CHAPTER 6. LINKS BETWEEN CODES AND PRIMITIVE GROUPS 65
• the block K1gi is incident with the point K2hj if and only ifK2hj∩K1gi ∼=Gi, i= 1, . . . , k where{G1, . . . , Gk} ⊂ {K2x∩K1y|x, y ∈G}.
Denote a 1−design constructed in this way by D(G, K2, K1;G1, . . . , Gk).
From the conjugacy class of a maximal subgroup K of a simple group G for example, using the method given earlier the authors postulated in [41] that one can obtain a regular graph, in the following way:
• the vertex set of the graph is cclG(K),
• the vertex Khi is adjacent to the vertex Khj if and only if Khi ∩Khj ∼= Gi, i = 1, . . . , k, where {G1, . . . , Gk} ⊂ {Kg ∩Kg0 | g, g0 ∈ G}. We denote this graph G(G, K;G1, . . . , Gk).
Gacts primitively on the set of vertices of G(G, K;G1, . . . , Gk).
Remark 6.9. Let ψ be an automorphism of a finite group G. Then the design D(G, K2, K1;G1, . . . , Gk) is isomorphic to D(G, ψ(K2), ψ(K1);G1, . . . , Gk), and the graph G(G, K;G1, . . . , Gk) is isomorphic to G(G, ψ(K);G1, . . . , Gk).
CHAPTER 6. LINKS BETWEEN CODES AND PRIMITIVE GROUPS 66 the group action ρ : G−→ GL(V) defined by ρ(g, x)7→ xg =ρ(g)(x), g ∈ G, x∈ V.
Extending linearly the induced G-action on V makesV into an FΩ-module called an FΩ− permutation module .
In the following we define how a permutation group G≤Sn acts onFΩ.
Definition 6.11. LetG≤Snandg ∈G,and letW ⊆FΩwithu= (u1, u2, . . . , uj)∈ FΩ. Then we make the following definitions:
g(u) = g((u1, u2, . . . , uj)) = (g(u1), g(u2), . . . , g(uj));
g(W) = {g(u)|u ∈W},
G·W = {g(u)|u ∈W, g ∈G}.
We now define the notion ofG-invariance for codes.
Definition 6.12. Let G ≤Sn and let C ⊆ FΩ. We say that C is invariant under G if G·C =C.
Remark 6.13. Notice that sinceCis a submodule ofFΩ we can talk about invariant codes. We will show in Lemma 6.16 that a group Gthat leaves a codeC invariant is related to the automorphism group of C. We define and discuss the automorphism group of a code and explore the interplay between Gand this automorphism group.
Definition 6.14. Let C be a code of length n. We define the automorphism group of C as
Aut(C) ={g ∈Sn|g(C) =C}.
The following lemma from [108] will be of use in the chapters which ensue.
Lemma 6.15. Let C be an [n, k, d] code and let {b1, b2, . . . , bk} be a basis for C and g ∈Sn. Then g ∈Aut(C) if and only if g(bi)∈C for all 1≤i≤k.
Proof: Let g ∈ Aut(C). Then by definition g(c) ∈ C for every c ∈ C and so the result. Conversely, let g(bi) ∈ C for 1 ≤ i ≤ k and let u ∈ C so that we can write u =Pk
i=1αibi for some scalars αi ∈F2. Then g(u) = g
k
X
i=1
αibi
!
=
k
X
i=1
αig(bi)∈C,
CHAPTER 6. LINKS BETWEEN CODES AND PRIMITIVE GROUPS 67 and from this expression we deduce that g(C) ⊆ C. In fact g(C) = C, and hence g ∈Aut(C).
Note though that Lemma 6.15 is not practical when finding the automorphism group of a code with large length. More sophisticated tools would be needed in such a case. We discuss next the relationship between Aut(C) and any group G which leaves C invariant.
Lemma 6.16. [108] If C is a code, then C is invariant under the group G if and only if G≤Aut(C).
Proof: Suppose that G ≤ Aut(C), then g(C) = C for all g ∈ G which in turn implies that G·C =C. Conversely, ifG·C =C,then
{g(u)|u ∈C, g ∈G}=C ⇒ g(u)∈C for all u∈C, g ∈G
⇒ g(C) =C for all g ∈G
⇒ G≤Aut(C).
Now we define the inner product on FΩ that we will be using throughout the thesis. TheF-vector spaceFΩ is equipped with a non-degenerate symmetric bilinear form h·,·i.Let u,v∈FΩ, then h·,·i : FΩ×FΩ−→Fwhere
hu,vi=X
x∈Ω
αxβx for all u=X
x∈Ω
αxx, v=X
x∈Ω
βxx∈FΩ is an inner product on FΩ.
The following is a reformulation of Definition 4.8.
Definition 6.17. Let W be a submodule of a permutation module FΩ. The dual code of W denoted by W⊥, is the orthogonal under the given inner product, that is W⊥ ={u∈FΩ| hw,vi= 0 for all w∈W}.
Lemma 6.18. Let C be a G-invariant code, then C⊥ is also G-invariant.
Proof: For any g ∈ G and any u = P
x∈Ωαxx and any v = P
x∈Ωβxx ∈ FΩ, we
CHAPTER 6. LINKS BETWEEN CODES AND PRIMITIVE GROUPS 68 have
hg(u), g(v)i =
* g(X
x∈Ω
αxx), g(X
x∈Ω
βxx) +
=
* X
x∈Ω
αxg(x),X
x∈Ω
βxg(x) +
= X
x∈Ω
αxβx
= hu,vi.
That is, the classical inner product on FΩ is G-invariant in the following sense:
hg(u), g(v)i=hu,vi.
Now, suppose that C is G-invariant and let w ∈ C⊥, u ∈C, and g ∈ G. Since C is G-invariant it follows that hg(w), ui = hg(w), g(w0)i for some w0 ∈ C. As we have seen earlier the inner product is preserved, so hg(w), g(w0)i =hw, w0i. By definition of dual code we get hw, w0i = 0. Therefore, hg(w), ui = 0 and g(w) ∈ C⊥ for all g ∈G. ThusC⊥ isG-invariant. In addition C⊥ is an FG-submodule.
6.3.1 Codes from quotient modules
Codes obtained from permutation representations of finite groups have been given particular attention in recent years. Given a representation of group elements of a group G by permutations one can work modulo p and obtain a representation of G on a vector spaceV overFp. The invariant subspaces (the subspaces ofV taken into themselves by every group element) are then all the binary codes C for which G is a subgroup of Aut(C). Similarly we could produce codes over fields of characteristic p, where p > 2. This modulo-theoretic technique has been used in [13, 14, 83]. In [83], Knapp and Schmid consider [n, k, d]q codes where the monomial automorphism group is a particular group. The groups examined were the alternating groups An, the symmetric groups Sn and the Mathieu groups written as permutation groups of degree n and associated with codes of length n. Important information about these codes can be obtained from the theory of modular representations of groups. Using these ideas, Calderbank and Wales in [22] construct a binary [176,22,50]2 code whose
CHAPTER 6. LINKS BETWEEN CODES AND PRIMITIVE GROUPS 69 automorphism group is the Higman-Sims (HS) group. Various arguments yield the Hoffman-Singleton graph on 50 vertices, a 2-(176,50,14) design discovered by G.
Higman, and the original rank-3 construction of HS.
Brooke in [13, 14] has found all codes obtainable this way from the primitive permutation representations of the simple groups P SU4(2) and P SU3(3). In particular are examined all binary codes arising from primitive permutation representations of these groups. The simple group P SU4(2) of order 25920 has an especially rich structure. It is the simple constituent of the groups Sp4(3), U4(2), Ω−(6,2), Ω+(5,3), and ofW(E6), the Weyl group of typeE6. In [13], representations of P SU4(2) on the 27 lines of the general cubic surface, on the root system of type E6 as well as some complex 4- and 5-dimensional representations are described.
These are used to construct the five primitive permutation representations of degrees 27, 36, 40, 40 and 45. These representations lead to 6, 10, 6, 10 and 22 codes respectively (excluding the zero code and the ambient space). These codes are all inequivalent except for the repetition code h1i and its dual which appear in both representations of degree 40. The group P SU3(3) has order 6040 and has four permutation representations of degrees 28, 36, 63 and 63 leading to 4, 10, 26 and 42 codes, respectively, all of which are inequivalent except for the repetition code and its dual appearing in both degree 63 representations. A detailed description of the corresponding modular representations over the field with two elements is presented.
In each case the complete lattice of submodules is given. Irreducible modules of degrees 1, 6, 8, and 14 are involved. Further the weight distribution of subcodes (that is, submodules) with respect to the standard basis is determined.
Taking G to be a permutation group of degree n, and V the corresponding F2- permutation module. The submodules of V can be regarded as being G-invariant binary linear codes in V, and one may therefore ask for the weight distribution of these codes. In [14] a search is carried out when (G, V) corresponds to one of the four primitive permutation modules associated with the simple unitary groupG=U3(3), of order 6048. The approach is to regard G as acting 2-transitively on a certain Steiner system S(2,4,28), and then to obtain the other primitive representations of
CHAPTER 6. LINKS BETWEEN CODES AND PRIMITIVE GROUPS 70 Gin terms of the action ofU3(3) on various geometric and algebraic objects that live in S(2,4,28). Of particular interest is the description of S(2,4,28) in terms of the Cayley integers and therefore provide an explicit isomorphism between U3(3) and G02(2).
In [95] a [275,22,100]2 self-orthogonal doubly-even code left invariant by the McLaughlin simple group McL was constructed. Later in [96] a much larger code with a large error correction capability was obtained. This code has parameters [2300,22,1024]2 and is left invariant by the simple group of Conway Co2. For a collected list of references and more details on codes from permutation representations, the reader is encouraged to consult [35] and [67, Section 7.4].
6.3.2 Codes from maximal submodules
We are interested in finding all G-invariant codes from the primitive permutation representations, hence we shall consider the permutation module obtained from the action of the group on the cosets of its maximal subgroups and thus explore the corresponding FΩ-submodules (in particular maximal submodules).
Given a permutation group G on a finite set Ω and a finite fieldF it is often of considerable interest to know the structure of the permutation module FΩ (that is, the vector space over F with basis Ω considered as anFGmodule). TheG-invariant submodules of FΩ can be regarded as linear codes in FΩ, (see Lemma 6.19) and one may therefore ask for the weight distribution of these codes. In this Section and further we combine the techniques discussed in Chapter 3 and Section 6.3 and propose a novel approach in which the modular irreducibles show up as submodules and not as factor submodules, and thus determine all binary codes invariant under a given group more directly, since we obtain explicit bases for the codes. Moreover, for each primitive representation of a given permutation group G,we use Meat-Axe recursively and Magma [27] to construct the associated permutation module over F2 and subsequently a chain of its maximal submodules. Each maximal submodule constitutes in turn the binary code that is invariant under G. After eliminating
CHAPTER 6. LINKS BETWEEN CODES AND PRIMITIVE GROUPS 71 isomorphic copies, we obtain a lattice of submodules, thus in some way responding to the classification and enumeration problems alluded to earlier. This approach can be seen as dual to that used in Section 6.3.1 with the advantage that the codes are intrinsically the submodules without having the tedious and cumbersome task of defining quotients of irreducible submodules.
Let G be a finite group and H = Gα where α ∈ Ω its maximal subgroup and consider the action of G on the set of cosets Ω = (G, G/Gα) where G/Gα = {gGα|g ∈ G}. We know that G acts transitively and primitively (see Theorem 2.6 and Theorem 2.16) in a natural way by left multiplication on Ω and the image of this action is a primitive permutation representation. The FΩ-permutation module over Fq corresponding to this representation is constructed as described above. We shall consider these permutation modules which are vector spaces to construct subspaces.
The G-invariant subspaces (i.e., submodules) of the permutation module give all the p-ary codes invariant under G.The approach offered by this section, which is at the core of the purpose of the thesis, is more inclusive than that presented in sections 6.2.1 and 6.2.2. The codes constructed using those methods are in general subcodes of the ones constructed using the method that we present in the ensuing section.
Since this thesis is concerned with binary codes we restrict our attention to the field F=F2 and prove the following lemma
Lemma 6.19. [108] Let G be a finite group and Ω a finite G-set. Then the F2G- submodules of FΩ are precisely the G-invariant codes (i.e., G-invariant subspaces of FΩ).
Proof: LetGbe a finite permutation group acting on a finite set Ω in the usual way.
LetV =FΩ be theFvector space with basis the elements of Ω. Letρ : G−→GL(V) be a representation of Ggiven by
ρ(g)(x) =g(x) for all g ∈G and x∈V.
We can consider V as theF2G-module obtained fromρ.LetS be anF2G-submodule of the permutation module V. Then by Definition 6.12 of G-invariant code (see also
CHAPTER 6. LINKS BETWEEN CODES AND PRIMITIVE GROUPS 72
Definition 3.5) we have X
g∈G
αgg
!
·S ∈ S for all X
g∈G
αgg ∈F2G and S ∈ S.
In particular,
g·S ∈ S for all g ∈G and S ∈ S.
Thus, for all g ∈ G and S ∈ S we obtain ρ(g)(S) ∈ S or g(S) ∈ S and so S is G-invariant. Conversely, if S is G-invariant, then for all g ∈ G and S ∈ S we have ρ(g)(S)∈ S. Therefore for scalarsαg ∈F2 we have
X
g∈G
αgρ(g)(S)∈ S
by linearity. This implies that X
g∈G
αgg
!
·S ∈ S.