A Class of Quantum LDPC Codes Constructed From Finite Geometries

Low-density parity check (LDPC) codes are a significant class of classical codes with many applications. Several good LDPC codes have been constructed using random, algebraic, and finite geometries approaches, with containing cycles of length at least six in their Tanner graphs. However, it is impossible to design a self-orthogonal parity check matrix of an LDPC code without introducing cycles of length four. In this paper, a new class of quantum LDPC codes based on lines and points of finite geometries is constructed. The parity check matrices of these codes are adapted to be self-orthogonal with containing only one cycle of length four. Also, the column and row weights, and bounds on the minimum distance of these codes are given. As a consequence, the encoding and decoding algorithms of these codes as well as their performance over various quantum depolarizing channels will be investigated.Comment: 5pages, 2 figure

Iterative Decoding and Turbo Equalization: The Z-Crease Phenomenon

Iterative probabilistic inference, popularly dubbed the soft-iterative paradigm, has found great use in a wide range of communication applications, including turbo decoding and turbo equalization. The classic approach of analyzing the iterative approach inevitably use the statistical and information-theoretical tools that bear ensemble-average flavors. This paper consider the per-block error rate performance, and analyzes it using nonlinear dynamical theory. By modeling the iterative processor as a nonlinear dynamical system, we report a universal "Z-crease phenomenon:" the zig-zag or up-and-down fluctuation -- rather than the monotonic decrease -- of the per-block errors, as the number of iteration increases. Using the turbo decoder as an example, we also report several interesting motion phenomenons which were not previously reported, and which appear to correspond well with the notion of "pseudo codewords" and "stopping/trapping sets." We further propose a heuristic stopping criterion to control Z-crease and identify the best iteration. Our stopping criterion is most useful for controlling the worst-case per-block errors, and helps to significantly reduce the average-iteration numbers.Comment: 6 page

Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms

Mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches, challenging established but often heuristic algorithms based on iterative message passing. Concepts from mathematical programming used in the context of decoding include linear, integer, and nonlinear programming, network flows, notions of duality as well as matroid and polyhedral theory. This survey article reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory. Published July 201

Stopping Set Distributions of Some Linear Codes

Stopping sets and stopping set distribution of an low-density parity-check code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let $C$ be a binary $[n,k]$ linear code with parity-check matrix $H$, where the rows of $H$ may be dependent. A stopping set $S$ of $C$ with parity-check matrix $H$ is a subset of column indices of $H$ such that the restriction of $H$ to $S$ does not contain a row of weight one. The stopping set distribution $\{T_i(H)\}_{i=0}^n$ enumerates the number of stopping sets with size $i$ of $C$ with parity-check matrix $H$. Note that stopping sets and stopping set distribution are related to the parity-check matrix $H$ of $C$. Let $H^{*}$ be the parity-check matrix of $C$ which is formed by all the non-zero codewords of its dual code $C^{\perp}$. A parity-check matrix $H$ is called BEC-optimal if $T_i(H)=T_i(H^*), i=0,1,..., n$ and $H$ has the smallest number of rows. On the BEC, iterative decoder of $C$ with BEC-optimal parity-check matrix is an optimal decoder with much lower decoding complexity than the exhaustive decoder. In this paper, we study stopping sets, stopping set distributions and BEC-optimal parity-check matrices of binary linear codes. Using finite geometry in combinatorics, we obtain BEC-optimal parity-check matrices and then determine the stopping set distributions for the Simplex codes, the Hamming codes, the first order Reed-Muller codes and the extended Hamming codes.Comment: 33 pages, submitted to IEEE Trans. Inform. Theory, Feb. 201

Self-concatenated code design and its application in power-efficient cooperative communications

In this tutorial, we have focused on the design of binary self-concatenated coding schemes with the help of EXtrinsic Information Transfer (EXIT) charts and Union bound analysis. The design methodology of future iteratively decoded self-concatenated aided cooperative communication schemes is presented. In doing so, we will identify the most important milestones in the area of channel coding, concatenated coding schemes and cooperative communication systems till date and suggest future research directions

The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure

Powerful Quantum Error Correction Codes (QECCs) are required for stabilizing and protecting fragile qubits against the undesirable effects of quantum decoherence. Similar to classical codes, hashing bound approaching QECCs may be designed by exploiting a concatenated code structure, which invokes iterative decoding. Therefore, in this paper we provide an extensive step-by-step tutorial for designing EXtrinsic Information Transfer (EXIT) chart aided concatenated quantum codes based on the underlying quantum-to-classical isomorphism. These design lessons are then exemplified in the context of our proposed Quantum Irregular Convolutional Code (QIRCC), which constitutes the outer component of a concatenated quantum code. The proposed QIRCC can be dynamically adapted to match any given inner code using EXIT charts, hence achieving a performance close to the hashing bound. It is demonstrated that our QIRCC-based optimized design is capable of operating within 0.4 dB of the noise limit