Low-Complexity LP Decoding of Nonbinary Linear Codes

Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has attracted much attention in the research community in the past few years. LP decoding has been derived for binary and nonbinary linear codes. However, the most important problem with LP decoding for both binary and nonbinary linear codes is that the complexity of standard LP solvers such as the simplex algorithm remains prohibitively large for codes of moderate to large block length. To address this problem, two low-complexity LP (LCLP) decoding algorithms for binary linear codes have been proposed by Vontobel and Koetter, henceforth called the basic LCLP decoding algorithm and the subgradient LCLP decoding algorithm. In this paper, we generalize these LCLP decoding algorithms to nonbinary linear codes. The computational complexity per iteration of the proposed nonbinary LCLP decoding algorithms scales linearly with the block length of the code. A modified BCJR algorithm for efficient check-node calculations in the nonbinary basic LCLP decoding algorithm is also proposed, which has complexity linear in the check node degree. Several simulation results are presented for nonbinary LDPC codes defined over Z_4, GF(4), and GF(8) using quaternary phase-shift keying and 8-phase-shift keying, respectively, over the AWGN channel. It is shown that for some group-structured LDPC codes, the error-correcting performance of the nonbinary LCLP decoding algorithms is similar to or better than that of the min-sum decoding algorithm.Comment: To appear in IEEE Transactions on Communications, 201

Simple Rate-1/3 Convolutional and Tail-Biting Quantum Error-Correcting Codes

Simple rate-1/3 single-error-correcting unrestricted and CSS-type quantum convolutional codes are constructed from classical self-orthogonal \F_4-linear and \F_2-linear convolutional codes, respectively. These quantum convolutional codes have higher rate than comparable quantum block codes or previous quantum convolutional codes, and are simple to decode. A block single-error-correcting [9, 3, 3] tail-biting code is derived from the unrestricted convolutional code, and similarly a [15, 5, 3] CSS-type block code from the CSS-type convolutional code.Comment: 5 pages; to appear in Proceedings of 2005 IEEE International Symposium on Information Theor

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

Entanglement Increases the Error-Correcting Ability of Quantum Error-Correcting Codes

If entanglement is available, the error-correcting ability of quantum codes can be increased. We show how to optimize the minimum distance of an entanglement-assisted quantum error-correcting (EAQEC) code, obtained by adding ebits to a standard quantum error-correcting code, over different encoding operators. By this encoding optimization procedure, we found several new EAQEC codes, including a family of [[n, 1, n; n-1]] EAQEC codes for n odd and code parameters [[7, 1, 5; 2]], [[7, 1, 5; 3]], [[9, 1, 7; 4]], [[9, 1, 7; 5]], which saturate the quantum singleton bound for EAQEC codes. A random search algorithm for the encoding optimization procedure is also proposed.Comment: 39 pages, 10 table

Correcting Quantum Errors with Entanglement

We show how entanglement shared between encoder and decoder can simplify the theory of quantum error correction. The entanglement-assisted quantum codes we describe do not require the dual-containing constraint necessary for standard quantum error correcting codes, thus allowing us to ``quantize'' all of classical linear coding theory. In particular, efficient modern classical codes that attain the Shannon capacity can be made into entanglement-assisted quantum codes attaining the hashing bound (closely related to the quantum capacity). For systems without large amounts of shared entanglement, these codes can also be used as catalytic codes, in which a small amount of initial entanglement enables quantum communication.Comment: 17 pages, no figure. To appear in Scienc

A linear construction for certain Kerdock and Preparata codes

The Nordstrom-Robinson, Kerdock, and (slightly modified) Pre\- parata codes are shown to be linear over \ZZ_4, the integers $\bmod~4$. The Kerdock and Preparata codes are duals over \ZZ_4, and the Nordstrom-Robinson code is self-dual. All these codes are just extended cyclic codes over \ZZ_4. This provides a simple definition for these codes and explains why their Hamming weight distributions are dual to each other. First- and second-order Reed-Muller codes are also linear codes over \ZZ_4, but Hamming codes in general are not, nor is the Golay code.Comment: 5 page

Extra Shared Entanglement Reduces Memory Demand in Quantum Convolutional Coding

We show how extra entanglement shared between sender and receiver reduces the memory requirements for a general entanglement-assisted quantum convolutional code. We construct quantum convolutional codes with good error-correcting properties by exploiting the error-correcting properties of an arbitrary basic set of Pauli generators. The main benefit of this particular construction is that there is no need to increase the frame size of the code when extra shared entanglement is available. Then there is no need to increase the memory requirements or circuit complexity of the code because the frame size of the code is directly related to these two code properties. Another benefit, similar to results of previous work in entanglement-assisted convolutional coding, is that we can import an arbitrary classical quaternary code for use as an entanglement-assisted quantum convolutional code. The rate and error-correcting properties of the imported classical code translate to the quantum code. We provide an example that illustrates how to import a classical quaternary code for use as an entanglement-assisted quantum convolutional code. We finally show how to "piggyback" classical information to make use of the extra shared entanglement in the code.Comment: 7 pages, 1 figure, accepted for publication in Physical Review

Enhanced Feedback Iterative Decoding of Sparse Quantum Codes

Decoding sparse quantum codes can be accomplished by syndrome-based decoding using a belief propagation (BP) algorithm.We significantly improve this decoding scheme by developing a new feedback adjustment strategy for the standard BP algorithm. In our feedback procedure, we exploit much of the information from stabilizers, not just the syndrome but also the values of the frustrated checks on individual qubits of the code and the channel model. Furthermore we show that our decoding algorithm is superior to belief propagation algorithms using only the syndrome in the feedback procedure for all cases of the depolarizing channel. Our algorithm does not increase the measurement overhead compared to the previous method, as the extra information comes for free from the requisite stabilizer measurements.Comment: 10 pages, 11 figures, Second version, To be appeared in IEEE Transactions on Information Theor