Iterative Programming of Noisy Memory Cells

This paper studies a model, which was first presented by Bunte and Lapidoth, that mimics the programming operation of memory cells. Under this paradigm it is assumed that the cells are programmed sequentially and individually. The programming process is modeled as transmission over a channel, while it is possible to read the cell state in order to determine its programming success, and in case of programming failure, to reprogram the cell again. Reprogramming a cell can reduce the bit error rate, however this comes with the price of increasing the overall programming time and thereby affecting the writing speed of the memory. An iterative programming scheme is an algorithm which specifies the number of attempts to program each cell. Given the programming channel and constraints on the average and maximum number of attempts to program a cell, we study programming schemes for maximizing the number of bits that can be reliably stored in the memory. We extend the results by Bunte and Lapidoth and study this problem when the programming channel is either the BSC, the BEC, or the {Z} channel, where we focus on a specific broad family of programming schemes. For the BSC and the BEC our analysis is also extended for the case where the error probabilities on consecutive writes are not necessarily the same. Lastly, we also study a related model which is motivated by the synthesis process of DNA molecules.