Two-Dimensional RC/SW Constrained Codes: Bounded Weight and Almost Balanced Weight

In this work, we study two types of constraints on two-dimensional binary arrays. Given
$p\in [{0,1}],\epsilon \in [{0,1/2}]$
, we study 1) the
$p$
-bounded constraint: a binary vector of size
$n$
is said to be
$p$
-bounded if its weight is at most
$pn$
, and 2) the
$\epsilon $
-balanced constraint: a binary vector of size
$n$
is said to be
$\epsilon $
-balanced if its weight is within
$\big [(1/2-\epsilon)n, (1/2+\epsilon)n\big]$
. 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
$\epsilon $
-balanced constraint) is enforced over every row and every column, regarded as 2D row-column (RC) constrained codes; or over every window (where each window refers to as a subarray consisting of consecutive rows and consecutive columns), regarded as 2D sliding-window (SW) constrained codes. While low-complexity designs have been proposed in the literature, mostly focusing on 2D RC constrained codes where
$p=1/2$
and
$\epsilon =0$
, this work provides efficient coding methods that work for both 2D RC constrained codes and 2D SW constrained codes, and more importantly, the methods are applicable for arbitrary values of
$p$
and
$\epsilon $
. Furthermore, for certain values of
$p$
and
$\epsilon $
, we show that, for sufficiently large array size, there exists linear-time encoding/decoding algorithm that incurs at most one redundant bit.