Explicit and Efficient WOM Codes of Finite Length

Abstract

Write-once memory (WOM) is a storage device consisting of binary cells that can only increase their levels. A $t$ -write WOM code is a coding scheme that makes it possible to write $t$ times to a WOM without decreasing the levels of any of the cells. The sum-rate of a WOM code is the ratio between the total number of bits written to the memory during the $t$ writes and the number of cells. It is known that the maximum possible sum-rate of a $t$ -write WOM code is $\log (t+1)$ . This is also an achievable upper bound, both by information-theoretic arguments and through explicit constructions. While existing constructions of WOM codes are targeted at the sum-rate, we consider here two more figures of merit. The first one is the complexity of the encoding and decoding maps. The second figure of merit is the convergence rate , defined as the minimum code length $n(\delta)$ required to reach a point that is $\delta $ -close to the capacity region. One of our main results in this paper is a capacity-achieving construction of two-write WOM codes which has polynomial encoding/decoding complexity while the block length $n(\delta)$ required to be $\delta $ -close to capacity is significantly smaller than existing constructions. Using these two-write WOM codes, we then obtain three-write WOM codes that approach a sum-rate of 1.809 at relatively short block lengths. We also provide several explicit constructions of finite length three-write WOM codes; in particular, we achieve a sum-rate of 1.716 by using only 93 cells. Finally, we modify our two-write WOM codes to construct $\epsilon $ -error WOM codes of high rates and small probability of failure.