Design of Non-Binary Quasi-Cyclic LDPC Codes by ACE Optimization

An algorithm for constructing Tanner graphs of non-binary irregular quasi-cyclic LDPC codes is introduced. It employs a new method for selection of edge labels allowing control over the code's non-binary ACE spectrum and resulting in low error-floor. The efficiency of the algorithm is demonstrated by generating good codes of short to moderate length over small fields, outperforming codes generated by the known methods.Comment: Accepted to 2013 IEEE Information Theory Worksho

LDPC Codes over the q-ary Multi-Bit Channel

In this paper, we introduce a new channel model termed as the q-ary multi-bit channel. This channel models a memory device, where q-ary symbols (q=2^s) are stored in the form of current/voltage levels. The symbols are read in a measurement process, which provides a symbol bit in each measurement step, starting from the most significant bit. An error event occurs when not all the symbol bits are known. To deal with such error events, we use GF(q) low-density parity-check (LDPC) codes and analyze their decoding performance. We start with iterative-decoding threshold analysis and derive optimal edge-label distributions for maximizing the decoding threshold. We later move to a finite-length iterative-decoding analysis and propose an edge-labeling algorithm for the improved decoding performance. We then provide a finite-length maximum-likelihood decoding analysis for both the standard non-binary random ensemble and LDPC ensembles. Finally, we demonstrate by simulations that the proposed edge-labeling algorithm improves the finite-length decoding performance by orders of magnitude

LDPC Codes over the q-ary Multi-Bit Channel

In this paper, we introduce a new channel model termed as the q-ary multi-bit channel. This channel models a memory device, where q-ary symbols (q=2^s) are stored in the form of current/voltage levels. The symbols are read in a measurement process, which provides a symbol bit in each measurement step, starting from the most significant bit. An error event occurs when not all the symbol bits are known. To deal with such error events, we use GF(q) low-density parity-check (LDPC) codes and analyze their decoding performance. We start with iterative-decoding threshold analysis and derive optimal edge-label distributions for maximizing the decoding threshold. We later move to a finite-length iterative-decoding analysis and propose an edge-labeling algorithm for the improved decoding performance. We then provide a finite-length maximum-likelihood decoding analysis for both the standard non-binary random ensemble and LDPC ensembles. Finally, we demonstrate by simulations that the proposed edge-labeling algorithm improves the finite-length decoding performance by orders of magnitude

A Combinatorial Methodology for Optimizing Non-Binary Graph-Based Codes: Theoretical Analysis and Applications in Data Storage

Non-binary (NB) low-density parity-check (LDPC) codes are graph-based codes that are increasingly being considered as a powerful error correction tool for modern dense storage devices. Optimizing NB-LDPC codes to overcome their error floor is one of the main code design challenges facing storage engineers upon deploying such codes in practice. Furthermore, the increasing levels of asymmetry incorporated by the channels underlying modern dense storage systems, e.g., multi-level Flash systems, exacerbates the error floor problem by widening the spectrum of problematic objects that contributes to the error floor of an NB-LDPC code. In a recent research, the weight consistency matrix (WCM) framework was introduced as an effective combinatorial NB-LDPC code optimization methodology that is suitable for modern Flash memory and magnetic recording (MR) systems. The WCM framework was used to optimize codes for asymmetric Flash channels, MR channels that have intrinsic memory, in addition to canonical symmetric additive white Gaussian noise channels. In this paper, we provide an in-depth theoretical analysis needed to understand and properly apply the WCM framework. We focus on general absorbing sets of type two (GASTs) as the detrimental objects of interest. In particular, we introduce a novel tree representation of a GAST called the unlabeled GAST tree, using which we prove that the WCM framework is optimal in the sense that it operates on the minimum number of matrices, which are the WCMs, to remove a GAST. Then, we enumerate WCMs and demonstrate the significance of the savings achieved by the WCM framework in the number of matrices processed to remove a GAST. Moreover, we provide a linear-algebraic analysis of the null spaces of WCMs associated with a GAST. We derive the minimum number of edge weight changes needed to remove a GAST via its WCMs, along with how to choose these changes. Additionally, we propose a new set of problematic objects, namely oscillating sets of type two (OSTs), which contribute to the error floor of NB-LDPC codes with even column weights on asymmetric channels, and we show how to customize the WCM framework to remove OSTs. We also extend the domain of the WCM framework applications by demonstrating its benefits in optimizing column weight 5 codes, codes used over Flash channels with soft information, and spatially-coupled codes. The performance gains achieved via the WCM framework range between 1 and nearly 2.5 orders of magnitude in the error floor region over interesting channels

A Combinatorial Methodology for Optimizing Non-Binary Graph-Based Codes: Theoretical Analysis and Applications in Data Storage

Non-binary (NB) low-density parity-check (LDPC) codes are graph-based codes that are increasingly being considered as a powerful error correction tool for modern dense storage devices. Optimizing NB-LDPC codes to overcome their error floor is one of the main code design challenges facing storage engineers upon deploying such codes in practice. Furthermore, the increasing levels of asymmetry incorporated by the channels underlying modern dense storage systems, e.g., multi-level Flash systems, exacerbates the error floor problem by widening the spectrum of problematic objects that contributes to the error floor of an NB-LDPC code. In a recent research, the weight consistency matrix (WCM) framework was introduced as an effective combinatorial NB-LDPC code optimization methodology that is suitable for modern Flash memory and magnetic recording (MR) systems. The WCM framework was used to optimize codes for asymmetric Flash channels, MR channels that have intrinsic memory, in addition to canonical symmetric additive white Gaussian noise channels. In this paper, we provide an in-depth theoretical analysis needed to understand and properly apply the WCM framework. We focus on general absorbing sets of type two (GASTs) as the detrimental objects of interest. In particular, we introduce a novel tree representation of a GAST called the unlabeled GAST tree, using which we prove that the WCM framework is optimal in the sense that it operates on the minimum number of matrices, which are the WCMs, to remove a GAST. Then, we enumerate WCMs and demonstrate the significance of the savings achieved by the WCM framework in the number of matrices processed to remove a GAST. Moreover, we provide a linear-algebraic analysis of the null spaces of WCMs associated with a GAST. We derive the minimum number of edge weight changes needed to remove a GAST via its WCMs, along with how to choose these changes. Additionally, we propose a new set of problematic objects, namely oscillating sets of type two (OSTs), which contribute to the error floor of NB-LDPC codes with even column weights on asymmetric channels, and we show how to customize the WCM framework to remove OSTs. We also extend the domain of the WCM framework applications by demonstrating its benefits in optimizing column weight 5 codes, codes used over Flash channels with soft information, and spatially-coupled codes. The performance gains achieved via the WCM framework range between 1 and nearly 2.5 orders of magnitude in the error floor region over interesting channels