Designing floating codes for expected performance

Flavio Chierichetti, Hilary Finucane, Zhenming Liu, Michael Mitzenmacher

Abstract: Floating codes are codes designed to store multiple values in a Write Asymmetric Memory, with applications to flash memory. In this model, a memory consists of a block of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> cells, with each cell in one of q states {0,1,...,q -1}. The cells are used to represent k variable values from an ¿-ary alphabet. Cells can move from lower values to higher values easily, but moving any cell from a higher value to a lower value requires first resetting the entire block to an all 0 state. Reset operations are to be avoided; generally a block can only experience a large but finite number of resets before wearing out entirely. A code here corresponds to a mapping from cell states to variable values, and a transition function that gives how to rewrite cell states when a variable is changed. Previous work has focused on developing codes that maximize the worst-case number of variable changes, or equivalently cell rewrites, that can be experienced before resetting. In this paper, we introduce the problem of maximizing the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">expected</i> number of variable changes before resetting, given an underlying Markov chain that models variable changes. We demonstrate that codes designed for expected performance can differ substantially from optimal worst-case codes, and suggest constructions for some simple cases. We then study the related question of the performance of random codes, again focusing on the issue of expected behavior.