Improved Decoding of Staircase Codes: The Soft-aided Bit-marking (SABM) Algorithm

Staircase codes (SCCs) are typically decoded using iterative bounded-distance decoding (BDD) and hard decisions. In this paper, a novel decoding algorithm is proposed, which partially uses soft information from the channel. The proposed algorithm is based on marking certain number of highly reliable and highly unreliable bits. These marked bits are used to improve the miscorrection-detection capability of the SCC decoder and the error-correcting capability of BDD. For SCCs with $2$-error-correcting Bose-Chaudhuri-Hocquenghem component codes, our algorithm improves upon standard SCC decoding by up to $0.30$~dB at a bit-error rate (BER) of $10^{-7}$. The proposed algorithm is shown to achieve almost half of the gain achievable by an idealized decoder with this structure. A complexity analysis based on the number of additional calls to the component BDD decoder shows that the relative complexity increase is only around $4\%$ at a BER of $10^{-4}$. This additional complexity is shown to decrease as the channel quality improves. Our algorithm is also extended (with minor modifications) to product codes. The simulation results show that in this case, the algorithm offers gains of up to $0.44$~dB at a BER of $10^{-8}$.Comment: 10 pages, 12 figure

Deriving the Normalized Min-Sum Algorithm from Cooperative Optimization

The normalized min-sum algorithm can achieve near-optimal performance at decoding LDPC codes. However, it is a critical question to understand the mathematical principle underlying the algorithm. Traditionally, people thought that the normalized min-sum algorithm is a good approximation to the sum-product algorithm, the best known algorithm for decoding LDPC codes and Turbo codes. This paper offers an alternative approach to understand the normalized min-sum algorithm. The algorithm is derived directly from cooperative optimization, a newly discovered general method for global/combinatorial optimization. This approach provides us another theoretical basis for the algorithm and offers new insights on its power and limitation. It also gives us a general framework for designing new decoding algorithms.Comment: Accepted by IEEE Information Theory Workshop, Chengdu, China, 200

Fast performance estimation of block codes

Importance sampling is used in this paper to address the classical yet important problem of performance estimation of block codes. Simulation distributions that comprise discreteand continuous-mixture probability densities are motivated and used for this application. These mixtures are employed in concert with the so-called g-method, which is a conditional importance sampling technique that more effectively exploits knowledge of underlying input distributions. For performance estimation, the emphasis is on bit by bit maximum a-posteriori probability decoding, but message passing algorithms for certain codes have also been investigated. Considered here are single parity check codes, multidimensional product codes, and briefly, low-density parity-check codes. Several error rate results are presented for these various codes, together with performances of the simulation techniques

Spatially Coupled Codes and Optical Fiber Communications: An Ideal Match?

In this paper, we highlight the class of spatially coupled codes and discuss their applicability to long-haul and submarine optical communication systems. We first demonstrate how to optimize irregular spatially coupled LDPC codes for their use in optical communications with limited decoding hardware complexity and then present simulation results with an FPGA-based decoder where we show that very low error rates can be achieved and that conventional block-based LDPC codes can be outperformed. In the second part of the paper, we focus on the combination of spatially coupled LDPC codes with different demodulators and detectors, important for future systems with adaptive modulation and for varying channel characteristics. We demonstrate that SC codes can be employed as universal, channel-agnostic coding schemes.Comment: Invited paper to be presented in the special session on "Signal Processing, Coding, and Information Theory for Optical Communications" at IEEE SPAWC 201

Effects of noise on quantum error correction algorithms

It has recently been shown that there are efficient algorithms for quantum computers to solve certain problems, such as prime factorization, which are intractable to date on classical computers. The chances for practical implementation, however, are limited by decoherence, in which the effect of an external environment causes random errors in the quantum calculation. To combat this problem, quantum error correction schemes have been proposed, in which a single quantum bit (qubit) is ``encoded'' as a state of some larger number of qubits, chosen to resist particular types of errors. Most such schemes are vulnerable, however, to errors in the encoding and decoding itself. We examine two such schemes, in which a single qubit is encoded in a state of $n$ qubits while subject to dephasing or to arbitrary isotropic noise. Using both analytical and numerical calculations, we argue that error correction remains beneficial in the presence of weak noise, and that there is an optimal time between error correction steps, determined by the strength of the interaction with the environment and the parameters set by the encoding.Comment: 26 pages, LaTeX, 4 PS figures embedded. Reprints available from the authors or http://eve.physics.ox.ac.uk/QChome.htm