Two-dimensional weight-constrained codes through enumeration bounds

For a rational /spl alpha//spl isin/(0,1), let /spl Ascr//sub n/spl times/m,/spl alpha// be the set of binary n/spl times/m arrays in which each row has Hamming weight /spl alpha/m and each column has Hamming weight /spl alpha/n, where /spl alpha/m and /spl alpha/n are integers. (The special case of two-dimensional balanced arrays corresponds to /spl alpha/=1/2 and even values for n and m.) The redundancy of /spl Ascr//sub n/spl times/m,/spl alpha// is defined by /spl rho//sub n/spl times/m,/spl alpha//=nmH(/spl alpha/)-log/sub 2/|/spl Ascr//sub n/spl times/m,/spl alpha//| where H(x)=-xlog/sub 2/x-(1-x)log/sub 2/(1-x). Bounds on /spl rho//sub n/spl times/m,/spl alpha// are obtained in terms of the redundancies of the sets /spl Ascr//sub /spl Lscr/,/spl alpha// of all binary /spl Lscr/-vectors with Hamming weight /spl alpha//spl Lscr/, /spl Lscr//spl isin/{n,m}. Specifically, it is shown that /spl rho//sub n/spl times/m,/spl alpha///spl les/n/spl rho//sub m,/spl alpha//+m/spl rho//sub n,/spl alpha// where /spl rho//sub /spl Lscr/,/spl alpha//=/spl Lscr/H(/spl alpha/)-log/sub 2/|/spl Ascr//sub /spl Lscr/,/spl alpha//| 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 /spl Ascr//sub n/spl times/m,/spl alpha// at a rate H(/spl alpha/)-(/spl rho//sub m,/spl alpha///m).