Structured Random Linear Codes (SRLC): Bridging the Gap between Block and Convolutional Codes

Several types of AL-FEC (Application-Level FEC) codes for the Packet Erasure Channel exist. Random Linear Codes (RLC), where redundancy packets consist of random linear combinations of source packets over a certain finite field, are a simple yet efficient coding technique, for instance massively used for Network Coding applications. However the price to pay is a high encoding and decoding complexity, especially when working on $GF(2^8)$, which seriously limits the number of packets in the encoding window. On the opposite, structured block codes have been designed for situations where the set of source packets is known in advance, for instance with file transfer applications. Here the encoding and decoding complexity is controlled, even for huge block sizes, thanks to the sparse nature of the code and advanced decoding techniques that exploit this sparseness (e.g., Structured Gaussian Elimination). But their design also prevents their use in convolutional use-cases featuring an encoding window that slides over a continuous set of incoming packets. In this work we try to bridge the gap between these two code classes, bringing some structure to RLC codes in order to enlarge the use-cases where they can be efficiently used: in convolutional mode (as any RLC code), but also in block mode with either tiny, medium or large block sizes. We also demonstrate how to design compact signaling for these codes (for encoder/decoder synchronization), which is an essential practical aspect.Comment: 7 pages, 12 figure

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