Explicit constructions of finite-length WOM codes

Write-once memory (WOM) is a storage device consisting of binary cells which can only increase their levels. A t-write WOM code is a coding scheme which allows to write t times to the WOM without decreasing the levels of the cells. The sum-rate of a WOM code is the ratio between the total number of bits written to the memory and the number of cells. It is known that the maximum sum-rate of a t-write WOM code is log(t + 1). This is also an achievable upper bound both by information theory arguments and explicit WOM code constructions. While existing constructions of WOM codes were targeted to increase the sum-rate, we consider here two more figures of merit in evaluating the constructions. The first one is the complexity of the encoding and decoding maps of the code. The second one is called the convergence rate, and is defined to be the minimum code length n(ε) in order to reach e close to a point in the capacity region. One of our main results in the paper is a specific capacity achieving construction for two-write WOM codes which has polynomial complexity and relatively short block length to be ε close to the capacity. Using these two-write WOM codes, we obtain three-write WOM codes that approach sum-rate 1.809 with relatively short block lengths. Finally, we provide another construction of three-write WOM that achieves sum-rate 1.71 by using only 100 cells.