2.1. Necessary conditions
A hypothetical intersection array that does not satisfy one of the first three of the following conditions for some n does not satisfy it for all n. All other arrays are included in the small-value tables, grouped by covering radius (2,3,4,5,6,7,8,1), provided the values are small. For some arrays, the interweight-distribution condition gives a nontrivial lower bound on n (the trivial one is maxi(Si,i-1+Si,i+1)); such arrays are indicated by "!" in the "lower bound" column. For some arrays, calculation of the interweight distributions give negative values for rather large (hypothetically, for all; a prove is wanted) values of n; such arrays are indicated by "99+" in the "lower bound" column.
Monotonicity
For H(n,q): if i<j, then bi>bj and ci<cj. This is easily proved for two neighbor vertices A of color i and B of color i+1: If A has a neighbor X of color i-1, then X has two common neighbors with B, A and some Y(X). The color of Y(X) must be i. Since B has at least one additional neighbor of color i, A, we have ci+1>ci. It should be noted that this property does not necessarily hold for completely regular codes in distance-regular graphs in general. For example, the dodecahedron has a c.r.c (a face) with i.a. [1,2,1;1,2,1].
Integer cardinalities
The cardinality of each cell must be integer. The cardinalities can be calculated from the quotient matrix. For H(n,q): if the proportion between the cell cardinalities is s0:s1:...:sρ, then (s0+s1+...+sρ)/GCD(s0,s1,...,sρ) must divide a power of q.
Weight distribution
An equitable partition with quotient matrix S of a distance-regular graph G is an equitable partition of the corresponding distance-w graph for every w up to the diameter of G. For H(n,q), the corresponding quotient matrix is Sw=Kw(K1-1(S)), where Kw is the Krawtchouk polynomial of degree w. This matrix must be integer (it is "always" non-negative provided S is). This test is stronger than the previous one and also "monotonic" in n. That is, if the matrix Si is not integer and i is minimum with this property, then for the next dimension the corresponding matrices are calculated as Tw=Sw+(q-1)Sw-1, and Ti is not integer as well (it is implied that S satisfies the eigenvalues condition below).
Eigenvalues
The eigenvalues of the quotient matrix must be eigenvalues of the graph. For a completely regular code in H(n,2), the difference between the two largest eigenvalues must be the largest difference between neighbour eigenvalues [KLM10].
Interweight distribution
Given the quotient matrix, we can calculate the interweight distributions of a hypothetical equitable partition of H(n,2) [Vas09], [Kro14]. All the values must be nonnegative (they are "always" integer provided the weight-distribution condition is satisfied). For some intersection arrays, this gives a lower bound on the dimension of the Hamming graph containing a corresponding CR code.
ρ=1, the correlation-immunity bound
For an equitable partition with nonsymmetric (b≠c) quotient matrix [[a,b],[c,d]], the following inequality holds: a≥(3c-b)/4 [FDF07b]. In other words, θ≥-n/3, where θ=a-c=d-b is the second eigenvalue and n=a+b=c+d is the hypercube dimension. In [Hyun10], this bound is someway generalized to equitable partitions of hypercubes into more than 2 cells: if all but one eigenvalues of the quotient matrix are less than -n/3, then the matrix is not feasible (there are no equitable partitions with this quotient matrix). However, there are no known intersection arrays (ρ>1) that are rejected by this reason.
References
[Hyun10] Hyun. A bound on equitable partitions of the Hamming space. IEEE Trans. Inf. Theory 56(5) 2010, 2109-2111. DOI 10.1109/TIT.2010.2043773
[KLM10] J. H. Koolen, W. S. Lee, W. J. Martin. Characterizing completely regular codes from an algebraic viewpoint. Contemp. Math. 531 (2010), 223-242 http://arxiv.org/abs/0911.1828
[Kro14] D. S. Krotov. On Calculation of the Interweight Distribution of an Equitable Partition. J. Algebr.Comb. 40(2) 2014, 373-386. DOI 10.1007/s10801-013-0492-3
[Vas08] A. Yu. Vasil'eva. Linear binary completely regular codes with distance 2. Computational Technologies in Electrical and Electronics Engineering, 2008. SIBIRCON 2008. IEEE Region 8 International Conference on. 12-15. DOI 10.1109/SIBIRCON.2008.4602615
[Vas09] A. Yu. Vasil’eva. Local and interweight spectra of completely regular codes and of perfect colorings. Probl. Inf. Transm. 45(2), 2009, 151-157. DOI 10.1134/S0032946009020069, translated from Probl. Peredachi Inf. 45(2) 2009, 84-90.