Error-Correction Coding and Decoding: Bounds, Codes, Decoders, Analysis and Applications

Coding; Communications; Engineering; Networks; Information Theory; Algorithm

A STUDY OF ERASURE CORRECTING CODES

This work focus on erasure codes, particularly those that of high performance, and the related decoding algorithms, especially with low computational complexity. The work is composed of different pieces, but the main components are developed within the following two main themes. Ideas of message passing are applied to solve the erasures after the transmission. Efficient matrix-representation of the belief propagation (BP) decoding algorithm on the BEG is introduced as the recovery algorithm. Gallager's bit-flipping algorithm are further developed into the guess and multi-guess algorithms especially for the application to recover the unsolved erasures after the recovery algorithm. A novel maximum-likelihood decoding algorithm, the In-place algorithm, is proposed with a reduced computational complexity. A further study on the marginal number of correctable erasures by the In-place algoritinn determines a lower bound of the average number of correctable erasures. Following the spirit in search of the most likable codeword based on the received vector, we propose a new branch-evaluation- search-on-the-code-tree (BESOT) algorithm, which is powerful enough to approach the ML performance for all linear block codes. To maximise the recovery capability of the In-place algorithm in network transmissions, we propose the product packetisation structure to reconcile the computational complexity of the In-place algorithm. Combined with the proposed product packetisation structure, the computational complexity is less than the quadratic complexity bound. We then extend this to application of the Rayleigh fading channel to solve the errors and erasures. By concatenating an outer code, such as BCH codes, the product-packetised RS codes have the performance of the hard-decision In-place algorithm significantly better than that of the soft-decision iterative algorithms on optimally designed LDPC codes

Multi-Trial Guruswami–Sudan Decoding for Generalised Reed–Solomon Codes

An iterated refinement procedure for the Guruswami--Sudan list decoding algorithm for Generalised Reed--Solomon codes based on Alekhnovich's module minimisation is proposed. The method is parametrisable and allows variants of the usual list decoding approach. In particular, finding the list of \emph{closest} codewords within an intermediate radius can be performed with improved average-case complexity while retaining the worst-case complexity.Comment: WCC 2013 International Workshop on Coding and Cryptography (2013