Iterative Algebraic Soft-Decision List Decoding of Reed-Solomon Codes

In this paper, we present an iterative soft-decision decoding algorithm for Reed-Solomon codes offering both complexity and performance advantages over previously known decoding algorithms. Our algorithm is a list decoding algorithm which combines two powerful soft decision decoding techniques which were previously regarded in the literature as competitive, namely, the Koetter-Vardy algebraic soft-decision decoding algorithm and belief-propagation based on adaptive parity check matrices, recently proposed by Jiang and Narayanan. Building on the Jiang-Narayanan algorithm, we present a belief-propagation based algorithm with a significant reduction in computational complexity. We introduce the concept of using a belief-propagation based decoder to enhance the soft-input information prior to decoding with an algebraic soft-decision decoder. Our algorithm can also be viewed as an interpolation multiplicity assignment scheme for algebraic soft-decision decoding of Reed-Solomon codes.Comment: Submitted to IEEE for publication in Jan 200

Correcting Charge-Constrained Errors in the Rank-Modulation Scheme

We investigate error-correcting codes for a the rank-modulation scheme with an application to flash memory devices. In this scheme, a set of n cells stores information in the permutation induced by the different charge levels of the individual cells. The resulting scheme eliminates the need for discrete cell levels, overcomes overshoot errors when programming cells (a serious problem that reduces the writing speed), and mitigates the problem of asymmetric errors. In this paper, we study the properties of error-correcting codes for charge-constrained errors in the rank-modulation scheme. In this error model the number of errors corresponds to the minimal number of adjacent transpositions required to change a given stored permutation to another erroneous one—a distance measure known as Kendall’s τ-distance.We show bounds on the size of such codes, and use metric-embedding techniques to give constructions which translate a wealth of knowledge of codes in the Lee metric to codes over permutations in Kendall’s τ-metric. Specifically, the one-error-correcting codes we construct are at least half the ball-packing upper bound

Perfectly Secure Steganography: Capacity, Error Exponents, and Code Constructions

An analysis of steganographic systems subject to the following perfect undetectability condition is presented in this paper. Following embedding of the message into the covertext, the resulting stegotext is required to have exactly the same probability distribution as the covertext. Then no statistical test can reliably detect the presence of the hidden message. We refer to such steganographic schemes as perfectly secure. A few such schemes have been proposed in recent literature, but they have vanishing rate. We prove that communication performance can potentially be vastly improved; specifically, our basic setup assumes independently and identically distributed (i.i.d.) covertext, and we construct perfectly secure steganographic codes from public watermarking codes using binning methods and randomized permutations of the code. The permutation is a secret key shared between encoder and decoder. We derive (positive) capacity and random-coding exponents for perfectly-secure steganographic systems. The error exponents provide estimates of the code length required to achieve a target low error probability. We address the potential loss in communication performance due to the perfect-security requirement. This loss is the same as the loss obtained under a weaker order-1 steganographic requirement that would just require matching of first-order marginals of the covertext and stegotext distributions. Furthermore, no loss occurs if the covertext distribution is uniform and the distortion metric is cyclically symmetric; steganographic capacity is then achieved by randomized linear codes. Our framework may also be useful for developing computationally secure steganographic systems that have near-optimal communication performance.Comment: To appear in IEEE Trans. on Information Theory, June 2008; ignore Version 2 as the file was corrupte

Perfectly Secure Steganography: Capacity, Error Exponents, and Code Constructions

An analysis of steganographic systems subject to the following perfect undetectability condition is presented in this paper. Following embedding of the message into the covertext, the resulting stegotext is required to have exactly the same probability distribution as the covertext. Then no statistical test can reliably detect the presence of the hidden message. We refer to such steganographic schemes as perfectly secure. A few such schemes have been proposed in recent literature, but they have vanishing rate. We prove that communication performance can potentially be vastly improved; specifically, our basic setup assumes independently and identically distributed (i.i.d.) covertext, and we construct perfectly secure steganographic codes from public watermarking codes using binning methods and randomized permutations of the code. The permutation is a secret key shared between encoder and decoder. We derive (positive) capacity and random-coding exponents for perfectly-secure steganographic systems. The error exponents provide estimates of the code length required to achieve a target low error probability. We address the potential loss in communication performance due to the perfect-security requirement. This loss is the same as the loss obtained under a weaker order-1 steganographic requirement that would just require matching of first-order marginals of the covertext and stegotext distributions. Furthermore, no loss occurs if the covertext distribution is uniform and the distortion metric is cyclically symmetric; steganographic capacity is then achieved by randomized linear codes. Our framework may also be useful for developing computationally secure steganographic systems that have near-optimal communication performance.Comment: To appear in IEEE Trans. on Information Theory, June 2008; ignore Version 2 as the file was corrupte