Aziz Mouzali, Fatiha Merazka
In this paper, we provise an implementation of five, seven and nine-qubits error correcting codes on a classical computer using the quantum simulator Feynman program. We also compare the three codes by computing the fidelity when double errors occurs in a depolarizing channel. As triple errors and more are considered very unlikely, it has negligible effect on the next results
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Performing active quantum error correction to protect fragile quantum states highly depends on the correctness of error information--error syndromes. To obtain reliable error syndromes using imperfect physical circuits, we propose the idea of quantum data-syndrome (DS) codes that are capable of correcting both data qubits and syndrome bits errors. We study fundamental properties of quantum DS codes and provide several CSS-type code constructions of quantum DS codes.
Short Blocklength Error Correction Codes for Continuous-Variable Quantum Key Distribution
提出一种两步纠错方案，允许在连续变量量子密钥分发中使用短块长纠错码，从而在140公里距离上将密钥速率提升至7.3倍。
The hull of linear codes plays an important role in quantum information and coding theory. In the present paper, by investigating the Galois hulls of generalized Reed-Solomon (GRS) codes and extended GRS codes over the finite field Fq, we give several new families of MDS codes with Galois hulls of arbitrary dimensions that are not obtained before. Some of them generalize the ones in the literature [13]. As a consequence, using these MDS codes with Galois hulls of arbitrary dimensions, we construct nine new families of MDS entanglement-assisted quantum error-correcting codes (EAQECCs) with flex…
Advancing Finite-Length Quantum Error Correction Using Generalized Bicycle Codes
本文研究广义自行车码在有限长度下的量子纠错性能，证明其通过灵活构造和高行权重设计可达到或超越量子Tanner码等主流码族，并兼顾高速率与高效译码。
In this paper, for the purposes of information transmission and network error correction simultaneously, three classes of important linear network codes in network coding, linear multicast/broadcast/dispersion codes are generalized to linear network error correction coding, i.e., linear network error correction multicast/broadcast/dispersion codes. We further propose the (weakly, strongly) extended Singleton bounds for these new classes of codes, and define the optimal codes satisfying the corresponding Singleton bounds with equality, which are called multicast/broadcast/dispersion MDS codes r…
In this paper, we consider quantum error correction over depolarizing channels with non-binary low-density parity-check codes defined over Galois field of size $2^p$ . The proposed quantum error correcting codes are based on the binary quasi-cyclic CSS (Calderbank, Shor and Steane) codes. The resulting quantum codes outperform the best known quantum codes and surpass the performance limit of the bounded distance decoder. By increasing the size of the underlying Galois field, i.e., $2^p$, the error floors are considerably improved.
Recently, locally repairable codes (LRCs) with local erasure correction constraints that are unequal and disjoint have been proposed. In this work, we study the same topic and provide some improved and additional results.
We establish several relations between quantum error correction (QEC) and tensor network (TN) methods of quantum many-body physics. We exhibit correspondences between well-known families of QEC codes and TNs, and demonstrate a formal equivalence between decoding a QEC code and contracting a TN. We build on this equivalence to propose a new family of quantum codes and decoding algorithms that generalize and improve upon quantum polar codes and successive cancellation decoding in a natural way.
In this paper, two upper bounds on the achievable code rate of linear block codes for multiple phased-burst correction (MPBC) are presented. One bound is constrained to a maximum correctable cyclic burst length within every subblock, or equivalently a constraint on the minimum error free length or gap within every phased-burst. This bound, when reduced to the special case of a bound for single burst correction (SBC), is shown to be the Abramson bound when the cyclic burst length is less than half the block length. The second MPBC bound is developed without the minimum error free gap constraint…
On Minimum Distances for Error Correction and Detection of Generalized Network Code
研究了通用网络码中纠错与检测的最小距离差异，提出了通用网络通道和编码方案，系统定义了纠错和检测距离，并通过引入距离及其细化版本，完成对联合纠错和检测的完整表征。
Generalized Concatenated (GC), also known as Integrated Interleaved (II) Codes, are studied from an erasure correction point of view making them useful for Redundant Arrays of Independent Disks (RAID) types of architectures combining global and local properties. The fundamental erasure-correcting properties of the codes are proven and efficient encoding and decoding algorithms are provided. Although less powerful than the recently developed PMDS codes, this implementation has the advantage of allowing generalization to any range of parameters while the size of the field is much smaller than th…
A new class of exact-repair regenerating codes is constructed by combining two layers of erasure correction codes together with combinatorial block designs, e.g., Steiner systems, balanced incomplete block designs and t-designs. The proposed codes have the "uncoded repair" property where the nodes participating in the repair simply transfer part of the stored data directly, without performing any computation. The layered error correction structure makes the decoding process rather straightforward, and in general the complexity is low. We show that this construction is able to achieve performan…
In this paper, we investigate an artificial-intelligence (AI) driven approach to design error correction codes (ECC). Classic error correction code was designed upon coding theory that typically defines code properties (e.g., hamming distance, subchannel reliability, etc.) to reflect code performance. Its code design is to optimize code properties. However, an AI-driven approach doesn't necessarily rely on coding theory any longer. Specifically, we propose a constructor-evaluator framework, in which the code constructor is realized by AI algorithms and the code evaluator provides code performa…
Recently, network error correction coding (NEC) has been studied extensively. Several bounds in classical coding theory have been extended to network error correction coding, especially the Singleton bound. In this paper, following the research line using the extended global encoding kernels proposed in \cite{zhang-correction}, the refined Singleton bound of NEC can be proved more explicitly. Moreover, we give a constructive proof of the attainability of this bound and indicate that the required field size for the existence of network maximum distance separable (MDS) codes can become smaller f…
The relation between the girth and the error correction capability of column-weight-three LDPC codes is investigated. Specifically, it is shown that the Gallager A algorithm can correct $g/2-1$ errors in $g/2$ iterations on a Tanner graph of girth $g \geq 10$.
In the practical network communications, many internal nodes in the network are required to not only transmit messages but decode source messages. For different applications, four important classes of linear network codes in network coding theory, i.e., linear multicast, linear broadcast, linear dispersion, and generic network codes, have been studied extensively. More generally, when channels of communication networks are noisy, information transmission and error correction have to be under consideration simultaneously, and thus these four classes of linear network codes are generalized to li…
Use of an error correction code in a given transmission channel can be regarded as the statistical experiment. Therefore, powerful results from the theory of comparison of experiments can be applied to compare the performances of different error correction codes. We present results on the comparison of block error correction codes using the representation of error correction code as a linear experiment. In this case the code comparison is based on the Loewner matrix ordering of respective code matrices. Next, we demonstrate the bit-error rate code performance comparison based on the representa…
We present an asymptotic limit between correctable and uncor-rectable errors on the Reed-Muller codes of any order. This limit is theoretical and does not depend of any decoding algorithm.
In this paper, we investigate the error correction capability of column-weight-three LDPC codes when decoded using the Gallager A algorithm. We prove that the necessary condition for a code to correct $k \geq 5$ errors is to avoid cycles of length up to $2k$ in its Tanner graph. As a consequence of this result, we show that given any $α>0, \exists N $ such that $\forall n>N$, no code in the ensemble of column-weight-three codes can correct all $αn$ or fewer errors. We extend these results to the bit flipping algorithm.
We correct errors in Section III-B of the above-titled paper. The corrections are related to the joint distribution of the inputs to the main decoder which corresponds to a branch in the code trellis for the main decoder. Note that as far as the distribution of the input corresponding to a single code symbol is concerned, the results in Section III-B are correct. We have noticed that alpha in Proposition 12 (beta in Proposition 14) has another meaning, which has made it possible to derive the joint distribution of the inputs corresponding to the whole branch. Corrections are based on this impo…
In this work, we introduce convolutional codes for network-error correction in the context of coherent network coding. We give a construction of convolutional codes that correct a given set of error patterns, as long as consecutive errors are separated by a certain interval. We also give some bounds on the field size and the number of errors that can get corrected in a certain interval. Compared to previous network error correction schemes, using convolutional codes is seen to have advantages in field size and decoding technique. Some examples are discussed which illustrate the several possibl…
We investigate the relation between the girth and the guaranteed error correction capability of $γ$-left regular LDPC codes when decoded using the bit flipping (serial and parallel) algorithms. A lower bound on the number of variable nodes which expand by a factor of at least $3 γ/4$ is found based on the Moore bound. An upper bound on the guaranteed correction capability is established by studying the sizes of smallest possible trapping sets.
Coherent network error correction is the error-control problem in network coding with the knowledge of the network codes at the source and sink nodes. With respect to a given set of local encoding kernels defining a linear network code, we obtain refined versions of the Hamming bound, the Singleton bound and the Gilbert-Varshamov bound for coherent network error correction. Similar to its classical counterpart, this refined Singleton bound is tight for linear network codes. The tightness of this refined bound is shown by two construction algorithms of linear network codes achieving this bound.…
In this work, we develop a space-time block code for noncoherent communication using techniques from the field of quantum error correction. We decompose the multiple-input multiple-output (MIMO) channel into operators from quantum mechanics, and design a non-coherent space time code using the quantum stabilizer formalism. We derive an optimal decoder, and analyze the former through a quantum mechanical lens. We compare our approach to a comparable coherent approach and a noncoherent differential approach, achieving comparable or better performance.
CRC Codes as Error Correction Codes
本文证明借助GRAND及其变体可将CRC码用于低时延纠错，短CRC码性能不逊于BCH码并优于Polar/CA-Polar码，且无需改造现有编码器。
The rapidly improving performance of modern hardware renders convolutional codes obsolete, and allows for the practical implementation of more sophisticated correction codes such as low density parity check (LDPC) and turbo codes (TC). Both are decoded by iterative algorithms, which require a disproportional computational effort for low channel noise. They are also unable to correct higher noise levels, still below the Shannon theoretical limit. In this paper, we discuss an enhanced version of a convolutional-like decoding paradigm which adopts very large spaces of possible system states, of t…
We consider data transmission over a network where each edge is an erasure channel and where the inner nodes transmit a random linear combination of their incoming information. We distinguish two channel models in this setting, the row and the column erasure channel model. For both models we derive the symbol erasure correction capabilities of spread codes and compare them to other known codes suitable for those models. Furthermore, we explain how to decode these codes in the two channel models and compare their decoding complexities. The results show that, depending on the application and the…
Linear codes over finite extension fields have widespread applications in theory and practice. In some scenarios, the decoder has a sequential access to the codeword symbols, giving rise to a hierarchical erasure structure. In this paper we develop a mathematical framework for hierarchical erasures over extension fields, provide several bounds and constructions, and discuss potential applications in distributed storage and flash memories. Our results show intimate connection to Universally Decodable Matrices, as well as to Reed-Solomon and Gabidulin codes.
This article is focused on some variations of Reed-Muller codes that yield improvements to the rate for a prescribed decoding performance under the Berlekamp-Massey-Sakata algorithm with majority voting. Explicit formulas for the redundancies of the new codes are given.
In network communication, the source often transmits messages at several different information rates within a session. How to deal with information transmission and network error correction simultaneously under different rates is introduced in this paper as a variable-rate network error correction problem. Apparently, linear network error correction MDS codes are expected to be used for these different rates. For this purpose, designing a linear network error correction MDS code based on the existing results for each information rate is an efficient solution. In order to solve the problem more…
For communication systems with heavy burst noise, an optimal Forward Error Correction (FEC) scheme is expected to have a large burst error correction capacity while simultaneously owning moderate random error correction capability. This letter presents a new FEC scheme based on multiple-symbol interleaved Reed-Solomon codes and an associated two-pass decoding algorithm. It is shown that the proposed multi-symbol interleaved coding scheme can achieve nearly twice as much as the burst error correction capability of conventional symbol-interleaved Reed-Solomon codes with the same code length and …
Cosine Annealing Optimized Denoising Diffusion Error Correction Codes
本文提出用余弦退火优化去噪扩散纠错码，通过在反向扩散中调整方差，降低长码字解码的误码率并提升迭代效率。
In practical communication and computation systems, errors occur predominantly in adjacent positions rather than in a random manner. In this paper, we develop a stabilizer formalism for quantum burst error correction codes (QBECC) to combat such error patterns in the quantum regime. Our contributions are as follows. Firstly, we derive an upper bound for the correctable burst errors of QBECCs, the quantum Reiger bound (QRB). This bound generalizes the quantum Singleton bound for standard quantum error correction codes (QECCs). Secondly, we propose two constructions of QBECCs: one by heuristic c…
Recently Reed-Solomon (RS) codes were shown to possess a repair scheme that supports repair of failed nodes with optimal repair bandwidth. In this paper, we extend this result in two directions. First, we propose a new repair scheme for the RS codes constructed in [Tamo-Ye-Barg, {\em IEEE Transactions on Information Theory}, vol. 65, May 2019] and show that our new scheme is robust to erroneous information provided by the helper nodes while maintaining the optimal repair bandwidth. Second, we construct a new family of RS codes with optimal access for the repair of any single failed node. We al…
In this paper we study codes for correcting deletable errors in binary words, where each bit is either retained, substituted, erased or deleted and the total number of errors is much smaller compared to the length of the codeword. We construct codes capable of correcting errors in the received codeword in real-time with small delay and low probability of decoding error.
DNA-Correcting Codes: End-to-end Correction in DNA Storage Systems
本文提出DNA校正码，通过集成聚类、重建和纠错三个检索步骤，并引入新距离度量，给出充分必要条件、界与构造，确保DNA存储系统输出唯一。
Homomorphic Quantum Error Correction
研究同态量子加密与量子纠错的代数兼容性，给出稳定子码保持加密编码态在码空间内的充要条件，并针对非Clifford门实现提出两种解决方案。
A New Class of Geometric Analog Error Correction Codes for Crossbar Based In-Memory Computing
针对电阻交叉开关上的模拟内存计算，提出一类几何模拟纠错码，通过几何分析表征其m-高度分布，可处理多个异常值。
Physical Unclonable Functions can be used for secure key generation in cryptographic applications. It is explained how methods from coding theory must be applied in order to ensure reliable key regeneration. Based on previous work, we show ways how to obtain better results with respect to error probability and codeword length. Also, an example based on Generalized Concatenated codes is given, which improves upon used coding schemes for PUFs.
The relation between the girth and the guaranteed error correction capability of $γ$-left regular LDPC codes when decoded using the bit flipping (serial and parallel) algorithms is investigated. A lower bound on the size of variable node sets which expand by a factor of at least $3 γ/4$ is found based on the Moore bound. An upper bound on the guaranteed error correction capability is established by studying the sizes of smallest possible trapping sets. The results are extended to generalized LDPC codes. It is shown that generalized LDPC codes can correct a linear fraction of errors under the p…
Motivated by applications to distributed storage, Gopalan \textit{et al} recently introduced the interesting notion of information-symbol locality in a linear code. By this it is meant that each message symbol appears in a parity-check equation associated with small Hamming weight, thereby enabling recovery of the message symbol by examining a small number of other code symbols. This notion is expanded to the case when all code symbols, not just the message symbols, are covered by such "local" parity. In this paper, we extend the results of Gopalan et. al. so as to permit recovery of an erased…
Arithmetic Operators over Finite Field GF($2^m$) for Error Correction Codes Application
本文系统考察了有限域GF(2^m)上的除法、乘法、幂运算、乘法逆元、加法和减法等算术运算，阐明其对实现BCH码和Reed-Solomon码纠错技术的关键作用。
Error correction is an indispensable component when Physical Unclonable Functions (PUFs) are used in cryptographic applications. So far, there exist schemes that obtain helper data, which they need within the error correction process. We introduce a new scheme, which only uses an error correcting code without any further helper data. The main idea is to construct for each PUF instance an individual code which contains the initial PUF response as codeword. In this work we use LDPC codes, however other code classes are also possible. Our scheme allows a trade-off between code rate and cryptograp…
Codes for rank modulation have been recently proposed as a means of protecting flash memory devices from errors. We study basic coding theoretic problems for such codes, representing them as subsets of the set of permutations of $n$ elements equipped with the Kendall tau distance. We derive several lower and upper bounds on the size of codes. These bounds enable us to establish the exact scaling of the size of optimal codes for large values of $n$. We also show the existence of codes whose size is within a constant factor of the sphere packing bound for any fixed number of errors.
We propose (k,k') stabilizing codes, which is a type of delayless error correction codes that are useful for control over networks with erasures. For each input symbol, k output symbols are generated by the stabilizing code. Receiving any k' of these outputs guarantees stability. Thus, the system to be stabilized is taken into account in the design of the erasure codes. Our focus is on LTI systems, and we construct codes based on independent encodings and multiple descriptions. The theoretical efficiency and performance of the codes are assessed, and their practical performances are demonstrat…
We consider rank modulation codes for flash memories that allow for handling arbitrary charge-drop errors. Unlike classical rank modulation codes used for correcting errors that manifest themselves as swaps of two adjacently ranked elements, the proposed \emph{translocation rank codes} account for more general forms of errors that arise in storage systems. Translocations represent a natural extension of the notion of adjacent transpositions and as such may be analyzed using related concepts in combinatorics and rank modulation coding. Our results include derivation of the asymptotic capacity o…
Convolutional codes are error-correcting linear codes that utilize shift registers to encode. These codes have an arbitrary block size and they can incorporate both past and current information bits. DNA codes represent DNA sequences and are defined as sets of words comprised of the alphabet A, C, T, G satisfying certain mathematical bounds and constraints. The application of convolutional code models to DNA codes is a growing field of biocomputation. As opposed to block codes, convolutional codes factor in nearby information bits, which makes them an optimal model for representing biological …
A conjugate code pair is defined as a pair of linear codes either of which contains the dual of the other. A conjugate code pair represents the essential structure of the corresponding Calderbank-Shor-Steane (CSS) quantum error-correcting code. It is known that conjugate code pairs are applicable to quantum cryptography. In this work, a polynomial construction of conjugate code pairs is presented. The constructed pairs achieve the highest known achievable rate on additive channels, and are decodable with algorithms of polynomial complexity.
Polar codes are a class of capacity achieving error correcting codes that has been recently selected for the next generation of wireless communication standards (5G). Polar code decoding algorithms have evolved in various directions, striking different balances between error-correction performance, speed and complexity. Successive-cancellation list (SCL) and its incarnations constitute a powerful, well-studied set of algorithms, in constant improvement. At the same time, different implementation approaches provide a wide range of area occupations and latency results. 5G puts a focus on improve…
A single source network is said to be memory-free if all of the internal nodes (those except the source and the sinks) do not employ memory but merely send linear combinations of the symbols received at their incoming edges on their outgoing edges. In this work, we introduce network-error correction for single source, acyclic, unit-delay, memory-free networks with coherent network coding for multicast. A convolutional code is designed at the source based on the network code in order to correct network-errors that correspond to any of a given set of error patterns, as long as consecutive errors…
We investigate a novel class of quantum error correcting codes to correct errors on both qubits and higher-state quantum systems represented as qudits. These codes arise from an original graph-theoretic representation of sets of quantum errors. In this new framework, we represent the algebraic conditions for error correction in terms of edge avoidance between graphs providing a visual representation of the interplay between errors and error correcting codes. Most importantly, this framework supports the development of quantum codes that correct against a predetermined set of errors, in contras…
Iterative decoders used for decoding low-density parity-check (LDPC) and moderate-density parity-check (MDPC) codes are not characterized by a deterministic decoding radius and their error rate performance is usually assessed through intensive Monte Carlo simulations. However, several applications, like code-based cryptography, need guaranteed low values of the error rate, which are infeasible to assess through simulations, thus requiring the development of theoretical models for the error rate of these codes under iterative decoding. Some models of this type already exist, but become computat…
We propose $(k,k')$ stabilizing codes, which is a type of delayless error correction codes that are useful for control over networks with erasures. For each input symbol, $k$ output symbols are generated by the stabilizing code. Receiving at least $k'$ of these outputs guarantees stability. Thus, both the system to be stabilized and the channel is taken into account in the design of the erasure codes. Receiving more than $k'$ outputs further improves the performance of the system. In the case of i.i.d. erasures, we further demonstrate that the erasure code can be constructed such that stabilit…
The deletion channel is known to be a notoriously diffcult channel to design error-correction codes for. In spite of this difficulty, there are some beautiful code constructions which give some intuition about the channel and about what good deletion codes look like. In this tutorial we will take a look at some of them. This document is a transcript of my talk at the coding theory reading group on some interesting works on deletion channel. It is not intended to be an exhaustive survey of works on deletion channel, but more as a tutorial to some of the important and cute ideas in this area. Fo…
Unraveling codes: fast, robust, beyond-bound error correction for DRAM
本文提出解缠里德-所罗门码，结合交织码的速度与全块码的鲁棒性，实现DRAM超越界限的快速可靠纠错。
One of the main obstacles to the wider use of the modern error-correction codes is that, due to the complex behavior of their decoding algorithms, no systematic method which would allow characterization of the Bit-Error-Rate (BER) is known. This is especially true at the weak noise where many systems operate and where coding performance is difficult to estimate because of the diminishingly small number of errors. We show how the instanton method of physics allows one to solve the problem of BER analysis in the weak noise range by recasting it as a computationally tractable minimization problem…
Barcodes like QR Codes have made that encoded messages have entered our everyday life, what suggests to attach them a second layer of information: directly available to human receiver for informational or marketing purposes. We will discuss a general problem of using codes with chosen statistical constrains, for example reproducing given grayscale picture using halftone technique. If both sender and receiver know these constrains, the optimal capacity can be easily approached by entropy coder. The problem is that this time only the sender knows them - we will refer to these scenarios as constr…
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