Explicit and Efficient WOM Codes of Finite Length

Write-once memory (WOM) is a storage device consisting of binary cells that can only increase their levels. A <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> <italic>-write WOM code</italic> is a coding scheme that makes it possible to write <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> times to a WOM without decreasing the levels of any of the cells. The <italic>sum-rate</italic> of a WOM code is the ratio between the total number of bits written to the memory during the <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> writes and the number of cells. It is known that the maximum possible sum-rate of a <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-write WOM code is <inline-formula> <tex-math notation="LaTeX">$\log (t+1)$ </tex-math></inline-formula>. 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 <italic>complexity</italic> of the encoding and decoding maps. The second figure of merit is the <italic>convergence rate</italic>, defined as the minimum code length <inline-formula> <tex-math notation="LaTeX">$n(\delta)$ </tex-math></inline-formula> required to reach a point that is <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">$n(\delta)$ </tex-math></inline-formula> required to be <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">$\epsilon $ </tex-math></inline-formula>-error WOM codes of high rates and small probability of failure.

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