want to have as few invariants as possible, we consider those representations ρ where the quadratic group ρ Φ (Φ) is as large as possible: For self-dual codes over fields of odd characteristic, we focus on self-dual codes that contain the all-ones vector 1 = (1, . . ., 1) and in even characteristic on the generalized doubly even codes.

Decomposable codes. For codes
Clearly the genus-m weight enumerator of C 1 ⊕ C 2 is just the product of the genus-m weight enumerators of C 1 and C 2 .
In the following, we only list representatives of the equivalence classes of indecomposable codes, i.e., codes that cannot be written as a non-trivial orthogonal sum.The trivial code T N,N is the double of the zero code T N,N = Double( 0 N ).

Enumeration of equivalence classes of codes.
In this section, we sketch some methods to enumerate all self-dual isotropic codes in V N .

The mass formula.
In many situations, the unitary group (in the examples below, the orthogonal group of the quadratic space 1 ⊥ / 1 ) acts transitively on the set of codes of Type (N, N ); their number t N is the number of isotropic subspaces in a certain finite geometry.If C 1 , . . ., C hN represent the Sym N × Sym N -orbits of codes of Type (N, N ), then we obtain the following mass formula: where Aut(C i ) is the stabiliser in Sym N ×Sym N of the code C i ≤ V N +N .This formula can be used to check the completeness of a list of pairwise inequivalent codes but also yields a method to enumerate these codes by partitioning the orbit of the unitary group into Sym N × Sym N -orbits.

Kneser neighbors.
A more efficient method to enumerate self-dual codes is the Kneser neighbor method [5] (see also [7] for an application to codes and [10] for a survey).
Starting from one self-dual code C (e.g., C = T N,N ), one enumerates all Aut(C)-orbits of neighbors, i.e., those equivalence classes of codes D of Type ρ that intersect C in a maximal subcode.Continuing with the neighbors, one successively enumerates all codes up to equivalence.

Using projections.
Denote by π 1 : V N +N → V N the projection onto the first N components and by π 2 the projection onto the last N coordinates.For a self-dual isotropic code C of Type (N, N ), the kernel of π 2 is the dual of the image of π 1 .
There is a shortcut to classify the codes C ≤ F N +N q of Type (N, N ) for which π 1 (C) = F N q .If the involution is trivial, then each such code has a generator matrix (I N |A) with AA tr = I N .Equivalence of codes translates into the action of Sym hence yield generator matrices for representatives of the equivalence classes of codes.If one only wants to classify codes that contain the all-ones vector of length 2N , one needs to additionally assume that the sum of the rows of A is the all-ones vector 1 of length N , i.e., 1A = 1.As 1g = 1 for all permutation matrices g ∈ Sym N , this property is invariant on the Sym N -double cosets.7.3.Doubly even binary self-dual codes.This is the most important example and has been considered in [1].The Type of these codes is denoted by 2 II in [9, Section 2.3.2], has conductor 8, and the associated Clifford-Weil group is the complex Clifford group.The indecomposable codes of length (N, N ), i.e., of Type 2 where the last code has generator matrix (I 4 |2J 4 − I 4 ) and J 4 is the all-ones matrix.
7.6.Quinary codes that contain the all-ones vector.We consider the analogous Type as in the previous section, where the underlying field is F 5 and the conductor is 5.We obtain the following list for small lengths: N is also a CG-module, where the action is given by f • g : w → f (wg −1 ) for all w ∈ W N , g ∈ G, f ∈ W * N .

5 .Remark 5 . 1 .
Clifford-Weil groups and full weight enumerators.Let ρ = (V, ρ M , ρ Φ , β) be a finite representation of a form ring and put v := |V |.The group algebra CV is a v-dimensional complex vector space with basis (b w : w ∈ V ).The full weight enumerator of a code in ρ is defined as fwe(C) := c∈C b c ∈ CV.The associated Clifford-Weil group C(ρ) is a group of linear operators on CV whose generators are explicitly given in [9, Definition 5.3.1];these generators are the obvious transformations that stabilise fwe(C) for any code of Type ρ.In particular, the full weight enumerators of codes of Type ρ are invariant under C(ρ) (see [9, Theorem 5.5.1]).The Weight Enumerator Conjecture [9, Conjecture 5.5.2]states that in this general situation the fixed space of C(ρ) is spanned by the full weight enumerators of codes of Type ρ.

Theorem 6 . 3 .
Assume that ρ satisfies the Weight Enumerator Conjecture 5.1 and that a satisfies the sign condition.Then the space of invariants of C(ρ) in K[x • Γ] d is spanned by the Γ-conjugate complete weight enumerators of self-dual isotropic codes in ρ a .

7. 1 . 7 . 1 . 1 .
Codes of Type (N, N ).The trivial codes.The standard norm z m := || || 2 on C v m is an invariant for the full unitary group U v m (C).In fact, z N m is the genus-m complex conjugate weight enumerator of the code T N,N with generator matrix (I N |I N ), the trivial code of Type (N, N ).

7. 1 . 3 .
The doubling construction.There is another general construction of codes in ρ (N,N ) which we call the doubling construction: Let C be a self-orthogonal isotropic code C in ρ N and put D := C ⊥ ⊇ C. Then Double(C) := {(d, d + c) | d ∈ D, c ∈ C} is a code of Type ρ (N,N ) .If C is a self-dual isotropic code in ρ N , i.e., C = D, then Double(C) = {(c 1 , c 2 ) | c 1 , c 2 ∈ C}.