Two-dimensional weight-constrained codes through enumeration bounds

For a rational alpha is an element of (0, 1), let A(n x m, alpha) be the se t of binary n x m arrays in which each row has Hamming weight alpha m and e ach column has Hamming weight alpha n, where alpha m and alpha n are intege rs. (The special case of two-dimensional balanced arrays corresponds to alp ha = 1/2 and even values for n and m,) The redundancy of A(n x m, alpha) is defined by rho(n x m, alpha) = nmH(alpha) - log(2) \A(n x m, alpha)\ where H(x) = -x log(2) x - (1 - x) log(2)(1 - x), Bounds on rho(n x m, alpha) are obtained in terms of the redundancies of th e sets A(l, alpha) of all binary e-vectors with Hamming weight alpha l, l i s an element of {n, m}, Specifically, it is shown that rho(n x m, alpha) less than or equal to nP(m, alpha) + m rho(n, alpha) where rho(l, alpha) = lH(alpha) - log(2) \A(l, alpha)\ and that this bound is tight up to an additive term O(n + log m), A polynom ial-time coding algorithm is presented that maps unconstrained input sequen ces into A(n x m, alpha) at a rate H(alpha) - (rho(m, alpha)/m).