Families of nested completely regular codes and distance-regular graphs

In this paper infinite families of linear binary nested completely regular codes are constructed. They have covering radius $\rho$ equal to $3$ or $4$, and are $1/2^i$-th parts, for $i\in\{1,\ldots,u\}$ of binary (respectively, extended binary) Hamming codes of length $n=2^m-1$ (respectively, $2^m$), where $m=2u$. In the usual way, i.e., as coset graphs, infinite families of embedded distance-regular coset graphs of diameter $D$ equal to $3$ or $4$ are constructed. In some cases, the constructed codes are also completely transitive codes and the corresponding coset graphs are distance-transitive.


Introduction
Let F q denote the finite field with q ≥ 2 elements, q being a prime power.
For a vector x ∈ F n q denote by wt(x) its Hamming weight (i.e., the number of its nonzero positions). For every two vectors x = (x 1 , . . . , x n ) and y = (y 1 , . . . , y n ) from F n q denote by d(x, y) the Hamming distance between x and y (i.e., the number of positions i, where x i = y i ). We use the standard notation [n, k, d] for a binary linear code C of length n, dimension k and minimum distance d over the binary field F 2 .
The automorphism group Aut(C) of C consists of all n × n binary permutation matrices M, such that cM ∈ C for all c ∈ C. Note that the automorphism group Aut(C) coincides with the subgroup of the symmetric group S n consisting of all n! permutations of the n coordinate positions which send C into itself. Aut(C) acts in a natural way over the set of cosets of C: π(C + v) = C + π(v) for every v ∈ F n 2 and π ∈ Aut(C). For any v ∈ F n 2 its distance to the code C is d(v, C) = min x∈C {d(v, x)} and the covering radius of the code C is ρ = max v∈F n 2 {d(v, C)}. Let J = {1, 2, . . . , n} be the set of coordinate positions of vectors from F n 2 . Denote by Supp(x) the support of the vector x = (x 1 , . . . , x n ) ∈ F n 2 , i.e., Supp(x) = {j ∈ J : x j = 0}. Say that two vectors x, y ∈ F n 2 are neighbors if d(x, y) = 1 and also say that vector x covers vector y if Supp(y) ⊆ Supp(x). Definition 1.2 [14] A binary linear code C with covering radius ρ is completely transitive if Aut(C) has ρ + 1 orbits when acts on the cosets of C.
Since two cosets in the same orbit have the same weight distribution, it is clear that any completely transitive code is completely regular.
Existence and enumeration of completely regular and completely transi- tive codes are open hard problems (see [5,7,11,14,8] and references there).
The purpose of this paper is to construct nested infinite families of completely regular codes with covering radius ρ = 3 and ρ = 4. When m is growing the length of the chain of these nested codes (with constant covering radius) is also growing. For length n = 2 m − 1, where m = 2u, each family is formed by u nested completely regular codes of length n with the same covering radius ρ = 3. The last code in the nested family, so the code with the smallest cardinality is a 1/2 u -th part of a Hamming code of length n. These last codes are known to be completely regular codes due to Calderbank and Goethals [6,10]. These nested families of completely regular codes and their extended codes induces infinite families of embedded distance-regular coset graphs with diameters 3 and 4, which also give interesting families of embedded covering graphs. We point out that in some cases such completely regular codes are also completely transitive and hence the corresponding coset graphs are also distance transitive.

Preliminary results
Definition 2.1 Let C be a binary code of length n and let ρ be its covering radius. We say that C is uniformly packed in the wide sense, i.e., in the sense of [1], if there exist rational numbers β 0 , . . . , β ρ such that for any where α k (v) is the number of codewords at distance k from v.
Let C be a linear code. Denote by s the number of nonzero weights in its dual code C ⊥ . Following [7], we call s the external distance of C.
Lemma 2.2 Let C be a code with covering radius ρ and external distance s. Then: Next, following [5], we give some facts on distance-regular graphs. Let Γ be a finite connected simple graph (i.e., undirected, without loops and multiple edges). Let d(γ, δ) be the distance between two vertices γ and δ (i.e., the number of edges in the minimal path between γ and δ). The diameter D of Γ is its largest distance. Two vertices γ and δ from Γ are An automorphism of a graph Γ is a permutation π of the vertex set of Γ if any two of its vertices are adjacents.
A connected graph Γ with diameter D ≥ 3 is called antipodal if the graph Γ D is a disjoint union of cliques [5]. Such a graph is imprimitive by definition. In this case, the folded graph, or antipodal quotient of Γ is defined as the graphΓ, whose vertices are the maximal cliques (which are called fibres) of Γ D , with two adjacent if and only if there is an edge between them in Γ. If, in addition, each edge γ ∈ Γ has the same valency as its image under folding, then Γ is called an antipodal covering graph ofΓ. If, moreover, all fibres of Γ D have the same size r, then Γ is also called an antipodal r-cover ofΓ. Lemma 2.5 [5,12] Let C be a linear completely regular code with covering radius ρ and intersection array (b 0 , . . . , b ρ−1 ; c 1 , . . . c ρ ) and let Γ C be the coset graph of C. Then Γ C is distance-regular of diameter D = ρ with the same intersection array. If C is completely transitive, then Γ C is distance-transitive.
Definition 2.6 A set T of vectors v ∈ F n 2 of weight w is a t-design, denoted by T (n, w, t, λ), if for any vector z ∈ F n 2 of weight t, 1 ≤ t ≤ w, there are precisely λ vectors v i , i = 1, . . . , λ from T (n, w, t, λ), each of them covering z.
The following well known fact directly follows from the definition of completely regular code. Given a code C with minimum distance d = 2e + 1, denote by C * the extended code, i.e., the code obtained from C by adding an overall parity checking position. In [2] it has been shown when an extension of an uniformly packed code is again uniformly packed. If this happens the extended code C * has the following property.

Lemma 2.8 [2]
Let C be a uniformly packed code of length n with odd minimum distance d and let C d be a t-design T (n, d, t, λ). If the extended code C * is uniformly packed, then the set C * d+1 is a (t+1)-design T (n+1, d+1, t+1, λ).
Now we give a lemma, which is an strengthening of a result from [3].

Lemma 2.9 [3]
Let C be a completely regular linear code of length n = 2 m −1 with minimum distance d = 3, covering radius ρ = 3 and intersection array Then the extended code C * is completely regular with covering radius ρ * = 4 and intersection array (n + 1, n, b 1 , 1; 1, c 2 , n, n + 1), if and only if Proof. Let C be given by a parity check matrix H. The parity check matrix H * of the extended code C * is obtained from H by adding the zero column and then the all-one vector. From the condition w 1 + w 3 = 2 w 2 = n + 1, we conclude that the external distance s * of C * equals s * = s + 1 = 4.
To complete the proof, it is enough to compute the intersection array of Since C * has distance d * = 4 we have: Since codewords of weight 3 of C form a design T (n, 3, 1, λ) (Lemma 2.7) we have that b * 2 = b 1 = n − 1 − 2 λ (Theorem 1 in [3]). Now, we show that c * 2 = c 2 . Let x ∈ C(2). The number c 2 is the number of cases when the vector y of weight 3, at distance one from x, is covered by some codewords c ∈ C of weight 4. Consider C * and see that the vector x * = (0, x) is also in C * (2).
Since the set of codewords of weight 4 of C with zero parity check position is not changed, we conclude that, for this vector x * , we have c * 2 (x * ) = c 2 . Now, for the case when x * = (1, x) is of weight 2, we obtain the same value c * 2 (x * ) = c 2 , for the codewords of C * of weight 4 form a 2-design, i.e., the number of vectors y, at distance one from x, covering by some words from We can present the elements of F 2 m as elements in a quadratic extension Every element γ ∈ F 2 m can be presented as The matrix H m can also be written as the binary matrix of The above definition is the usual definition of determinant. For a homomorphism g : F 2 2 u −→ F 2 2 u and any two elements a and b from Take the matrix P m as the vertical join of H m and E m .
It is well known [6] that the code C (u) with parity check matrix P m is a cyclic binary completely regular code with covering radius ρ = 3, minimum distance d = 3 and dimension n − (m + u). The generator polynomial of The next lemma gives a new description for the code C (u) . From the first condition we have 0 = H m v T = i∈Iv γ i1 + γ i2 α, implying that i∈Iv γ i1 = 0 and i∈Iv γ i2 = 0. It also gives i∈Iv γ 2 i1 = 0 and i∈Iv γ 2 i2 = 0. Now consider the second one: Since r = 2 u + 1 and γ 2 u ik = γ ik for k = 1, 2, we obtain (recall that β = α r ) and, since H m v T = 0 and α + α r−1 = 0, we finally obtain is not only completely regular [6], but also completely transitive.
The above presentation implies that the general linear group GL 2 (2 u ) stabilizes C (u) .

Proposition 3.3
The automorphism group of C (u) contains the linear group Proof. Let Φ ∈ GL 2 (2 u ) and, as we said before, consider the associated permutation ϕ ∈ S n . We want to see that ϕ ∈ Aut(C (u) ). Let . Also we have i∈Iv γ 2 i1 = 0 and i∈Iv γ 2 i2 = 0. Now we have to prove that H m (ϕ(v)) T = S(ϕ(v)) = 0. We obtain The representatives in all cosets of weight 2 are vectors of weight two, which can be seen as pairs a, b, where a = (a 1 , Note that the maximum number of independent syndromes we can take is u, so the biggest code we can obtain is of dimension u + dim(C (u) ) = n − m, which is the Hamming code C (0) = H m . All the constructed codes contains C (u) and, at the same time, they are contained in the Hamming code C (0) .
The number of codes C (u−i) equals the number of subspaces of dimension i we can take in F u 2 , so the Gaussian binomial coefficient Taking all the possibilities, we are able to construct several nested families of codes between C (u) and C (0) = H m . In fact, it is easy to compute that there are All these codes C (i) are completely regular as we show later in Theorem 3.8. We have seen that C (u) and C (0) are completely transitive and, in addition we show that also C (1) is also a completely transitive code. Note that for m = 6 (so u = 3), all codes C (i) in the chain are completely transitive. Thus, the conjecture is true for this case.
Finally, we can prove that all codes C (i) are completely regular.
Proof. From the construction of codes C (i) we know that the codewords v of weight three are those such that H m v T = 0 and S(v) belongs to a fixed subspace A u−i ⊂ F u 2 of dimension u−i over F 2 . Hence, taking a fixed nonzero element γ = γ 1 + γ 2 α ∈ F 2 m every codeword of weight three covering this element γ is defined giving be its support. Then, since H m v T = 0 we have γ ′′ = γ + γ ′ and so Now we want to count how many codewords of weight three cover a fixed nonzero element γ ∈ F 2 m . We begin by counting how many γ ′ ∈ F 2 m gives  (1). Without loss of generality, we assume that x has weight one. Therefore, x has n − 1 neighbors of weight two.
Since ρ(C (i) ) = 3, we conclude that s(C (i) ) = 3 for any i ∈ {0, . . . , u}. As it was shown in [6], the dual code of C (u) has the following values in the weight spectrum: But any code C (i) contains the code C (u) as a subcode, implying that the dual (C (i) ) ⊥ is contained in (C (u) ) ⊥ . This, in turn, implies that any such code (C (i) ) ⊥ has the same weight spectrum as the code (C (u) ) ⊥ . Now the result follows from Lemma 2.9. ✷ The next theorem shows that the extended codes C (i) * are not only completely regular, but completely transitive.  Code C (i) * is the extension of C (i) by an overall parity check coordinate, which we assume is the 0th coordinate. Codewords in C (i) * have n = 2 m components and we can associate, at random and once for all, the coordinate ith with a vector w i ∈ F m 2 . Any vector w ∈ F m 2 define a permutation π w : {1, . . . , n} −→ {1, . . . , n} such that π w (i) = j, where w j = w + w i . Let T = {π w : w ∈ F m 2 } the set of all these permutations and note that T has a group structure isomorphic to the additive structure F m 2 . For each w ∈ F m 2 , As all codewords in C (i) * have even weight it is clear that T is a subgroup of Aut(C (i) * ). Indeed, let a = (a 0 , . . . , a n ) ∈ C (i) * , this means that a has an even number of nonzero components ( n i=0 a i = 0); Hence, π w (a) ∈ C (i) * .
Furthermore, T is a normal subgroup in Aut(C (i) * ). Indeed, for any φ ∈ Aut(C (i) * ) we have that φπ w φ −1 is again a translation π z , where z = φ(w).
For any φ ∈ Aut(C (i) * ), it is clear that we can find φ ′ ∈ Aut(C (i) ) fixing the extended coordinate and a vector y ∈ F m 2 , such that φ = φ ′ π y . Therefore, we have Aut(C (i) * )/T ∼ = Aut(C) and so Aut(C (i) * ) is the semidirect product of F m 2 and Aut(C (i) ) (obviously, we can identify T with F m 2 ). The first statement is proven.
completely transitive we show that all cosets of C (i) * in F 2 m 2 with the same minimum weight are in the same orbit by the action of Aut(C (i) * ).
The number of cosets of C (i) * is twice the cosets of C (i) . If C (i) + v is a coset of C (i) , where v is a representative vector of minimum weight then C (i) * + (0|v) and C (i) * + (1|v) are cosets of C (i) * . The cosets of C (i) * of weight 4 are of the form C (i) * + (1|v), where C (i) + v is a coset of weight 3 of C (i) . Since the cosets of weight 3 of C (i) are in the same Aut(C (i) )-orbit and Aut(C (i) ) ⊂ Aut(C (i) * ), it follows that all the cosets of weight 4 of C (i) * are in the same Aut(C (i) * )-orbit. Now consider the cosets of C (i) * of weight r ∈ {1, 2, 3}. They are of the form C (i) * + (0|v), where C (i) + v is a coset of weight r of C (i) and of the form Cosets of the same minimum weight in C (i) can be moved among them by Aut(C (i) ) and so, we need only to show that there exists an automorphism in Aut(C (i) * ) moving C (i) * + (0|v) to C (i) * + (1|v ′ ), where v, v ′ are at distance r and r − 1 from C (i) , respectively. Without loss of generality, we further assume that Supp(v ′ ) ⊂ Supp(v) and so, Supp(v) = Supp(v ′ ) ∪ {j}, for some index j ∈ {1, . . . , 2 m }. The automorphism π w j moves C (i) * + (0|v) to C (i) * + (1|v ′′ ), where Supp(v ′′ ) = {k : w k = w j + w s ; s ∈ Supp(v ′ )} and, finally, by using an automorphism from Aut(C (i) ) we can move from Nested antipodal distance-regular graphs and distance-transitive graphs of diameter 3 and 4 Denote by Γ (i) (respectively, Γ (i) * ) the coset graph, obtained from the code Since all cosets of weight 3 (respectively, of weight 4) of the Hamming code H m (respectively, of the extended Hamming code H * m ) belong to this code, we conclude that all graphs Γ (i) (respectively, Γ (i) * ) are antipodal. This means that for i > 0 all graphs Γ (i) and Γ (i) * are imprimitive.
We need the following statement from [9]. Then Γ is a r-fold covering graph of K n , for some r and n and recall that c 2 is the number of common neighbors of two vertices in Γ at distance two.
As a direct result of Lemma 4.1 and Theorem 3.8 we obtain the following new distance-regular and distance-transitive coset graphs. • Γ (i) is a subgraph of Γ (i+1) for all i = 0, 1, . . . , u − 1.
As for the codes that give rise to this graphs, we conjecture that the graphs Γ (i) are distance-transitive for i ∈ {2, . . . , u − 1} and 2 i ≤ u + 1.
Graphs Γ (u) and Γ (u) * are also known. The corresponding codes C (u) and C (u) * have been constructed by Kasami [10] and have been presented in a very symmetric form by Calderbank and Goethals [6]. They proved that these codes form an association scheme [7], which immediately implies the existence of the corresponding distance-regular graphs Γ (u) and Γ (u) * (Ch. 11 in [5]).
All graphs Γ (i) for i = 0, 1, . . . , u have been constructed by Godsil and Hensel using the Quotient Construction [9]. But it was not mentioned in all references above that some of these graphs are completely transitive. Besides, except for the graphs Γ (u) , it was not stated that these graphs can be constructed as coset graphs.
The graphs Γ (i) * for i = 2, . . . , u − 1 seems to be new; we could not find graphs with these parameters in the above mentioned literature.