$2$-Neighbour-Transitive Codes with Small Blocks of Imprimitivity

A code $C$ in the Hamming graph $\varGamma=H(m,q)$ is $2\it{\text{-neighbour-transitive}}$ if ${\rm Aut}(C)$ acts transitively on each of $C=C_0$, $C_1$ and $C_2$, the first three parts of the distance partition of $V\varGamma$ with respect to $C$. Previous classifications of families of $2$-neighbour-transitive codes leave only those with an affine action on the alphabet to be investigated. Here, $2$-neighbour-transitive codes with minimum distance at least $5$ and that contain"small"subcodes as blocks of imprimitivity are classified. When considering codes with minimum distance at least $5$, completely transitive codes are a proper subclass of $2$-neighbour-transitive codes. Thus, as a corollary of the main result, completely transitive codes satisfying the above conditions are also classified.


Introduction
Classifying classes of codes is an important task in error correcting coding theory. The parameters of perfect codes over prime power alphabets have been classified; see [31] or [34]. In contrast, for the classes of completely regular and s-regular codes, introduced by Delsarte [11] as a generalisation of perfect codes, similar classification results have only been achieved for certain subclasses. Recent results include [3,4,5,6]. For a survey of results on completely regular codes see [7]. Classifying families of 2-neighbour transitive codes has been the subject of [15,16].
A subset C of the vertex set V Γ of the Hamming graph Γ = H(m, q) is called a code, the elements of C are called codewords, and the subset C i of V Γ consisting of all vertices of H(m, q) having nearest codeword at Hamming distance i is called the set of i-neighbours of C. The definition of a completely regular code C involves certain combinatorial regularity conditions on the distance partition {C, C 1 , . . . , C ρ } of C, where ρ is the covering radius. The current paper concerns the algebraic analogues, defined directly below, of the classes of completely regular and s-regular codes. Note that the group Aut(C) is the setwise stabiliser of C in the full automorphism group of H(m, q); precise definitions of notations are available in Section 2.
A variant of the above concept of complete transitivity was introduced for linear codes by Solé [29], with the above definition first appearing in [23]. Note that non-linear completely transitive codes do indeed exist; see [21]. Completely transitive codes form a subfamily of completely regular codes, and s-neighbour-transitive codes are a sub-family of s-regular codes, for each s. It is hoped that studying 2-neighbour-transitive codes will lead to a better understanding of completely transitive and completely regular codes. Indeed a classification of 2-neighbour-transitive codes would have as a corollary a classification of completely transitive codes.
Completely transitive codes have been studied in [6,13], for instance. Neighbourtransitive codes are investigated in [17,19,20]. The class of 2-neighbour-transitive codes is the subject of [15,16], and the present work comprises part of the second author's PhD thesis [24]. Recently, codes with 2-transitive actions on the coordinate entries of vertices in the Hamming graph have been used to construct families of codes that achieve capacity on erasure channels [26], and many 2-neighbour-transitive codes indeed admit such an action; see Proposition 6.
Each vertex of H(m, q) is of the form α = (α 1 , . . . , α m ), where the entries α i come from an alphabet Q of size q. A typical automorphism of H(m, q) is a composition of two automorphisms, each of a special type. An automorphism of the first type corresponds to an m-tuple (h 1 , . . . , h m ) of permutations of Q and maps a vertex α to (α h 1 1 , . . . , α hm m ). An automorphism of the second type corresponds to a permutation of the set M = {1, . . . , m} of subscripts and simply permutes the entries of vertices, for example the map corresponding to the permutation (123) of M maps α = (α 1 , α 2 , α 3 , α 4 , . . . , α m ) to (α 3 , α 1 , α 2 , α 4 , . . . , α m ). (More details are given in Section 2.1.) It is sometimes helpful to distinguish between entries in the different positions, and we will refer to the set of entries occurring in position i as Q i . Then maps of the second type may be viewed as permuting the subsets Q 1 , . . . , Q m between themselves in the same way that they permute M .
For a subgroup X of automorphisms of H(m, q), the subgroup K of X consisting of all elements which fix each of the Q i setwise is a normal subgroup of X and is the kernel of the action X induces on M . Also, for each i ∈ M , the set of elements of X which fix Q i setwise forms a subgroup X i of X, and X i induces a subgroup of permutations of Q i , which we denote by X Q i i . (Again, more details are given in Section 2.1.) It was shown in [15] that the family of (X, 2)-neighbour-transitive codes can be subdivided into three disjoint sub-families, according properties of the group X (see Proposition 7). We now introduce these sub-families in Definition 2.
Definition 2. Let C be a code in H(m, q) and let X Aut(C). Moreover, let Q i , X i and K be as described above. Then C is 1. X-entry-faithful if X acts faithfully on M , that is, K = 1, 2. X-alphabet-almost-simple if K = 1, X acts transitively on M , and X Q i i is a 2transitive almost-simple group, and, 3. X-alphabet-affine if K = 1, X acts transitively on M , and X Q i i is a 2-transitive affine group.
Those (X, 2)-neighbour transitive codes that are also X-entry-faithful and have minimum distance at least 5 are classified in [15]; while those that are X-alphabet-almostsimple and have minimum distance at least 3 are classified in [16]. Hence, in this paper we study X-alphabet-affine codes. For such graphs q = p d for some prime p and positive integer d, and we identify the vertex set of the Hamming graph H(m, q) with the (dm)dimensional vector space V = F dm p . For a nontrivial subspace W of V we denote by T W the group of translations by elements of W ; recall for a subgroup X of Aut(H(m, q)) (the automorphism group of H(m, q)), we denote by K the kernel of the action of the group X on M ; and we note that K = X ∩ B where B ∼ = S m q is the base group of Aut(H(m, q)); see Section 2.
Definition 3. Let q = p d , V = F dm p be as above and let W be a non-trivial F p -subspace of V . An (X, 2)-neighbour-transitive extension of W is an (X, 2)-neighbour-transitive code C containing 0 such that T W X, where T W is the group of translations by elements of W , and W is fixed setwise by K = X ∩ B. Note that T W X and 0 ∈ C means that W ⊆ C. If C = W then the extension is said to be non-trivial.
The main result of this paper, below, classifies all (X, 2)-neighbour-transitive extensions of W , supposing W is a k-dimensional F p -subspace of V , where k d. Theorem 4 may be seen as a sequel of [15,Theorem 1.1] where, rather than assuming that the kernel K of the action of X on M is trivial, the condition k d limits the size of W , implicitly restricting the possibilities for K. The motivation for assuming k d comes largely from [22,Problem 6.5.4], which proposes investigating hypotheses similar to those of Theorem 4, but in the context of X-completely transitive codes in H(m, 2); see also Corollary 5. Corollary 5. Let C be an X-completely transitive code in H(m, 2) with minimum distance δ 5 such that K = X ∩ B = Diag m (S 2 ). Then C is equivalent to one of the codes appearing in Theorem 4, each of which is indeed completely transitive. Section 2 introduces the notation used throughout the paper and Section 3 proves the main results.

Notation and preliminaries
Let the set of coordinate entries M and the alphabet Q be sets of sizes m and q, respectively, both m and q integers at least 2. The vertex set V Γ of the Hamming graph Γ = H(m, q) consists of all functions from the set M to the set Q, usually expressed as m-tuples. Let Q i ∼ = Q be the copy of the alphabet in the entry i ∈ M so that the vertex set of H(m, q) is identified with the Cartesian product An edge exists between two vertices if and only if they differ as m-tuples in exactly one entry. Note that S × will denote the set S \ {0} for any set S containing 0. In particular, Q will usually be a vector-space here, and hence contains the zero vector. A code C is a subset of V Γ . If α is a vertex of H(m, q) and i ∈ M then α i refers to the value of α in the i-th entry, that is, α i ∈ Q i , so that α = (α 1 , . . . , α m ) when M = {1, . . . , m}. For more in depth background material on coding theory see [10] or [28].
Let α, β be vertices and C be a code in a Hamming graph H(m, q) with 0 ∈ Q a distinguished element of the alphabet. A summary of important notation regarding codes in Hamming graphs is contained in Table 1.
Note that if the minimum distance δ of a code C satisfies δ 2s, then the set of s-neighbours C s satisfies C s = ∪ α∈C Γ s (α) and if δ 2s + 1 this is a disjoint union. This fact is crucial in many of the proofs below; it is often assumed that δ 5, in which case every element of C 2 is distance 2 from a unique codeword.
A linear code is a code C in H(m, q) with alphabet Q = F q a finite field, so that the vertices of H(m, q) form a vector space V , such that C is an F q -subspace of V . Given α, β ∈ V , the usual inner product is given by α, β = i∈M α i β i . The dual code of C is The Singleton bound (see [11, 4.3.2]) is a well known bound for the size of a code C in H(m, q) with minimum distance δ, stating that |C| q m−δ+1 . For a linear code C this may be stated as A vertex or an entire code from a Hamming graph H(m, q) may be projected into a smaller Hamming graph H(k, q). For a subset J = {j 1 , . . . , j k } ⊆ M the projection of α, with respect to J, is π J (α) = (α j 1 , . . . , α j k ). For a code C the projection of C, with respect to J, is π J (C) = {π J (α) | α ∈ C}.

Automorphisms of a Hamming graph
The automorphism group Aut(Γ ) of the Hamming graph is the semi-direct product B L, . Note that B and L are called the base group and the top group, respectively, of Aut(Γ ). Since we identify Q i with Q, we also identify Sym Then h and σ act on α explicitly via: For reasons of readability we often write the image α σ as (α 1σ −1 , . . . , α mσ −1 ). The automorphism group of a code C in Γ = H(m, q) is Aut(C) = Aut(Γ ) C , the setwise stabiliser of C in Aut(Γ ). Let G be a group acting on a set Ω, ω ∈ Ω and S ⊆ Ω. Then, 1. G ω denotes the subgroup of G stabilising ω, (For more background and notation on permutation groups see, for instance, [12].) In particular, let X Aut(Γ ). Then: 2. K = K ∩ B is the kernel X (M ) of the action of X on entries and is precisely the subgroup of X fixing M point-wise.
3. If i ∈ M , then X i denotes the stabiliser of the entry i and any x ∈ X i is of the form hσ (h ∈ B and σ ∈ L) where σ fixes i ∈ M . So x = hσ induces the permutation h i ∈ Sym(Q i ) on the alphabet Q i . This defines a homomorphism from X i to Sym(Q i ) and we denote the image of this homomorphism by X Q i i . We refer to X Q i i as the action on the alphabet. It is worth mentioning that coding theorists often consider more restricted groups of automorphisms, such as the group PermAut

For any
The elements of this group are called pure permutations on the entries of the code.
Two codes C and C in H(m, q) are said to be equivalent if there exists some x ∈ Aut(Γ ) such that C x = {α x | α ∈ C} = C . Equivalence preserves many of the important properties in coding theory, such as minimum distance and covering radius, since Aut(Γ ) preserves distances in H(m, q).

s-Neighbour-transitive codes
This section presents preliminary results regarding (X, s)-neighbour-transitive codes, defined in Definition 1. The next results give certain 2-homogeneous and 2-transitive actions associated with an (X, 2)-neighbour-transitive code. Proposition 6. [15, Proposition 2.5] Let C be an (X, s)-neighbour-transitive code in H(m, q) with minimum distance δ, where δ 3 and s 1. Then for α ∈ C and i min{s, δ−1 2 }, the stabiliser X α fixes setwise and acts transitively on Γ i (α). In particular, the action of X α on M is i-homogeneous. Lemma 8. Let C be an (X, 2)-neighbour-transitive code in H(m, q) with δ 5 and 0 ∈ C, and let i, j ∈ M be distinct. Then the following hold: 1. The stabiliser X 0,i,j acts transitively on each of the sets Q × i and Q × j .
2. Moreover, X 0,i,j has at most two orbits on Q × i × Q × j , and if X 0,i,j has two orbits on Q × i × Q × j then both orbits are the same size and X 0 acts 2-transitively on M .
3. The order of X 0 , and hence |X|, is divisible by m 2 (q − 1) 2 . 4. If |X 0 | = m 2 then q = 2. Proof. Now X 0 acts transitively on Γ 2 (0), by Proposition 6, since δ 5. Since |Γ 2 (0)| = m 2 (q − 1) 2 , parts 3 and 4 hold. Also, we have that the stabiliser X 0,{i,j} of the subset {i, j} ⊆ M is transitive on the set of weight 2 vertices with support {i, j}. Hence X 0,i,j has at most two orbits on Q × i × Q × j and if there are two they have equal size. Note that if X 0,i,j has one orbit on Q × i × Q × j then X 0,i,j acts transitively on each of Q × i and Q × j . Suppose that X 0,i,j has two orbits on Q × i × Q × j , and hence that X 0,i,j = X 0,{i,j} . By Proposition 6, X 0 acts 2-homogeneously on M . Since X 0,i,j = X 0,{i,j} , we have that X 0 is in fact 2-transitive on M , proving part 2. Let k be the number of X 0,i,j -orbits on and hence X 0,i,j has the same number of orbits on each of Q × i and Q × j . Since each orbit of X 0,i,j on Q × i × Q × j is contained in the Cartesian product of an orbit on Q × i with an orbit on Q × j , it follows that X 0,i,j has at least k 2 orbits on Q × i × Q × j . However, k 2 implies k 2 4, contradicting part 2, and hence part 1 holds.
The concept of a design, introduced below, comes up frequently in coding theory. Let α ∈ H(m, q) and 0 ∈ Q. A vertex ν of H(m, q) is said to be covered by α if ν i = α i for every i ∈ M such that ν i = 0. When q = 2, that is, in the case of a binary design, then each vertex of D can be interpreted as the characteristic vector of a k-element subset of {1, . . . , m}. Thus the set D can be thought of as a collection of k-element subsets of {1, . . . , m}, called blocks, and the "covering condition" becomes the condition that each s-element subset of {1, . . . , m} is contained in exactly λ blocks. With this interpretation, these structures are usually called combinatorial designs.
The following equations can be found, for instance, in [30]. Let D be a combinatorial s-(m, k, λ) design with |D| = b blocks and let r be the number of blocks containing any given point. Then mr = bk, r(k − 1) = λ(m − 1) and the electronic journal of combinatorics 27(1) (2020), #P1.42 The definition below is required in order to state the remaining two results of this section.
Definition 10. Let C be a code in H(m, q) with covering radius ρ, and s be an integer with 0 s ρ. Then, 1. C is s-regular if, for each i ∈ {0, 1, . . . , s}, each k ∈ {0, 1, . . . , m}, and every vertex ν ∈ C i , the number |Γ k (ν) ∩ C| depends only on i and k, and, 2. C is completely regular if C is ρ-regular. 1. δ = 11 and C is equivalent to the binary repetition code, 2. δ = 5 and C is equivalent to the punctured Hadamard code P, or 3. δ = 6 and C is equivalent to the even weight subcode E of P.

Extensions of the binary repetition code
In this section it will be shown that the hypotheses of Theorem 4 imply that W is the binary repetition code in H(m, q). From there, all (X, 2)-neighbour-transitive extensions of the binary repetition code are classified. First, a more general result regarding (X, 2)neighbour-transitive codes. Note that a system of imprimitivity for the action of a group G on a set Ω is a non-trivial partition of Ω preserved by G, and a part of the partition is called a block of imprimitivity.
Lemma 14. Suppose C is an (X, 2)-neighbour-transitive code with δ 5 and that ∆ is a block of imprimitivity for the action of X on C. Then ∆ is an (X ∆ , 2)-neighbour-transitive code with minimum distance δ ∆ 5.
Corollary 15. Let C be an (X, 2)-neighbour-transitive extension of W such that C has minimum distance δ 5. Then W is a block of imprimitivity for the action of X on C and W is (X W , 2)-neighbour-transitive with minimum distance δ W 5.
Proof. Now, K = K W is normal in X and T W K W is transitive on W from which it follows that W is an orbit of K on C and hence, by [12, Theorem 1.6A (i)], is a block of imprimitivity for the action of X on C. Thus, the result is implied by Lemma 14.
The next result shows that the binary repetition code is the only 2-neighbour-transitive code which is a k-dimensional F p -subspace of V = F dm p , identified with the vertex set of H(m, p d ), such that 1 k d.
Lemma 16. Let q = p d and V = F dm p be the vertex set of the Hamming graph H(m, q) and let W be a k-dimensional F p -subspace of V , with 1 k d, such that W is an (X, 2)-neighbour-transitive code with minimum distance δ 5. Then q = 2 and W is the binary repetition code in H(m, 2).
Proof. We claim that δ = m. As any (X, 2)-neighbour transitive code is also 2-regular, by Lemma 11, and 0 ∈ W , proving the claim implies the result, by [15,Lemma 2.15]. Suppose for a contradiction that δ < m. It follows that there exists a weight δ codeword α ∈ W and distinct i, j ∈ M such that α i = 0 and α j = 0. Now, X 0,i,j acts transitively on Q × j , by Lemma 8, so that for all non-zero a ∈ F d p there exists some x a ∈ X 0,i,j such that α xa ∈ W with (α xa ) j = a. As a ranges over all non-zero a ∈ F d p this gives p d − 1 distinct codewords. Since |W | = p k p d , and 0 ∈ W , it follows that |W | = p d and k = d. Note that since α i = 0 and x a ∈ X 0,i,j this implies that every element of W has i-th entry 0. By Proposition 6, X 0 is, in particular, transitive on M . Hence, there exists some y = hσ ∈ X 0 , with h ∈ B and σ ∈ L, such that j σ = i. Thus α y ∈ W with (α y ) i = 0. This gives a contradiction, proving the claim that δ = m.
Lemma 16 implies part 1 of Theorem 4 and also that, given the hypotheses of Theorem 4, it can be assumed that q = 2 and W is the repetition code in H(m, 2).
Proof. Let W be the repetition code in H(m, 2). If x = hσ ∈ X 0 , with h ∈ B and σ ∈ L, then q = 2 implies h i = 1 for all i ∈ M . Thus X 0 ∼ = X M 0 . By Corollary 15, W is a block of imprimitivity for the action of X on C, from which it follows that X W = T W X 0 , since T W acts transitively on W . Thus, X 0 ∼ = X M 0 = X M W and K = T W . Proof. First, note that ω ∈ W if and only if ω i = ω j for all i, j ∈ M . Since C = W there exists a codeword α ∈ C \ W and distinct i, j ∈ M such that α i = 0 and α j = 1, since otherwise α ∈ W . Note that this implies that δ = m. Let J = {i, j} ⊆ M and consider the projection code P = π J (C). Now, π J (W ) = {(0, 0), (1, 1)} ⊆ P and π J (α) = (0, 1) ∈ P . Also, β = α + (1, . . . , 1) ∈ C, since T W X, which implies π J (β) = (1, 0) ∈ P . Thus, P is the complete code in the Hamming graph H(2, 2). By [15,Corollary 2.6], X {i,j} acts transitively on C, from which it follows that X P {i,j} acts transitively on P . Thus |P | = 4 divides |X P {i,j} | and hence also divides |X|. By Lemma 17, K = T W so that |K| = 2. Thus 2 divides |X/K|. Proposition 6 and [12, Exercise 2.1.11] then imply that X/K = X M is 2-transitive.
By Corollary 15, W is (X W , 2)-neighbour-transitive. Thus, by Proposition 6, X M W is 2-homogeneous on M . Suppose X M W is 2-transitive on M . Since X P W,{i,j} contains K and interchanges i and j, |X P W,{i,j} | is divisible by 4. Now, |X P {i,j} | 8, since Aut(H(2, 2)) = (S 2 × S 2 ) S 2 . Furthermore, |X P {i,j} : X P W,{i,j} | = 2, since X P {i,j} acts transitively on P . Thus X P {i,j} = (S 2 × S 2 ) S 2 , and so |X P i,j | = 4. Let H be the kernel of the action of X i,j on P . Since the only non-identity element of K = T W acts non-trivially on P , we deduce that |K P | = 2 and H ∩ K = 1. Hence, Therefore, X M i,j has a quotient of size 2, since |X P i,j /K P | = 2, and thus H M is a normal subgroup of X M i,j of index 2.
the electronic journal of combinatorics 27(1) (2020), #P1.42 The socle of a finite group is the product of all its minimal normal subgroups. If C is an (X, 2)-neighbour-transitive extension of the binary repetition code W in H(m, 2) then the next two results show that the socles of X M and X M W cannot be equal and that the socle of X M cannot be A m .
Lemma 19. Let W be the repetition code in H(m, 2) and C be a non-trivial (X, 2)neighbour-transitive extension of W with δ 5. Then soc(X/K) = soc(X W /K).

Proof.
Let H X such that K < H and H/K = soc(X/K). Note that this implies that H X. By Lemma 17, X W = K X 0 . Suppose H/K = soc(X W /K), and note that by Lemma     The main theorem can now be proved.
Proof of Theorem 4. Suppose C is an (X, 2)-neighbour-transitive extension of W with δ 5, where W is a k-dimensional F p -subspace of V = F dm p and 1 k d. By Lemma 16, W is the binary repetition code (not just an equivalent copy of it, since 0 ∈ W ) and thus q = 2. If C = W then C is a trivial extension of W and outcome 1 holds. Suppose the extension is non-trivial. Then, by Lemma Table 2. Now T W X implies that if there exists some weight k codeword in C, then there is also a weight m − k codeword. Thus δ m/2 and δ 5 implies m 10. In particular, X M = PSL 3 (2) or AGL 3 (2). Suppose X M ∼ = PSL 2 (11) and m = 11. Then δ = 5 and, by Proposition 13, C is either the punctured Hadamard code P or the even weight subcode E of the punctured Hadamard code. The even weight subcode of the punctured Hadamard code is not invariant under T W , so C = E. Moreover, as in the proof of [15,Proposition 4.3], the only copy of PSL 2 (11) in Aut(P) fixes 0, and hence X M  (11). Then, by Proposition 13, C is either the punctured Hadamard code P or the even weight subcode of P. The even weight subcode of P is not invariant under T W , so C = P. The automorphism group of P is X = Aut(P) ∼ = 2 × M 11 with X 0 ∼ = PSL 2 (11) and K = T W . By Finally, the proof of Corollary 5 is given below.
Proof of Corollary 5. Suppose C is X-completely transitive with minimum distance δ 5 such that K = Diag m (S 2 ), and assume that 0 ∈ C. The fact that δ 5 implies that C 2 is non-empty and thus C is (X, 2)-neighbour-transitive. Since K X and X acts transitively on C, it follows from Lemma 14 that the orbit ∆ = 0 K of 0 under K is an (X ∆ , 2)-neighbour-transitive code. Since K = Diag m (S 2 ) we have that |∆| = 2 and ∆ has minimum distance m. Thus, since any 2-neighbour-transitive code is 2-regular, [15, the electronic journal of combinatorics 27(1) (2020), #P1.42 Lemma 2.15] implies that ∆ is the binary repetition code in H(m, 2). Hence, q = 2, Q ∼ = Z 2 and C satisfies the hypotheses of Theorem 4, and so is one of the codes listed there. The binary repetition code has automorphism group Diag m (S 2 ) Sym(M ) and is seen to be completely transitive by identifying the vertices of H(m, 2) with the subsets of M . By [18,Theorem 1.1], the Hadamard code of length 12 and its punctured code are completely transitive. This completes the proof.