Expander Codes Research Articles

Generalized Belief Propagation Algorithms for Decoding of Surface Codes

Belief propagation (BP) is well-known as a low complexity decoding algorithm with a strong performance for important classes of quantum error correcting codes, e.g. notably for the quantum low-density parity check (LDPC) code class of random expander codes. However, it is also well-known that the performance of BP breaks down when facing topological codes such as the surface code, where naive BP fails entirely to reach a below-threshold regime, i.e. the regime where error correction becomes useful. Previous works have shown, that this can be remedied by resorting to post-processing decoders outside the framework of BP. In this work, we present a generalized belief propagation method with an outer re-initialization loop that successfully decodes surface codes, i.e. opposed to naive BP it recovers the sub-threshold regime known from decoders tailored to the surface code and from statistical-mechanical mappings. We report a threshold of 17% under independent bit-and phase-flip data noise (to be compared to the ideal threshold of 20.6%) and a threshold value of 14% under depolarizing data noise (compared to the ideal threshold of 18.9%), which are on par with thresholds achieved by non-BP post-processing methods.

Improved Decoding of Expander Codes

We study the classical expander codes, introduced by Sipser and Spielman [1]. Given any constants 0 < α, ε < 1/2, and an arbitrary bipartite graph with <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> vertices on the left, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> < <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> vertices on the right, and left degree <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> such that any left subset <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> of size at most α <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> has at least (1–ε)| <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> | <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</i> neighbors, we show that the corresponding linear code given by parity checks on the right has distance at least roughly αN/2ε. This is strictly better than the best known previous result of 2(1 – ε )α <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> [2, 3] whenever ε < 1/2, and improves the previous result significantly when ε is small. Furthermore, we show that this distance is tight in general, thus providing a complete characterization of the distance of general expander codes. Next, we provide several efficient decoding algorithms, which vastly improve previous results in terms of the fraction of errors corrected, whenever ε < 1/4. Finally, we also give a bound on the list-decoding radius of general expander codes, which beats the classical Johnson bound in certain situations (e.g., when the graph is almost regular and the code has a high rate). Our techniques exploit novel combinatorial properties of bipartite expander graphs. In particular, we establish a new size-expansion tradeoff, which may be of independent interests.

A new formula for the minimum distance of an expander code

An expander code is a binary linear code whose parity-check matrix is the bi-adjacency matrix of a bipartite expander graph. We provide a new formula for the minimum distance of such codes. We also provide a new proof of the result that $2(1-\varepsilon) \gamma n$ is a lower bound of the minimum distance of the expander code given by an $(m,n,d,\gamma,1-\varepsilon)$ expander bipartite graph.

Linear-Time Erasure List-Decoding of Expander Codes

We give a linear-time erasure list-decoding algorithm for expander codes. More precisely, let r > 0 be any integer. Given an inner code <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> of length d, and a d-regular bipartite expander graph G with n vertices on each side, we give an algorithm to list-decode the code <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> = <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> (G, <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> ) of length nd from approximately δδ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> nd erasures in time n·poly (d2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sup> /δ), where δ and δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> are the relative distance and the r'th generalized relative distance of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , respectively. To the best of our knowledge, this is the first linear-time algorithm that can list-decode expander codes from erasures beyond their (designed) distance of approximately δ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> nd. To obtain our results, we show that an approach similar to that of (Hemenway and Wootters, Information and Computation, 2018) can be used to obtain such an erasure-list-decoding algorithm with an exponentially worse dependence of the running time on r and δ; then we show how to improve the dependence of the running time on these parameters.

Constant overhead quantum fault tolerance with quantum expander codes

The threshold theorem is a seminal result in the field of quantum computing asserting that arbitrarily long quantum computations can be performed on a faulty quantum computer provided that the noise level is below some constant threshold. This remarkable result comes at the price of increasing the number of qubits (quantum bits) by a large factor that scales polylogarithmically with the size of the quantum computation we wish to realize. Minimizing the space overhead for fault-tolerant quantum computation is a pressing challenge that is crucial to benefit from the computational potential of quantum devices. In this paper, we study the asymptotic scaling of the space overhead needed for fault-tolerant quantum computation. We show that the polylogarithmic factor in the standard threshold theorem is in fact not needed and that there is a fault-tolerant construction that uses a number of qubits that is only a constant factor more than the number of qubits of the ideal computation. This result was conjectured by Gottesman who suggested to replace the concatenated codes from the standard threshold theorem by quantum error-correcting codes with a constant encoding rate. The main challenge was then to find an appropriate family of quantum codes together with an efficient classical decoding algorithm working even with a noisy syndrome. The efficiency constraint is crucial here: bear in mind that qubits are inherently noisy and that faults keep accumulating during the decoding process. The role of the decoder is therefore to keep the number of errors under control during the whole computation. On a technical level, our main contribution is the analysis of the SMALL-SET-FLIP decoding algorithm applied to the family of quantum expander codes . We show that it can be parallelized to run in constant time while correcting sufficiently many errors on both the qubits and the syndrome to keep the error under control. These tools can be seen as a quantum generalization of the BIT-FLIP algorithm applied to the (classical) expander codes of Sipser and Spielman.

Encryption scheme based on expanded Reed-Solomon codes

We present a code-based public-key cryptosystem, in which we use Reed-Solomon codes over an extension field as secret codes and disguise it by considering its shortened expanded code over the base field. Considering shortened expanded codes provides a safeguard against distinguisher attacks based on the Schur product. Moreover, without using a cyclic or a quasi-cyclic structure we obtain a key size reduction of nearly $45 \%$ compared to the classic McEliece cryptosystem proposed by Bernstein et al.

Optimal quantum subsystem codes in two dimensions

Given any two classical codes with parameters $[n_1,k,d_1]$ and $[n_2,k,d_2]$, we show how to construct a quantum subsystem code in 2-dimensions with parameters $[[N,K,D]]$ satisfying $N\le 2n_1n_2$, $K=k$, and $D=\min(d_1,d_2)$. These quantum codes are in the class of generalized Bacon-Shor codes introduced by Bravyi. We note that constructions of good classical codes can be used to construct quantum codes that saturate Bravyi's bound $KD=O(N)$ on the code parameters of 2-dimensional subsystem codes. One of these good constructions uses classical expander codes. This construction has the additional advantage of a linear time quantum decoder based on the classical Sipser-Spielman flip decoder. Finally, while the subsystem codes we create do not have asymptotic thresholds, we show how they can be gauge-fixed to certain hypergraph product codes that do.

Linear-time list recovery of high-rate expander codes

We show that expander codes, when properly instantiated, are high-rate list recoverable codes with linear-time list recovery algorithms. List recoverable codes have applications to constructing efficiently list-decodable codes, as well as in compressed sensing and group testing. Previous list recoverable codes with linear-time decoding algorithms have all had rate at most 1/2; in contrast, our codes can have rate 1−ε for any ε>0. We can plug our high-rate codes into a construction of Alon and Luby (1996), recently highlighted by Meir (2014) to obtain linear-time list recoverable codes of arbitrary rates R, which approach the optimal trade-off between the number of non-trivial lists provided and the rate of the code.A slight strengthening of our result would imply linear-time and optimally list-decodable codes for all rates. Thus, our result is a step in the direction of solving this important problem.

Fast Decoding of Expander Codes

Expander codes are Tanner codes defined on sparse graphs that have good expansion properties. Sipser and Spielman (1996) showed that there is a linear-time decoding algorithm for expander codes when the vertex expansion is at least 3/4 and the number of errors corrected is a constant fraction of the code length. Later, Feldman et al. (2007) gave a decoding algorithm that allows the expansion to be 2/3 + 1/(3c), where $c$ is the left degree of the underlying bipartite graph, at the expense of polynomial-time decoding complexity. Recently, Viderman (2013) further improved the expansion parameter to $2/3 - 1/(6c)$ , and the decoding algorithm runs in linear time. These results are for expander codes whose inner codes are parity-check codes. By using stronger inner codes, Chilappagari et al. (2010) showed that there is a linear-time decoding algorithm for every vertex expansion greater than 1/2. In this paper, it is shown that for every vertex expansion, there is a linear-time decoding algorithm for expander codes (using inner codes with minimum distance depending on the vertex expansion), and that the number of errors corrected is a constant fraction of the code length.

Associative Memory Using Dictionary Learning and Expander Decoding

An associative memory is a framework of content-addressable memory that stores a collection of message vectors (or a dataset) over a neural network while enabling a neurally feasible mechanism to recover any message in the dataset from its noisy version. Designing an associative memory requires addressing two main tasks: 1) learning phase: given a dataset, learn a concise representation of the dataset in the form of a graphical model (or a neural network), 2) recall phase: given a noisy version of a message vector from the dataset, output the correct message vector via a neurally feasible algorithm over the network learnt during the learning phase. This paper studies the problem of designing a class of neural associative memories which learns a network representation for a large dataset that ensures correction against a large number of adversarial errors during the recall phase. Specifically, the associative memories designed in this paper can store dataset containing exp(n) n-length message vectors over a network with O(n) nodes and can tolerate Ω(n / polylog) adversarial errors. This paper carries out this memory design by mapping the learning phase and recall phase to the tasks of dictionary learning with a square dictionary and iterative error correction in an expander code, respectively.

ICD-10-based expanded code set for use in cleft lip/palate research and surveillance.

On October 1, 2015, the United States required use of the Clinical Modification of the International Classification of Diseases, 10th Revision (ICD-10-CM) for diagnostic coding. The ICD-10-CM code set is limited to gross categories for cleft lip and/or cleft palate (using only four of a possible seven characters). Herein, a clinically useful expansion of the ICD-10-CM code set is proposed to improve the diagnostic accuracy necessary for individual clinical, research, and statistical projects that require it. (This is similar to how the Centers for Disease Control and Prevention/British Paediatric Association Code served to extend the ICD-9 code base.) RESULTS: Our proposed expansion does not replace the required use of ICD-10-CM for clinical, administrative, or financial transactions. Rather, it is offered as an optional set of cleft codes that could be used in parallel to document true classification-level data with phenotypic accuracy. The expanded set is "collapsible" into the official ICD-10-CM codes; this improves compatibility of the expanded codes that would be contained in research and epidemiologic databases with the standard codes from hospital electronic medical record systems and administrative billing data. Birth Defects Research (Part A) 106:905-914, 2016. © 2016 Wiley Periodicals, Inc.

Local correctability of expander codes

In this work, we present the first local-decoding algorithm for expander codes. This yields a new family of constant-rate codes that can recover from a constant fraction of errors in the codeword symbols, and where any symbol of the codeword can be recovered with high probability by reading Nε symbols from the corrupted codeword, where N is the block-length of the code.Expander codes, introduced by Sipser and Spielman, are formed from an expander graph G=(V,E) of degree d, and an inner code of block-length d over an alphabet Σ. Each edge of the expander graph is associated with a symbol in Σ. A string in ΣE will be a codeword if for each vertex in V, the symbols on the adjacent edges form a codeword in the inner code.We show that if the inner code has a smooth reconstruction algorithm in the noiseless setting, then the corresponding expander code has an efficient local-correction algorithm in the noisy setting. Instantiating our construction with inner codes based on finite geometries, we obtain novel locally decodable codes with rate approaching one. This provides an alternative to the multiplicity codes of Kopparty, Saraf and Yekhanin (STOC '11) and the lifted codes of Guo, Kopparty and Sudan (ITCS '13).

Photonic circuits for iterative decoding of a class of low-density parity-check codes

Photonic circuits in which stateful components are coupled via guided electromagnetic fields are natural candidates for resource-efficient implementation of iterative stochastic algorithms based on propagation of information around a graph. Conversely, such message=passing algorithms suggest novel circuit architectures for signal processing and computation that are well matched to nanophotonic device physics. Here, we construct and analyze a quantum optical model of a photonic circuit for iterative decoding of a class of low-density parity-check (LDPC) codes called expander codes. Our circuit can be understood as an open quantum system whose autonomous dynamics map straightforwardly onto the subroutines of an LDPC decoding scheme, with several attractive features: it can operate in the ultra-low power regime of photonics in which quantum fluctuations become significant, it is robust to noise and component imperfections, it achieves comparable performance to known iterative algorithms for this class of codes, and it provides an instructive example of how nanophotonic cavity quantum electrodynamic components can enable useful new information technology even if the solid-state qubits on which they are based are heavily dephased and cannot support large-scale entanglement.

OPTIMAL FOLDING OF DATA FLOW GRAPHS BASED ON FINITE PROJECTIVE GEOMETRY USING VECTOR SPACE PARTITIONING

A number of computations exist, especially in area of error-control coding and matrix computations, whose underlying data flow graphs are based on finite projective-geometry (PG) based balanced bipartite graphs. Many of these applications of finite projective geometry are actively being researched upon, especially in coding theory. Almost all these applications need large bipartite graphs, whose nodes represent parallel computations. To reduce its implementation cost, reducing amount of system/hardware resources during design is an important engineering objective. In this context, we present a scheme to reduce resource utilization while designing systems modeled using PG-based graphs. In such systems, the number of processing units is equal to the number of vertices, each performing an atomic computation. We present a novel way of partitioning the vertex set assigned to various atomic computations, into blocks. Each block of partition is then assigned to a processing unit. A processing unit performs the computations corresponding to the vertices in the block assigned to it in a sequential fashion, thus creating the effect of folding the overall computation. The symmetric properties of projective space lattices enable us to develop a conflict-free communication schedule. We employed the technique of coset decomposition of a finite field for partitioning. The folding scheme achieves the best possible throughput, in lack of any overhead of shuffling data across memories while scheduling another computation on the same processing unit. We first provide a scheme for a finite projective space of dimension five, and the corresponding schedules. This specific scheme is then generalized for arbitrary finite projective spaces. Both the folding schemes have been verified by both simulation as well as hardware prototyping. For example, a semi-parallel decoder architecture for a new class of expander codes was designed and implemented using this scheme, with potential deployment in DVD-R/CD-ROM drives.

Compressive sampling photoacoustic tomography based on edge expander codes and TV regularization

A new photoacoustic (PA) signal sampling and image reconstruction method, called compressive sampling PA tomography (CSPAT), is recently proposed to make low sampling rate and high-resolution PA tomography possible. A key problem within the CSPAT framework is the design of optic masks. We propose to use edge expander codes-based masks instead of the conventional random distribution masks, and efficient total variation (TV) regularization-based model to formulate the associated problem. The edge expander codes-based masks, corresponding to non-uniform sampling schemes, are validated by both theoretical analysis and results from computer simulations. The proposed method is expected to enhance the capability of CSPAT for reducing the number of measurements and fast data acquisition.

OPTIMAL FOLDING OF DATA FLOW GRAPHS BASED ON FINITE PROJECTIVE GEOMETRY USING VECTOR SPACE PARTITIONING

A number of computations exist, especially in area of error-control coding and matrix computations, whose underlying data flow graphs are based on finite projective-geometry (PG)-based balanced bipartite graphs. Many of these applications of finite projective geometry are actively being researched upon, especially in coding theory. Almost all these applications need large bipartite graphs, whose nodes represent parallel computations. To reduce its implementation cost, reducing amount of system/hardware resources during design is an important engineering objective. In this context, we present a scheme to reduce resource utilization while designing systems modeled using PG-based graphs. In such systems, the number of processing units is equal to the number of vertices, each performing an atomic computation. We present a novel way of partitioning the vertex set assigned to various atomic computations, into blocks. Each block of partition is then assigned to a processing unit. A processing unit performs the computations corresponding to the vertices in the block assigned to it in a sequential fashion, thus creating the effect of folding the overall computation. The symmetric properties of projective space lattices enable us to develop a conflict-free communication schedule. We employed the technique of coset decomposition of a finite field for partitioning. The folding scheme achieves the best possible throughput, in lack of any overhead of shuffling data across memories while scheduling another computation on the same processing unit. We first provide a scheme for a finite projective space of dimension five, and the corresponding schedules. This specific scheme is then generalized for arbitrary finite projective spaces. Both the folding schemes have been verified by both simulation as well as hardware prototyping. For example, a semi-parallel decoder architecture for a new class of expander codes was designed and implemented using this scheme, with potential deployment in DVD-R/CD-ROM drives.

Composition of semi-LTCs by two-wise tensor products

In this paper, we obtain a composition theorem that allows us to construct locally testable codes (LTCs) by repeated two-wise tensor products. This is the first composition theorem showing that repeating the two-wise tensor operation any constant number of times still results in a locally testable code, improving upon previous results which only worked when the tensor product was applied once. To obtain our results, we define a new tester for tensor products. Our tester uses the distribution of the "inner tester" associated with the base code to sample rows and columns of the product code. This construction differs from previously studied testers for tensor product codes which sampled rows and columns uniformly. We show that if the base code is any LTC or any expander code, then the code obtained by taking the repeated two-wise tensor product of the base code with itself is locally testable. In particular, this answers a question posed in the paper of Dinur et al. (2006) by expanding the class of allowed base codes to include all LTCs and not just so-called uniform LTCs whose associated tester queries all codeword entries with equal probability. One corollary of our composition theorem is a simple construction of high-rate LTCs with sublinear query complexity. More formally, we show that for every ?, ? > 0 there exists a family of LTCs over the binary field with query complexity n? and rate at least 1 ? ?.

Linear-time decoding of regular expander codes

Sipser and Spielman (IEEE IT, [1996]) showed that any c , d )-regular expander code with expansion parameter &gt;¾ is decodable in linear time from a constant fraction of errors. Feldman et al. (IEEE IT, [2007]) proved that expansion parameter &gt;⅔ + 1/3c is sufficient to correct a constant fraction of errors in polynomial time using LP decoding. In this work, we give a simple combinatorial algorithm that achieves even better parameters. In particular, our algorithm runs in linear time and works for any expansion parameter &gt;⅔ − 1/6c. We also prove that our decoding algorithm can be executed in logarithmic time on a linear number of parallel processors.

LP decoding of codes with expansion parameter above 2/3

A code C⊆F2n is a (c,δ,ϵ)-expander code if it has a Tanner graph, where every variable node has degree c, and every subset of variable nodes L0 such that |L0|⩽δn has at least ϵc|L0| neighbors.Feldman et al. (2007) [3] proved that LP decoding corrects 3ϵ−22ϵ−1⋅(δn−1) errors of (c,δ,ϵ)-expander code, where ϵ>23+13c.In this paper, we provide a slight consolidation of their work and show that this result holds for every expansion parameter ϵ>23.

Correcting a Fraction of Errors in Nonbinary Expander Codes With Linear Programming

A linear-programming decoder for \emph{nonbinary} expander codes is presented. It is shown that the proposed decoder has the maximum-likelihood certificate properties. It is also shown that this decoder corrects any pattern of errors of a relative weight up to approximately 1/4 \delta_A \delta_B (where \delta_A and \delta_B are the relative minimum distances of the constituent codes).