Abstract—An LDPC code is proposed for flash memories based on rank modulation. In contrast to previous approaches, this enables the use of long ECCs with fixed-length modulation codes. For ECC design, the rank modulation scheme is treated as part of an equivalent channel. A probabilistic model of the equivalent channel is derived and a simple high-SNR approximation is given. LDPC codes over integer rings and finite fields are designed for the approximate channel and a low-complexity symbol-flipping verification-based (SFVB) message-passing decoding algorithm is proposed to take advantage of the channel structure. Density evolution (DE) is used to calculate decoding thresholds and simulations are used to compare the low-complexity decoder with sum-product decoding. I.

Abstract In this paper I present a class of parallel and serially concatenated linear block codes, focusing specifically on the case of repeat accumulate codes and the more general irregular repeat accumulate codes. The analytical tractability of repeat accumulate codes has enabled the first derivations of coding theorems for any class of turbo-like codes. Experimentally, repeat accumulate codes allow linear-time encoding and decoding with performance comparable to turbo and LDPC codes. 1 Introduction Turbo codes were introduced in 1993 [1] as a practical code construction whose performance approaches Shannon's theoretical limit for the capacity of noisy channels, which was first introduced in 1948 [2]. It has led to the subsequent rediscovery of low-density parity-check codes [3], and the discovery of the connection between iterative decoding and belief propagation [4][5]. There has been a recent explosion of interest in such codes defined on sparse random graphs. [6][7]

A statistical theory of turbo-codes, treated as a special family of convolutional codes with a low-density parity-check matrix, is developed. The basic ideas are: the representation of the transposed parity-check matrix of turbo-codes as a product of a sparse (multiple) scrambler matrix and the transposed parity-check matrix of the basic code, the introduction of a special statistical ensemble of scramblers having Markov properties, and numerical analysis of the recurrent equations describing the average performance of turbo-codes over this ensemble.