Deriving Good LDPC Convolutional Codes from LDPC Block Codes

Low-density parity-check (LDPC) convolutional codes are capable of achieving excellent performance with low encoding and decoding complexity. In this paper we discuss several graph-cover-based methods for deriving families of time-invariant and time-varying LDPC convolutional codes from LDPC block codes and show how earlier proposed LDPC convolutional code constructions can be presented within this framework. Some of the constructed convolutional codes significantly outperform the underlying LDPC block codes. We investigate some possible reasons for this "convolutional gain," and we also discuss the --- mostly moderate --- decoder cost increase that is incurred by going from LDPC block to LDPC convolutional codes.Comment: Submitted to IEEE Transactions on Information Theory, April 2010; revised August 2010, revised November 2010 (essentially final version). (Besides many small changes, the first and second revised versions contain corrected entries in Tables I and II.

Convolutional and tail-biting quantum error-correcting codes

Rate-(n-2)/n unrestricted and CSS-type quantum convolutional codes with up to 4096 states and minimum distances up to 10 are constructed as stabilizer codes from classical self-orthogonal rate-1/n F_4-linear and binary linear convolutional codes, respectively. These codes generally have higher rate and less decoding complexity than comparable quantum block codes or previous quantum convolutional codes. Rate-(n-2)/n block stabilizer codes with the same rate and error-correction capability and essentially the same decoding algorithms are derived from these convolutional codes via tail-biting.Comment: 30 pages. Submitted to IEEE Transactions on Information Theory. Minor revisions after first round of review

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

Convolutional coding techniques for data protection

Results of research on the use of convolutional codes in data communications are presented. Convolutional coding fundamentals are discussed along with modulation and coding interaction. Concatenated coding systems and data compression with convolutional codes are described