Construction of Fixed Rate Non-Binary WOM Codes Based on Integer Programming

Abstract

In this paper, we propose a construction of non-binary WOM (Write-Once-Memory) codes for WOM storages such as flash memories. The WOM codes discussed in this paper are fixed rate WOM codes where messages in a fixed alphabet of size M can be sequentially written in the WOM storage at least t * -times. In this paper, a WOM storage is modeled by a state transition graph. The proposed construction has the following two features. First, it includes a systematic method to determine the encoding regions in the state transition graph. Second, the proposed construction includes a labeling method for states by using integer programming. Several novel WOM codes for q level flash memories with 2 cells are constructed by the proposed construction. They achieve the worst numbers of writes t * that meet the known upper bound in the range 4 ≤ q ≤ 8, M = 8. In addition, we constructed fixed rate non-binary WOM codes with the capability to reduce ICI (inter cell interference) of flash cells. One of the advantages of the proposed construction is its flexibility. It can be applied to various storage devices, to various dimensions (i.e, number of cells), and various kind of additional constraints. I. IntroductionRecent progress of storage media has been creating interests on coding techniques to ensure reliability of the media and to lengthen the life of storage media. Write-Once-Memory (WOM) codes are getting renewed interests as one of promising coding techniques for storage media. In the scenario of the binary WOM codes, the binary WOM storage (or channel) is assumed as follows. A storage cell has two states 0 or 1 and the initial state is 0. If a cell changes its state to 1, then it cannot be reset to 0 any more. Punch cards and optical disks are examples of the binary WOM storages. The celebrated work by Rivest and Shamir in 1982 [1] presented the first binary WOM codes and their codes induced subsequent active researches in the field of the binary WOM codes [2] [3] [4].A memory cell in recent flash memories has multiple levels such as 4 or 8 levels and the number of levels are expected to be increased further in the near future. This trend has produced motivation to the research activities on the non-binary WOM codes that are closely related to the multilevel flash memories [5] [6] [7] [11].There are two threads of researches on the non-binary WOM codes. The first one is variable rate codes and the other is fixed rate codes.The variable rate codes are the non-binary WOM codes such that message alphabets used in a sequence of writing processes are not necessarily identical. This means that writing rate can vary at each writing attempt. Fu and Vinck [8] proved the channel capacity of the variable rate non-binary WOM codes. Recently, Shpilka [9] proposed a capacity achieving construction of non binary WOM codes. Moreover, Gabrys et al. [5] presented a construction of the non-binary WOM codes based on known efficient binary-WOM codes.Although the variable rate codes are efficient because they can fully utilize the potential of a WO...