ρ=1 is a special case, as the cells of every equitable 2-partition (colors of a perfect 2-coloring) are completely regular codes with covering radius 1. The theory of perfect 2-colorings of the hypercubes, in its current state, was developed by D. G. Fon-Der-Flaass in three papers; plus, additional nonexistence results were recently proved in [KV20]. Paper [FDF07a] contains general constructions and some bounds; the bounds are not actual after [FDF07b]. In [FDF07b], a bound on correlation immunity of Boolean function is proven. A corollary for completely regular codes implies that for feasible intersection array [b;c] with b≠c the minimal dimension n satisfies n≥3(b+c)/4 (equivalently, the minimal eigenvalue θ1 of the quotient matrix satisfies θ1≥-n/3). The third paper [FDF07c] contains a construction of equitable partitions of the 12-cube with the quotient matrix [[3,9],[7,5]] and a proof of the nonexistence of an equitable partition of the 12-cube with the quotient matrix [[1,11],[5,7]]. The last proof was generalized in [KV20], where it is shown that for an equitable partition of the n-cube with the quotient matrix [[a,b],[c,d]], a+b=c+d=n, b≠c, n=3(b+c)/4, either b/gcd(b,c) or c/gcd(b,c) must be divisible by 3 (in particular, [[2,22],[10,14]] and [[5,19],[13,11]] do not exist).
The table contains all pairs [b;c] such that (b+c)/gcd(b,c) is a power of 2, b≥c, and b+c≤128. We omit the column that indicate the code cardinality, as the ratio between the cardinalities of the two cells of an equitable 2-partition is easily calculated as c:b. Next column is (n-θ1)/2, which equals (b+c)/2. The column with the lower bound contains max(b,c,3(b+c)/4+s) whenever b≠c and s=0 if bc/gcd(b,c)2 mod 3=0, otherwise s=1 (this s comes from the nonexistence proved in [KV20], marked by "!!" in the table). The upper bound on the minimal feasible n is obtained by one of the constructions *t, splitting, tH, where tH denotes the union of t disjoint perfect codes (e.g., cosets of the Hamming code). The only exceptional array is [9;7], for which there is a special construction [FDF07c].
[FDF07a] D. G. Fon-Der-Flaass. Perfect 2-colorings of a hypercube. Sib. Math. J. 48(4) 2007, 740–745. DOI 10.1007/s11202-007-0075-4. Translated from Sib. Mat. Zh. 48(4) 2007, 923-930.
[FDF07b] D. G. Fon-Der-Flaass. A bound on correlation immunity. Siberian Electronic Mathematical Reports 4, 2007, 133-135. PDF
[FDF07c] D. G. Fon-Der-Flaass. Perfect colorings of the 12-cube that attain the bound on correlation immunity. Siberian Electronic Mathematical Reports 4, 2007, 292-295 [Russian]. English translation: arXiv:1403.8091
[KV20] D. S. Krotov, K. V. Vorob'ev. On unbalanced Boolean functions with best correlation immunity. The Electronic Journal of Combinatorics 27(1) 2020, #P1.45. DOI 10.37236/8557