Two-dimensional weight-constrained codes through enumeration bounds

For a rational α∈(0,1), let 𝒜_n×m,α be the set of binary n×m arrays in which each row has Hamming weight αm and each column has Hamming weight αn, where αm and αn are integers. (The special case of two-dimensional balanced arrays corresponds to α=1/2 and even values for n and m.) The redundancy of 𝒜_n×m,α is defined by ρ_n×m,α=nmH(α)-log₂|𝒜 _n×m,α| where H(x)=-xlog₂x-(1-x)log₂(1-x). Bounds on ρ_n×m,α are obtained in terms of the redundancies of the sets 𝒜_ℒ,α of all binary ℒ-vectors with Hamming weight αℒ, ℒ∈{n,m}. Specifically, it is shown that ρ_n×m,α⩽nρ_m,α+mρ _n,α where ρ_ℒ,α=ℒH(α)-log₂|𝒜 _ℒ,α| and that this bound is tight up to an additive term O(n+log m). A polynomial-time coding algorithm is presented that maps unconstrained input sequences into 𝒜_n×m,α at a rate H(α)-(ρ_m,α/m)