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Phase-change memory (PCM) is a promising non-volatile solid-state memorytechnology. A PCM cell stores data by using its amorphous and crystallinestates. The cell changes between these two states using high temperature.However, since the cells are sensitive to high temperature, it is important,when programming cells, to balance the heat both in time and space.  In this paper, we study the time-space constraint for PCM, which wasoriginally proposed by Jiang et al. A code is called an\emph{$(\alpha,\beta,p)$-constrained code} if for any $\alpha$ consecutiverewrites and for any segment of $\beta$ contiguous cells, the total rewritecost of the $\beta$ cells over those $\alpha$ rewrites is at most $p$. Here,the cells are binary and the rewrite cost is defined to be the Hamming distancebetween the current and next memory states. First, we show a general upperbound on the achievable rate of these codes which extends the results of Jianget al. Then, we generalize their construction for $(\alpha\geq 1,\beta=1,p=1)$-constrained codes and show another construction for $(\alpha = 1,\beta\geq 1,p\geq1)$-constrained codes. Finally, we show that these twoconstructions can be used to construct codes for all values of $\alpha$,$\beta$, and $p$.

Time-Space Constrained Codes for Phase-Change Memories