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Two dimensional RC/Subarray Constrained Codes: Bounded Weight and Almost Balanced Weight

In this work, we study two types of constraints on two-dimensional binary arrays. In particular, given $p,ε>0$, we study (i) The $p$-bounded constraint: a binary vector of size $m$ is said to be $p$-bounded if its weight is at most $pm$, and (ii) The $ε$-balanced constraint: a binary vector of size $m$ is said to be $ε$-balanced if its weight is within $[(0.5-ε)*m,(0.5+ε)*m]$. Such constraints are crucial in several data storage systems, those regard the information data as two-dimensional (2D) instead of one-dimensional (1D), such as the crossbar resistive memory arrays and the holographic data storage. In this work, efficient encoding/decoding algorithms are presented for binary arrays so that the weight constraint (either $p$-bounded constraint or $ε$-balanced constraint) is enforced over every row and every column, regarded as 2D row-column (RC) constrained codes; or over every subarray, regarded as 2D subarray constrained codes. While low-complexity designs have been proposed in the literature, mostly focusing on 2D RC constrained codes where $p = 1/2$ and $ε= 0$, this work provides efficient coding methods that work for both 2D RC constrained codes and 2D subarray constrained codes, and more importantly, the methods are applicable for arbitrary values of $p$ and $ε$. Furthermore, for certain values of $p$ and $ε$, we show that, for sufficiently large array size, there exists linear-time encoding/decoding algorithm that incurs at most one redundant bit.