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On completely regular codes with minimum eigenvalue in geometric graphs

We prove that any completely regular code with minimum eigenvalue in any geometric graph G corresponds to a completely regular code in the clique graph of G. Studying the interrelation of these codes, a complete characterization of the completely regular codes in the Johnson graphs J(n,w) with covering radius w-1 and strength 1 is obtained. In particular this result finishes a characterization of the completely regular codes in the Johnson graphs J(n,3). We also classify the completely regular codes of strength 1 in the Johnson graphs J(n,4) with only one case for the eigenvalues left open.


Introduction
The geometric graphs are the distance-regular graphs, whose edges are parted into maximum cliques attaining Delsarte upper bound [12]. The Delsarte cliques are known to be completely regular codes in distance-regular graph. Examples of geometric graphs include such classical distance-regular graphs as: Hamming, Johnson, Grassmann, dual-polar graphs, etc. [5].
In recent papers [20], [9], [10], using various techniques, the classification problem for completely regular codes in Hamming, Johnson and Grassmann graphs is reduced to the classification of codes or combinatorial structures in simplier graphs. In a sense, this paper continues this trend as we reformulate the existence problem for the completely regular codes with minimum eigenvalue in any geometric graph as that for completely regular codes in its clique graph.
For a survey on completely regular codes in Johnson graphs we refer to [4]. In Johnson graphs, completely regular codes are tightly interrelated with important classes of combinatorial t-designs such as Steiner triple and quadruple systems [16], [1], [19]. We refer to maximum t such that a collection D of fixed size blocks is a t-design as the strength of D. The completely regular designs in J(n, w) were classified in the following cases, see [15], [18], [9], [22]: • strength 0 • strength 1 and minimum distance at least 2 • strength 1 and covering radius 1.
Definitions and a review of the previous results are given in Section 2. A general approach is developed in Section 3. We show that any completely regular code C with minimum eigenvalue in a geometric graph Γ implies that the set of maximum cliques having at least one vertex from C is a completely regular code in the clique graph of Γ. The proof is based on the fact that any Delsarte clique is a completely regular code in a distance-regular graph and the minimum eigenvalue of the distance-regular graph is not its eigenvalue (see [14,Proposition 2] and [5,Proposition 4.4.6]). The approach is implemented in Section 4, where we give a characterization of the completely regular codes in the Johnson graphs J(n, w) with covering radius w − 1 and strength 1. This, combined with the results of work [15] of Meyerowitz, finish the characterization of completely regular codes of covering radius w − 1 and any strength. We show that these codes for w = 3 are necessarily a part of the series from the work [13] of Praeger and Godsil (see also [18]), and there are no such codes for w ≥ 4. This result in combination with [15], [9], [11], [7] closes the classification problem of the completely regular codes in J(n, 3). The completely regular codes of strength 2 coincide with simple 2 − (n, 3, λ) designs [16]. Since these important combinatorial objects include Steiner triple systems, their complete classification up to isomorphism is far from being solved. However, we have their classification up to parameter λ, see work [7] of Dehon. The classification results for completely regular codes in the Johnson graphs are obtained following the general idea for the geometric graphs. The Johnson graph J(n, w − 1) can be viewed as the clique graph of the geometric graph J(n, w). From results of Section 3, any completely regular code in the Johnson graph J(n, w) with covering radius w − 1 and strength 1 naturally corresponds to a completely regular code in J(n, w − 1) with covering radius w − 2 and strength 1. The incidence relations between these codes impose constraints on their structure. In particular, once the code in J(n, w − 1) is known, one can reconstruct the corresponding code in J(n, w).
We also show that all completely regular codes in J(n, 4) of strength 1 are either part of known series [18], [13] or their eigenvalues are θ 0 (n, 4), θ 2 (n, 4), θ 3 (n, 4). In the latter case using several local and global arguments we were able to find significant restrictions on the intersection arrays of such codes.

Definitions and basic theory
2.1. Geometric graphs and their clique graphs. A number θ is called an eigenvalue of a graph Γ if there is a nonzero vector v indexed by the vertices of Γ such that where A is the adjacency matrix of Γ. We call such vector v a θ-eigenvector of Γ.
A regular graph of diameter d is called distance-regular if there is an array of integers {β 0 , . . . , β d−1 ; γ 1 , . . . , γ d }, such that for any vertices x and y at distance i, i ∈ {0, . . . , d} there are exactly β i neighbors of y at distance i + 1 from x and γ i neighbors of y at distance i − 1 from x. The array of integers {β 0 , . . . , β d−1 ; γ 1 , . . . , γ d } is called the intersection array of the distance-regular graph Γ. It is well known that any distance-regular graph of diameter d has exactly d + 1 distinct eigenvalues. We index these eigenvalues in descending order throughout the paper: where k is the valency of Γ. If K is a clique in a distance-regular graph Γ of valency k and diameter d, then [5,Proposition 4.4] The bound was established by Delsarte [8]. A clique that attains the above upper bound is called a Delsarte clique of Γ. Any Delsarte clique is a completely regular code in Γ [5,Proposition 4.4.6]. If there is a set K of Delsarte cliques in a distance-regular graph such that any edge of the graph is in a unique clique of K, the graph is called geometric with respect to K [12]. Note that there can be many ways to choose the set K. A generalization of the concepts of Delsarte clique graph and geometric graph to non-distance-regular case is known as completely regular clique graphs [21].
Let Γ be a geometric graph with respect to a set of Delsarte cliques K. The clique graph of Γ with respect to K is the graph Γ ′ such that The vertices of the Johnson graph J(n, w) are all w-subsets of {1, . . . , n}, where two w-subsets are adjacent if they meet in a (w − 1)-subset. Since the Johnson graphs J(n, w) and J(n, n − w) are isomorphic, throughout the text we assume that w ≤ n/2. The Johnson graph J(n, w), w ≤ n/2 is distance-regular of diameter w and its eigenvalues are Example 1. Given a (w − 1)-subset y of {1, . . . , n}, one can consider the collection of all w-subsets of {1, . . . , n} containing y, which is a Delsarte clique in the Johnson graph J(n, w) [5]. Moreover, the graph J(n, w) is geometric with respect to the set K of these cliques for all (w − 1)-subsets y of {1, . . . , n}. The clique graph of J(n, w) can be represented as the graph with (w − 1)-subsets of {1, . . . , n} as vertices with edges being subsets meeting in a (w − 2)-subset, i.e. it is actually the Johnson graph J(n, w − 1). A vertex x of J(n, w) is in the clique y of K if and only if the vertex x (as w-subset) contains the vertex y (as (w − 1)-subset). So we see that the incidence of the vertices of the geometric graph J(n, w) and its clique graph J(n, w − 1) is the ordinary subset inclusion relation. We also note that in a similar manner the Grassmann graph J q (n, w) can be viewed as a geometric graph whose clique graph is J q (n, w − 1).
For a vector u indexed by the vertices of the graph Γ, we define the vector I(u) indexed by the cliques from a set K such that for K ∈ K we have (I(u)) K = x∈K u x . This mapping a particular case of the following function.
For a bipartite graph Γ and a vector u indexed by the vertices of one part, define (I(u)) y = Corollary 1. Let Γ be a geometric graph with respect to a set of Delsarte cliques K, Γ ′ be the clique graph of Γ with respect to K. Let k be the valency of Γ, θ d be its minimum eigenvalue and u be a θ-eigenvector of Γ. Then the vector Proof. Consider the bipartite graph whose parts are the vertices of Γ and the cliques from K with incident vertices and cliques being the edges. From the definition of geometric graph we see that such incidence graph is biregular with valencies −θ d , 1 − k θ d and any pair of its vertices at distance 2 are adjacent to a unique vertex. By Theorem 1 when θ = θ d , for any θ-eigenvector u of one of the halved graphs Γ of the biparitite graph, the vector I(u) is always a (θ − θ d − 1 + k θ d )-eigenvector of the second halved graph Γ ′ . Again by Theorem 1 when u is any θ d -eigenvector of Γ, the vector I(u) is always the all-zero vector.

2.2.
Completely regular codes. Let C be a code, i.e. a collection of vertices in a regular graph Γ and χ C denote the characteristic vector of C in Γ. A vertex x is in C i if the minimum of the distances between x and the vertices of C is i. The maximum of these distances is called the covering radius of C and is denoted by ρ. The distance partition of the vertices of C with respect to C 0 = C is {C i : i ∈ {0, . . . , ρ}}. A code C is called completely regular if there are numbers α 0 , . . . , α ρ , β 0 , . . . , β ρ−1 , γ 1 , . . . , γ ρ such that any vertex of C i is adjacent to exactly α i , β i , γ i vertices of C i , C i+1 and C i−1 respectively. Note that α 0 , . . . , α ρ can be obtained from the remaining constants and the valency of the graph. The array {β 0 , . . . , β ρ−1 ; γ 1 , . . . , γ ρ } is called the intersection array of the completely regular code C.
For a completely regular code C consider the following tridiagonal (ρ + 1) × (ρ + 1) matrix which we call the intersection matrix of the completely regular code C. The eigenvalues of this matrix are called the eigenvalues of the completely regular code C. The set of the eigenvalues of a completely regular code C is denoted by Spec(C). Due to the tridiagonal structure of the above matrix, it has exactly ρ + 1 distinct eigenvalues and ρ + 1 different eigenvectors up to the multiplication by a scalar. The opposite code of a code C of covering radius ρ is the code C ρ . It is easy to see that C is a completely regular code if and only if the opposite code C ρ is completely regular. Moreover, if C is completely regular, the eigevalues of C and C ρ coincide. Remark 1. Again due to tridiagonal structure, we note that the first element of any eigenvector of the intersection matrix is never zero and no two consecutive elements in any eigenvector are zeros.  The following fact is well known and could be found, for example, in [14, Proposition 2]. Proposition 2. Let Γ be a distance-regular graph, θ d be its minimum eigenvalue, K be a Delsarte clique of Γ. Then K is a completely regular code in Γ with Spec(K) = Spec(Γ) \ θ d and the vector χ K is orthogonal to any θ d -eigenvector of Γ.
We finish this subsection with the following result for completely regular codes in the Johnson graphs.
Then the strength of C is i 1 − 1 and in particular ρ ≤ w − i 1 + 1.

Completely regular codes in geometric graphs with minimum eigenvalue
Theorem 3. Let Γ be a geometric graph with respect to a set of Delsarte cliques K and the minimum eigenvalue θ d , C be a completely regular code with covering radius ρ such that θ d ∈ Spec(C). Then there are nonzero numbers a i , i ∈ {0, . . . , ρ − 1} that depend only on i and the intersection arrays of C and Γ such that for any K ∈ K, Proof. Let v be a θ d -eigenvector of the intersection matrix of C. Let v Γ be the vector indexed by the vertices of [6,Theorem 4.5], which, in particular, implies the Lloyd's theorem (Theorem 2).
By definition of a geometric graph, for any i ∈ {0, . . . , ρ − 1}, any two adjacent vertices of C i and C i+1 are in a unique clique K of K. Note that since K is a clique and C 0 , . . . , C ρ is a distance partition, K is a subset By the choice of cliques K's any such K contains an edge between C i and C i+1 and we have that |K ∩ C i | = 0, |K|. In view of Remark 1, we see that v i and v i+1 cannot be both zeros, so equality (1) Then again by orthogonality of v Γ and χ K , we have that |K|v i = 0, which contradicts v i = 0. We conclude that any clique K ∈ K fulfills (1) with |K ∩ C i | = 0, |K| for some i.
We note that the eigenvector v of the intersection matrix of C is unique up to the multiplication by a scalar. This combined with fact that the size of a Delsarte clique K is uniquely determined by the intersection array of Γ and (1) imply that the size of K ∩ C i is uniquely determined by the intersection arrays of C and Γ.
For a completely regular code C in a geometric graph Γ with respect to a set K of Delsarte cliques consider the following collection of cliques: In the next lemmas we work towards showing that C ′ is a completely regular code in the clique graph of the geometric graph Γ. Lemma 1. Let Γ be a geometric graph with respect to a set K of Delsarte cliques and θ d be the minimum eigenvalue of Γ and let Γ ′ be the clique graph of Γ with respect to K. If C is a completely regular code with covering radius ρ in Γ and eigenvalue θ d , then the distance partition with respect to the code Proof. The proof of Theorem 3 implies that for any i ∈ {0, . . . , ρ− 1}, the set {K ∈ K : K ⊂ C i ∪C i+1 } actually consists of the cliques in K with edges between C i and C i+1 . Since C 0 , . . . , C ρ is the distance partition with respect to the completely regular code C, any edge between C i and C i+1 is incident to an edge between C i−1 and C i , for any i such that 1 ≤ i ≤ ρ − 1. In view of the above reference to the proof of Theorem 3, any clique in We obtain the required.
Proof. By the definition of the completely regular code C, since x is in C i , x has γ i neighbors in C i−1 , where γ i is an element of the intersection array of C.
On the other hand, by the definition of a geometric graph, the vertex x and any of its γ i neighbors in C i−1 are contained in only one clique from K. We note that all these cliques are in By the proof of Theorem 3, any another clique containing a vertex x of C i , contains vertices of both C i and C i+1 , i.e. this clique is in C ′ i . Since there is no C ′ −1 and C ′ ρ , we see that b 0 = 0 and b ρ equals the number of the cliques in K containing a fixed vertex. The number b ρ is k where |K| is the size of a Delsarte clique K in Γ and k is the valency of the graph Γ. We note that for i ∈ {1, . . . , ρ − 1}, b i cannot be 0 or −θ d . Indeed, any vertex in C i has neighbors in C i−1 and C i+1 because C is a completely regular code. Since the edgeset of Γ is parted into cliques of K, x belongs to at least one clique contained in C i−1 ∪ C i and at least one clique contained in C i ∪ C i+1 . In other words, taking into account the expression for C ′ i 's, x belongs to at least one clique in C ′ i−1 and C ′ i and Theorem 4. Let Γ be a geometric graph with respect to a set of Delsarte cliques K, valency k and minimum Proof. By Lemma 1, for the code . . , ρ − 1} is the distance partition of the vertices of the clique graph of Γ with respect to the code C ′ .
For any i ∈ {0, . . . , ρ} we consider a clique K ∈ C ′ i and find the number of cliques in Since any two cliques in K either meet in a unique vertex or are disjoint, the numbers of the cliques from C ′ i−1 , In order to find the spectra of C ′ , we now yield a relationship between the characteristic vectors of C and C ′ in the vertex sets of Γ and Γ ′ respectively. For this purpose we consider the linear transform I, defined before Theorem 1. The transform I of a vector u indexed by the vertices of Γ is defined as follows: We take u to be χ C . If the clique K is not in C ′ , by definition of C ′ , K does not contain vertices of C, then it contains exactly a 0 vertices of C by Theorem 3, so (I(χ C )) K = a 0 . We see that (2) a 0 χ C ′ = I(χ C ).
By Proposition 1 we have the decomposition where u θ is a θ-eigenvector of Γ. In view of Corollary 1, the vector We see that the characteristic vector χ C ′ of the completely regular code C ′ with covering radius ρ − 1 is a sum of ρ eigenvectors with pairwise distinct eigenvalues This combined with Proposition 1 applied to the completely regular code Corollary 2. Let C be a completely regular code in the Johnson graph J(n, w) with covering radius ρ and θ w (n, w) ∈ Spec(C). Then the following holds for the code Proof. We view the clique graph of the geometric graph J(n, w) as the graph J(n, w − 1), where the vertices of J(n, w − 1) represent the Delsarte cliques and vertex-clique inclusion relation between J(n, w) and J(n, w − 1) is the inclusion relation of (w − 1)-subset (clique) into w-subset (vertex). The complete regularity for C ′ follows from Theorem 5, whereas the existence of constants b i 's is proved in Lemma 2.
4. Characterization of completely regular codes of strength 1 in Johnson graphs 4.1. Constructions of completely regular codes of strength 1. In below we give the list of known compelely regular codes of strength 1 in the Johnson graphs [2], [13], [18]. Let Y 1 , . . . , Y q be a partition of {1, . . . , n} into q subsets of same size p: |Y i | = p for i = 1, . . . , q.

4.2.
Completely regular codes of strength 1 in Johnson graphs. We start with the following description of completely regular codes in Johnson graphs J(n, 2) with strength 1.

Proposition 4.
If C is a completely regular code of strength 1 in J(n, 2), C is a simple 1-(n, 2, λ)-design, where λ or n are even and vice versa.
Proof. The completely regular codes in J(n, w) of strength w − 1 coincide with the class of simple (w − 1) − (n, w, λ)-designs [16]. It is well known that such objects exist if and only if n or λ are even.
Proposition above and Theorem below complete the characterization of completely regular codes in J(n, w) of strength 1 and covering radius w − 1. For a vertex {j, m} in C ′ , consider its neighbors from C ′ . The consideration above implies that if {m, l} is in C ′ , the vertex {j, l} is also in C ′ . We see that the neighbors of {j, m} belonging to C ′ are vertices {j, l s }, {m, l s }, for s = 1, . . . , α 0 /2.
We show that for distinct s and t, the vertex {l s , l t } is in C ′ . Indeed {j, l s } and {j, l t } are in C ′ , forcing {l s , l t } to be in C ′ . We conclude that C ′ is actually the union of all 2-subsets of some (α 0 /2 + 2)-subsets. Note that by the argument after (3), we have α 0 ≥ 2. We now show that any two different such (α 0 /2 + 2)-subsets are disjoint. Suppose the opposite. Let the 2-subsets of Y 1 and Y 2 be in C ′ , |Y 1 | = |Y 2 | = α 0 /2+2 and α 0 /2+2 > |Y 1 ∩Y 2 | ≥ 1. Take an element j ∈ Y 1 \ Y 2 , and an element l ∈ Y 1 ∩ Y 2 , which exists due to |Y 1 ∩ Y 2 | ≥ 1. A vertex {j, l} of J(n, 2) is in C ′ , because it is a subset of Y 1 . It is adjacent to α 0 2-subsets of Y 1 . Moreover it it adjacent to α 0 /2 + 2 − |Y 2 ∩ Y 1 | 2-subsets of Y 2 that are not 2-subsets of Y 1 . These are obtained by replacing j in {j, l} with any element from Y 2 \ Y 1 . We see that {j, l} is adjacent to at least 3α 0 /2 + 2 − |Y 2 ∩ Y 1 | > α 0 vertices of C ′ because Y 1 and Y 2 are distinct subsets of size α 0 /2 + 2 and α 0 /2 + 2 > |Y 1 ∩ Y 2 |. This contradicts the fact that according to the definition of the intersection array of the completely regular code C ′ the number of the vertices of C ′ being adjacent to a fixed vertex of C ′ is α 0 .
Thus the code C ′ is the union of the 2-subsets of disjoint (α 0 /2 + 2)-subsets. If these (α 0 /2 + 2)-subsets do not partition {1, . . . , n}, there is an element j of {1, . . . , n} such that {j, l} is not a vertex of C ′ for any l. We see that C is not a 1-design, a contradiction.
From the considerations in the begining of the proof α 0 is nonzero, so |Y i | ≥ 3.

2.
We have shown that C ′ is the union of the 2-subsets of disjoint subsets {Y 1 , . . . , Y q } of a fixed size p, p = α 0 /2 + 2 ≥ 3. For such C ′ we reconstruct the initial code C in J(n, 3). By the second statement of Corollary 2 and taking into account that b 0 = 0, a 3-subset {j, l, m} is a vertex of C if and only if all its 2-subsets are in C ′ . From the obtained description of C ′ , we see that this is the case only when there is i: {j, l, m} is a subset of Y i , i.e. the code C is actually (GP. ) and therefore has covering radius 2, which contradicts that the covering radius of D is 3. If there are at least 3 such Y i 's we show that D is not completely regular. Indeed, it is not hard to see that the distance partition with respect to D is such that: By induction hypothesis on w, w ≥ 5, we have that there are no completely regular codes in J(n, w − 1), covering radius w − 2 and strength 1. If a completely regular code with covering radius w − 1 and strength 1 exists in J(n, w), then by Corollary 2 there is a completely regular code in J(n, w − 1) with ρ = w − 2 and strength 1, a contradiction. According to Proposition 3, any completely regular code of covering radius w − 1 has strength 0 or 1. The codes of strength 0 were completely characterized in [15]. From Theorem 5 we have the following.
Corollary 3. Let C be a completely regular code in J(n, w) of covering radius w − 1. Then it either has strength 1 and is described in Theorem 5 or it has strength 0 and characterized in [15].  (2) [9] If C has strength 1 and covering radius 1 then n is even. Moreover, for n = 6 the code C is the union of arbitrary collection of antipodal classes in J(6, 3); for n ≥ 8 and n = 10, the code C is (GP.1') for w = 3, (GP. 3)
Proof. Any completely regular code in J(n, 3) has strength t, for some t, 0 ≤ t ≤ 2 and covering radius ρ. By Proposition 3 we have ρ ≤ 3 − t. All completely regular codes in J(n, 3) for t = 0 were characterized in [15].
. We see that the only 4-subsets of {1, . . . , n} such that any its 3-subsets is in C ′ are the unions of Y i 's, i.e.
which is exactly the code (M.1) for w = 4.
According to Corollary 4 there are several completely regular codes in J(10, 3) having the same intersection array {9; 9} as the code (M.2) but not isomorphic to it. We show that regardless the code being (M.2) or not, the corresponding completely regular code C in J(10, 4) does not exist. There are constants In the end of this Section, we will focus on completely regular codes of strength 1 in J(n, 4) without minimum eigenvalue in the spectrum. Although we have not obtained a full characterization of these codes, we derive some significant restrictions on their feasible intersection arrays.
and distance partition {C 0 = C, C 1 , C 2 }. Then the following equalities hold: Proof. Let us count the number of the edges between C 0 and C 1 . On the one hand, it is equal to |C 0 |β 0 , but on the other hand, it is equal to |C 1 |γ 1 . For edges between C 1 and C 2 , we have that |C 1 |β 1 = |C 2 |γ 2 . Since C has covering radius 2, we also have that |C 0 | + |C 1 | + |C 2 | = n 4 . These three equations give us the required.
For our purposes we need to use the following well-known local structure of the Johnson graphs.
Proposition 5. Let x be a vertex of J(n, w) and let N (x) be the set of its neighbors. Then the graph induced on N (x) by J(n, w) is a w × (n − w)-grid with edges between vertices in a common row or column. Each row of the grid together with x forms a Delsarte clique in J(n, w).
for some distinct positive integers β 0 and γ 2 .
. Direct substitution of β 1 in (4) and (7) shows that the equality (7) takes place for all values of β 0 and γ 2 . Hence, the intersection matrix of the code is equal to for some positive integers β 0 and γ 2 . Let us consider the case β 0 = γ 2 . Clearly, the equality (8) implies that γ 2 = 2n − 2 (the case γ 2 = 3n − 6 contradicts to γ 1 > 0). After substitutions β 0 = γ 2 = 2n − 2 in (4) and (7) equation (7) can be written in the following form: Consequently, for β 0 = γ 2 , the intersection matrix of the code is equal to for some positive integer γ 1 . Our next goal is to prove that there are no completely regular codes with the intersection matrix M 2 . As usual, by {C 0 = C, C 1 , C 2 } we mean the distance partition of C. Lemma 3 guarantees us that By Proposition 5 we know that the graph induced by the set of the neighbors of an arbitrary vertex is 4×(n−4)grid with edges between vertices in a common row or column. Without loss of generality we can consider Take some x ∈ C and let N (x) be the grid corresponding to its neighbors. We know that |N (x) ∩ C 0 | = 2n − 14 and |N (x) ∩ C 1 | = 2n − 2. Since 2(n − 4) > 2n − 14, N (x) may contain zero or exactly one row consisting of vertices from C only. Case A. Suppose that for every y ∈ C we have that N (y) contains exactly one row consisting of vertices from C only. It means that we have a set of Delsarte cliques such that every code vertex belongs to exactly one of them. In other words, there exists a code C ′ in J(n, 3) such that C = {x : |x| = 4, ∃y ∈ C ′ , y ⊂ x}, and code distance of C ′ is at least 2.
Since every vertex of C ′ defines exactly n − 3 vertices of C, we conclude that C ′ = γ1n(n−1)

72
. It is known that the size of a code in J(n, 3) with code distance 2 is bounded above by n(n−1)

6
, which follows from the Delsarte bound for independent sets. Therefore, we conclude that γ 1 ≤ 12. By our assumption for every y ∈ C exactly one of N (y)'s rows consists of n − 4 code vertices. Therefore, at least one of remaining three rows contains at least ⌈ n−10 3 ⌉ (and less than n − 4) code vertices. Consequently, a vertex from C 1 from this row has at least ⌈ n−10 3 ⌉ + 2 neighbors from C, where one them is y itself, and the other is taken from the row consisting of vertices from C only. We have that (11) ⌈ n − 4 3 ⌉ ≤ γ 1 .

72
. By Proposition 3 the code C is a 1-design. In other words, the number of the vertices of C containing a fixed element from {1, 2, . . . , n} does not depend on the choice of the element. Clearly, this number equals 4 n |C| = γ1(n−1)(n−3)

18
. In above, we showed how C is defined by C ′ . It is easy to see, that C ′ also must be a 1-design, so the number of the vertices of C ′ containing a fixed element is equal to 3 n |C ′ | = γ1(n−1)

24
. Consider the set S ′ = {x ∈ C 2 : 1 ∈ x}. Since C 2 ia also a completely regular code of strength 1, |S ′ | = 4 n |C 2 | = (n−4−γ1)(n−1)(n−3) 18 . Since the intersection size of any two vertices from S and S ′ is most 1, for any x ∈ S ′ we have that x ∩ {2, 3, . . . , γ1(n−1) 12 + 1} = ∅. Consequently, As we established above, 1 ≤ γ 1 ≤ 12 and 10 ≤ n ≤ 40. Numerical verification of the fulfillment of conditions (12) and (13) for these values of n and γ 1 gives only two possible cases: n = 10, γ 1 = 8 and n = 13, γ 1 = 6. However, they both contradict to inequality (10). Case B. Suppose that for some x ∈ C, N (x) does not have rows consisting of vertices from C only (let us note, that this can be the case for other code vertices). Hence, there is a row with at least 2n−14 4 code vertices. By our assumption this row also contains a vertex from C 1 . Therefore, the vertex has at least 2n−14 4 + 1 neighbors from C 0 . As a result, we have that Analysis of these inequalities shows that it is possible in two cases: n is even, γ 1 = n−4 2 ; and n is odd, γ 1 = n−5 2 . We need to put 2n − 14 vertices from C in 4 × (n − 4)-grid such that any other vertex of the grid would have not more than n−6 2 neighbors from these vertices. Clearly, in the first case (i.e. when n is even and γ 1 = n−4 2 ) N (x) must contain at least one row with n−6 2 code vertices. The remaining 3n−22 2 code vertices must be located in other three rows in the same set of columns. In other words, N (x) contains a 4 × ( n−6 2 )-grid with 2n − 14 and 2 vertices from C and C 1 respectively. There are only three nonequivalent ways for distribution of C in a grid and for all of them one of these vertices from C 1 would have at least n−4 2 neighbors from C inside this grid, a contradiction. In the second case (i.e. when n is odd and γ 1 = n−5 2 ), the only possible configuration of code vertices in N (x) is a 4 × ( n−7 2 )-grid. Without loss of generality, we have that x = {1, 2, 3, 4} and Now we can repeat these arguments for all neighbors of x from C, then for all their neighbors from C and so on until then we have that the set L of all 4-subsets of a set D = {1, 2, 3, . . . , n+1 2 } is a subset of C (conceptually, this part of the proof is close to the one from Theorem 5). Let us note that all neighbors of vertices from L outside L must belong to C 1 .
Suppose that there is another vertex x ′ ∈ C \ L with the same type of N (x ′ ) ∩ C (without rows consisting of vertices from C only). Consequently, there is one more set L ′ of all 4-subsets of a set D ′ such that |D ′ | = n+1 2 . First of all, our previous arguments guarantee that |D ∩ D ′ | ≤ 3 (otherwise, D must coincide with D ′ ). If |D ∩ D ′ | = 3, a vertex z = (D ∩ D ′ ) ∪ {i} for i ∈ D \ D ′ has at least 2n − 14 + n−5 2 neighbors from C. However, z ∈ C and we get a contradiction. In case |D ∩ D ′ | = 2 we have that z ′ = (D ∩ D ′ ) ∪ {i} ∪ {j} for i ∈ D \ D ′ and j ∈ D ′ \ D must be a vertex from C 1 . Direct calculations show that z ′ has at least n − 5 neighbors from L ∪ L ′ , and this is not possible. Since D and D ′ are both of size n+1 2 , the last case here is |D ∩ D ′ | = 1. In this case, by direct calculations we have that {x ∈ C : D ∩ D ′ ⊆ x} consists of at least 2 vertices. On the other hand, since C is a 1-design, the size of this set is bounded above by n 4 |C| = (n−1)(n−3)(n−5)

36
. We obtain a contradiction.
We proved that for all y ∈ C \ L we have that N (y) contains exactly one row consisting of vertices from C only. Again we have a set of Delsarte cliques such that every code vertex (except the set L) belongs to exactly one of them. Therefore, there exists a code C ′ in J(n, 3) such that C \ L = {x : |x| = 4, ∃y ∈ C ′ , y ⊂ x}, and code distance of C ′ is at least 2. By direct calculations and from (9) . Since for any x ∈ L we have that N (x) \ L ⊆ C 1 , we conclude that for any y ∈ C ′ , |y ∩ D| = 1 or |y ∩ D| = 0. Let us consider S 1 = {y ∈ C ′ : |y ∩ D| = 1} and S 0 = {y ∈ C ′ : |y ∩ D| = 0}. Clearly, (14) |S 1 | + |S 0 | = |C ′ | = (5n − 3)(n − 1)(n − 5) 1152 .
The fact that C is a 1-design implies that Q = n−1 Since |S 0 | must be integer, n − 4 divides 135, so n equals 9, 13, 19, 31, 49 or 139. Among them, S 0 is integral for n = 9 and n = 13, but |C ′ | is non-integral for all of them. Finally, we proved that there are no completely regular codes with the intersection matrix M 2 .
Combining results from Theorems 6 and 7, we formulate the main result for completely regular codes in J(n, 4).

Concluding remarks
In this paper, we suggest an approach for tackling the existence problem of completely regular codes with minimum eigenvalue in geometric graphs. An approach showed promising results for the codes in Johnson graphs as we classified several series of completely regular codes in these graphs.
The method suggested in Section 3 can also work for completely regular codes in other geometric graphs. From this perspective we view the Grassmann graphs as the most natural series for this technique. We suppose that some of the ideas from Section 4 can work for completely regular codes in Grassmann graphs with covering radius greater than 1 with the minimum eigenvalue. In the case of binary field, the first open cases are completely regular codes in the Grassmann graphs J 2 (n, 3) with covering radius 2. From Section 3 we observe that these codes are in a natural correspondence with completely regular codes in J 2 (n, 2). The latter codes for some spectra were classified in [10] and one can use this restriction for classification of codes in J 2 (n, 3).
In Section 4 we also described all the completely regular codes of strength 1 in the Johnson graphs J(n, 4) with only one case for the eigenvalues θ 0 (n, 4), θ 2 (n, 4), θ 3 (n, 4) partially left open. As one may see from the proof of Theorem 7 the intersection matrix M 2 depends on two parameters instead of one (it also holds for M 1 ). Although ideas that we used for proving the nonexistence of codes with the intersection matrix M 1 may be also helpful for the case with the matrix M 2 for some particular parameters, in our opinion, such codes might exist.
As an example, we present the following intersection matrices of possible completely regular codes with covering radius 2 in J(n, 4) for small n (all divisibility conditions known to us are satisfied here):