Manabu Hagiwara, Justin Kong
We present new applications on $q$-binomials, also known as Gaussian binomial coefficients. Our main theorems determine cardinalities of certain error-correcting codes based on Varshamov-Tenengolts codes and prove a curious phenomenon relating to deletion sphere for specific cases.
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The deletion channel is the simplest point-to-point communication channel that models lack of synchronization. Input bits are deleted independently with probability d, and when they are not deleted, they are not affected by the channel. Despite significant effort, little is known about the capacity of this channel, and even less about optimal coding schemes. In this paper we develop a new systematic approach to this problem, by demonstrating that capacity can be computed in a series expansion for small deletion probability. We compute three leading terms of this expansion, and find an input di…
In this paper, we propose a polar coding based scheme for set reconciliation between two network nodes. The system is modeled as a well-known Slepian-Wolf setting induced by a fixed number of deletions. The set reconciliation process is divided into two phases: 1) a deletion polar code is employed to help one node to identify the possible deletion indices, which may be larger than the number of genuine deletions; 2) a lossless compression polar code is then designed to feedback those indices with minimum overhead. Our scheme can be viewed as a generalization of polar codes to some emerging net…
Coding for Composite DNA to Correct Substitutions, Strand Losses, and Deletions
本文针对复合DNA存储中的替换、链丢失和删除错误，提出编码技术，推导非渐近上界并给出可达界的显式构造。
Motivated by the sequence reconstruction problem from traces in DNA-based storage, we consider the problem of designing codes for the deletion channel when multiple observations (or traces) are available to the decoder. We propose simple binary and non-binary codes based on Varshamov-Tenengolts (VT) codes. The proposed codes split the codeword in blocks and employ a VT code in each block. The availability of multiple traces helps the decoder to identify deletion-free copies of a block, and to avoid mis-synchronization while decoding. The encoding complexity of the proposed scheme is linear in …
Quantum deletions, which are harder to correct than erasure errors, occur in many realistic settings. It is therefore pertinent to develop quantum coding schemes for quantum deletion channels. To date, not much is known about which explicit quantum error correction codes can combat quantum deletions. We note that {\em any} permutation-invariant quantum code that has a distance of $t+1$ can correct $t$ quantum deletions for any positive integer $t$ in both the qubit and the qudit setting. Leveraging on coding properties of permutation-invariant quantum codes under erasure errors, we derive corr…
We study the application of polar codes in deletion channels by analyzing the cascade of a binary erasure channel (BEC) and a deletion channel. We show how polar codes can be used effectively on a BEC with a single deletion, and propose a list decoding algorithm with a cyclic redundancy check for this case. The decoding complexity is $O(N^2\log N)$, where $N$ is the blocklength of the code. An important contribution is an optimization of the amount of redundancy added to minimize the overall error probability. Our theoretical results are corroborated by numerical simulations which show that th…
Motivated by DNA-based storage applications, we study the problem of reconstructing a coded sequence from multiple traces. We consider the model where the traces are outputs of independent deletion channels, where each channel deletes each bit of the input codeword \(\mathbf{x} \in \{0,1\}^n\) independently with probability \(p\). We focus on the regime where the deletion probability \(p \to 0\) when \(n\to \infty\). Our main contribution is designing a novel code for trace reconstruction that allows reconstructing a coded sequence efficiently from a constant number of traces. We provide theor…
We propose a new coding scheme, called the delayed coding (DC) scheme, for channels with insertion, deletion, and substitution (IDS) errors. The proposed scheme employs delayed encoding and non-iterative detection and decoding strategies to manage the transmission of multiple codewords in a linear code. In the DC scheme, a channel input sequence consists of subblocks of multiple codewords from the previous to current time instances. At the receiver side, the maximum a posteriori detection applies to the received sequences that contain information of the codeword at the current time instance, w…
The problem of reconstructing a source sequence with the presence of decoder side-information that is mis-synchronized to the source due to deletions is studied in a distributed source coding framework. Motivated by practical applications, the deletion process is assumed to be bursty and is modeled by a Markov chain. The minimum rate needed to reconstruct the source sequence with high probability is characterized in terms of an information theoretic expression, which is interpreted as the amount of information of the deleted content and the locations of deletions, subtracting "nature's secret"…
We present many new results related to reliable (interactive) communication over insertion-deletion channels. Synchronization errors, such as insertions and deletions, strictly generalize the usual symbol corruption errors and are much harder to protect against. We show how to hide the complications of synchronization errors in many applications by introducing very general channel simulations which efficiently transform an insertion-deletion channel into a regular symbol corruption channel with an error rate larger by a constant factor and a slightly smaller alphabet. We generalize synchroniza…
This work gives an explicit construction of a family of error correcting codes for the binary deletion channel and for the Poisson repeat channel. In the binary deletion channel with parameter $p$ (BDC$_p$) every bit is deleted independently with probability $p$. A lower bound of $(1-p)/9$ is known on the capacity of the BDC$_p$ \cite{mitzenmacher2006simple}, yet no explicit construction is known to achieve this rate. We give an explicit family of codes of rate $(1-p)/16$, for every $p$. This improves upon the work of Guruswami and Li \cite{guruswami2017efficiently} that gave a construction of…
A Deep Learning Based Decoder for Concatenated Coding over Deletion Channels
本文提出一种基于深度学习的译码器，利用BI-GRUs作为LLR估计器和外码译码器处理删除/替换信道上的串行级联码，其错误率与MAP检测相当且支持广泛的信道参数。
This paper presents a coding scheme for an insertion deletion substitution channel. We extend a previous scheme for the deletion channel where polar codes are modified by adding "guard bands" between segments. In the new scheme, each guard band is comprised of a middle segment of '1' symbols, and left and right segments of '0' symbols. Our coding scheme allows for a regular hidden-Markov input distribution, and achieves the information rate between the input and corresponding output of such a distribution. Thus, we prove that our scheme can be used to efficiently achieve the capacity of the ch…
We consider binary error correcting codes when errors are deletions. A basic challenge concerning deletion codes is determining $p_0^{(adv)}$, the zero-rate threshold of adversarial deletions, defined to be the supremum of all $p$ for which there exists a code family with rate bounded away from 0 capable of correcting a fraction $p$ of adversarial deletions. A recent construction of deletion-correcting codes [Bukh et al 17] shows that $p_0^{(adv)} \ge \sqrt{2}-1$, and the trivial upper bound, $p_0^{(adv)}\le\frac{1}{2}$, is the best known. Perhaps surprisingly, we do not know whether or not $p…
This paper considers insertion and deletion channels with the additional assumption that the channel input sequence is implicitly divided into segments such that at most one edit can occur within a segment. No segment markers are available in the received sequence. We propose code constructions for the segmented deletion, segmented insertion, and segmented insertion-deletion channels based on subsets of Varshamov-Tenengolts codes chosen with pre-determined prefixes and/or suffixes. The proposed codes, constructed for any finite alphabet, are zero-error and can be decoded segment-by-segment. We…
Racetrack memory is a new technology which utilizes magnetic domains along a nanoscopic wire in order to obtain extremely high storage density. In racetrack memory, each magnetic domain can store a single bit of information, which can be sensed by a reading port (head). The memory has a tape-like structure which supports a shift operation that moves the domains to be read sequentially by the head. In order to increase the memory's speed, prior work studied how to minimize the latency of the shift operation, while the no less important reliability of this operation has received only a little at…
Insertion Correcting Capability for Quantum Deletion-Correcting Codes
本文证明了任何量子t删除纠错码在特定条件下也能纠正t个插入和删除错误。提出了量子indel距离，并用此距离描述量子码的插入和删除纠错能力。
Stronger Polarization for the Deletion Channel
本文提出一种删除信道极化编码方案，通过在几乎所有极化层级放置保护带并对整个接收字进行网格译码，将错误概率衰减从立方根提升至平方根，达到与无记忆信道极化码相同的速率，同时保持容量可达和多项式译码复杂度。
Salami Slicing Trellis for Synchronization Errors in DNA Coding
针对DNA存储中同时存在替换、插入和删除错误的问题，提出萨拉米切片网格与跨链极化码结合的决策反馈译码方案，逼近信道容量。
The construction of deletion codes for the Levenshtein metric is reduced to the construction of codes over the integers for the Manhattan metric by run length coding. The latter codes are constructed by expurgation of translates of lattices. These lattices, in turn, are obtained from Construction~A applied to binary codes and $\Z_4-$codes. A lower bound on the size of our codes for the Manhattan distance are obtained through generalized theta series of the corresponding lattices.
Racetrack memory is a non-volatile memory engineered to provide both high density and low latency, that is subject to synchronization or shift errors. This paper describes a fast coding solution, in which delimiter bits assist in identifying the type of shift error, and easily implementable graph-based codes are used to correct the error, once identified. A code that is able to detect and correct double shift errors is described in detail.
We consider the maximum coding rate achievable by uniformly-random codes for the deletion channel. We prove an upper bound that's within 0.1 of the best known lower bounds for all values of the deletion probability $d,$ and much closer for small and large $d.$ We give simulation results which suggest that our upper bound is within 0.05 of the exact value for all $d$, and within $0.01$ for $d>0.75$. Despite our upper bounds, based on simulations, we conjecture that a positive rate is achievable with uniformly-random codes for all deletion probabilities less than 1. Our results imply impossibili…
The Insertion List-Decoding Capacity and an Improved Bound on the Deletion List-Decoding Capacity
研究插入和删除列表译码容量，给出插入列表译码容量公式，证明特定马尔可夫链码可达该容量，指出此方法在删除列表译码中不优于均匀随机编码，还给出删除列表译码容量的改进上界。
Algorithms of self-synchronizing single-deletion-correcting codes
本文提出一种无需分隔符的自同步单删除纠错码算法，利用互补信息集码的特性，实现连续传输中码字的识别与纠正，提升传输效率。
Error-correcting codes over sets, with applications to DNA storage, are studied. The DNA-storage channel receives a set of sequences, and produces a corrupted version of the set, including sequence loss, symbol substitution, symbol insertion/deletion, and limited-magnitude errors in symbols. Various parameter regimes are studied. New bounds on code parameters are provided, which improve upon known bounds. New codes are constructed, at times matching the bounds up to lower-or der terms or small constant factors.
Function-Correcting Codes for Insertion-Deletion Channel
针对插入和删除信道，提出函数纠正码框架，定义插入/删除/插入删除函数纠正码并证明等价性，推导最优冗余的上下界，并针对局部有界函数、VT综合征函数等给出冗余界。
In this paper, two moment balancing schemes, namely a variable index scheme and a fixed index scheme, for either single insertion/deletion error correction or multiple substitution error correction are introduced for coded sequences originally developed for correcting substitution errors only. By judiciously flipping bits of the original substitution error correcting code word, the resulting word is able to correct either a reduced number of substitution errors or a single insertion/deletion error. The number of flips introduced by the two schemes can be kept small compared to the code length.…
The noise model of deletions poses significant challenges in coding theory, with basic questions like the capacity of the binary deletion channel still being open. In this paper, we study the harder model of worst-case deletions, with a focus on constructing efficiently decodable codes for the two extreme regimes of high-noise and high-rate. Specifically, we construct polynomial-time decodable codes with the following trade-offs (for any eps > 0): (1) Codes that can correct a fraction 1-eps of deletions with rate poly(eps) over an alphabet of size poly(1/eps); (2) Binary codes of rate 1-O~(sqr…
Correcting Tail Deletions in Rank Modulated Composite Encoding for Data Storage in DNA
本文研究了在DNA数据存储中结合两种新的编码方法，利用复合DNA字母表和rank modulation码的特性，提出了一种基于排列的高效编码方案，并针对删除和插入码进行了界限和构造研究。
Shift-Interleave Coding for DNA-Based Storage: Correction of IDS Errors and Sequence Losses
该研究提出一种用于DNA存储的移位交织（SI）编码方案，通过移位、映射、交织处理低密度奇偶校验码码字，采用非迭代解码，可有效校正IDS错误与序列丢失。
In this paper motivated from subspace coding we introduce subspace-metric codes and subset-metric codes. These are coordinate-position independent pseudometrics and suitable for the folded codes. The half-Singleton upper bounds for linear subspace-metric codes and linear subset-metric codes are proved. Subspace distances and subset distances of codes are natural lower bounds for insdel distances of codes, and then can be used to lower bound the insertion-deletion error-correcting capabilities of codes. Our subspace-metric codes or subset-metric codes can be used to construct explicit well-stru…
Insertion and Deletion Correction in Polymer-based Data Storage
研究针对聚合物数据存储中插入和删除错误的校正问题，提出新的编码约束以提高鲁棒性。
In this paper, we study error-correcting codes for the storage of data in synthetic deoxyribonucleic acid (DNA). We investigate a storage model where data is represented by an unordered set of $M$ sequences, each of length $L$. Errors within that model are losses of whole sequences and point errors inside the sequences, such as substitutions, insertions and deletions. We propose code constructions which can correct these errors with efficient encoders and decoders. By deriving upper bounds on the cardinalities of these codes using sphere packing arguments, we show that many of our codes are cl…
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…
Consider two remote nodes (encoder and decoder), each with a binary sequence. The encoder's sequence $X$ differs from the decoder's sequence $Y$ by a small number of edits (deletions and insertions). The goal is to construct a message $M$, to be sent via a one-way error free link, such that the decoder can reconstruct $X$ using $M$ and $Y$. In this paper, we devise a coding scheme for this one-way synchronization model. The scheme is based on multiple layers of Varshamov-Tenengolts (VT) codes combined with off-the-shelf linear error-correcting codes, and uses a list decoder. We bound the expec…
The beta-binomial (BBM) channel model was recently proposed to model the overdispersed statistics of empirically observed bit errors in multi-level cell (MLC) flash memories. In this paper, we study the capacity of the BBM channel model for MLC flash memories. Using the compound channel approach, we first show that the BBM channel model capacity is zero. However, through empirical observation, this appears to be a very pessimistic estimate of the flash memory channel capacity. We propose a refined channel model called the truncated-support beta-binomial (TS-BBM) channel model and derive its ca…
We construct locally decodable codes (LDCs) to correct insertion-deletion errors in the setting where the sender and receiver share a secret key or where the channel is resource-bounded. Our constructions rely on a so-called "Hamming-to-InsDel" compiler (Ostrovsky and Paskin-Cherniavsky, ITS '15 & Block et al., FSTTCS '20), which compiles any locally decodable Hamming code into a locally decodable code resilient to insertion-deletion (InsDel) errors. While the compilers were designed for the classical coding setting, we show that the compilers still work in a secret key or resource-bounded set…
Recent efforts in coding theory have focused on building codes for insertions and deletions, called insdel codes, with optimal trade-offs between their redundancy and their error-correction capabilities, as well as efficient encoding and decoding algorithms. In many applications, polynomial running time may still be prohibitively expensive, which has motivated the study of codes with super-efficient decoding algorithms. These have led to the well-studied notions of Locally Decodable Codes (LDCs) and Locally Correctable Codes (LCCs). Inspired by these notions, Ostrovsky and Paskin-Cherniavsky (…
In this paper we study error-correcting codes for the storage of data in synthetic deoxyribonucleic acid (DNA). We investigate a storage model where a data set is represented by an unordered set of $M$ sequences, each of length $L$. Errors within that model are a loss of whole sequences and point errors inside the sequences, such as insertions, deletions and substitutions. We derive Gilbert-Varshamov lower bounds and sphere packing upper bounds on achievable cardinalities of error-correcting codes within this storage model. We further propose explicit code constructions than can correct errors…
Finite Blocklength Performance Bound for the DNA Storage Channel
本文提出了DNA存储信道的有限块长性能界（DT界），用于评估级联编码方案，并基于EXIT图优化LDPC外码，在2000符号内实现了88%至96%的归一化速率。
Error Correction for DNA Storage
该研究针对DNA存储因读写物理限制带来的全新编码理论挑战，介绍了删除/插入纠错码、切片信道编码、重复纠错码等相关编码问题。
Channels with Markov Synchronization Errors: Information Stability and Capacity Bounds
本文研究具有插入、删除和替换误差的信道，证明其信息稳定性，从而存在香农容量，特别应用于DNA存储。
Correcting a Single Deletion in Reads from a Nanopore Sequencer
本文研究纳米孔测序ℓ-read向量中的单个删除错误，给出log n−ℓ比特冗余下界，构造附加常数意义下最优纠错码，并证明两个含噪读取向量可零冗余重构。
Coding for Ordered Composite DNA Sequences
针对DNA存储信息容量提升需求，提出新型复合DNA信道模型，研究二进制下替换与单删除错误的复合序列重构，定义两类纠错码并给出界，还构造了单删除场景的系统码，探讨了信道容量开放问题。
On The Decoding Error Weight of One or Two Deletion Channels
本文研究一个或两个删除信道的最优解码错误权重，刻画了ML*解码器，并针对降级ML解码器推导了期望归一化距离的下界，该下界在长词和低删除概率下收敛至约(3q-1)/(q-1)p^2。
Flow watermarks efficiently link packet flows in a network in order to thwart various attacks such as stepping stones. We study the problem of designing good flow watermarks. Earlier flow watermarking schemes mostly considered substitution errors, neglecting the effects of packet insertions and deletions that commonly happen within a network. More recent schemes consider packet deletions but often at the expense of the watermark visibility. We present an invisible flow watermarking scheme capable of enduring a large number of packet losses and insertions. To maintain invisibility, our scheme u…
Embracing Errors Is More Efficient Than Avoiding Them Through Constrained Coding for DNA Data Storage
本文研究了DNA数据存储中的错误控制策略，发现在现有系统中包容替换错误比使用受约束编码避免错误更高效，且受约束编码在现有条件下并不具备优势。
Correcting One Deletion and One Substitution with a Constant Number of Reads
本文研究了设计(n,N;B)型重建码的问题，针对N={14,11,9,5}，通过减少冗余提高纠错能力，特别是当N=5时冗余降低至3logn+4。
In this paper, we present a coding-theoretic framework for message transmission over packet-switched networks. Network is modeled as a channel which can induce packet errors, deletions, insertions, and out of order delivery of packets. The proposed approach can be viewed as an extension of the one introduced by Koetter and Kschischang for networks based on random linear network coding. Namely, while their framework is based on subspace codes and designed for networks in which network nodes perform random linear combining of the packets, ours is based on the so-called subset codes, and is desig…
Marker-Delimited Codes for Short-Blocklength, High-Rate Coding over Multi-Read Edit Channels
针对DNA存储多读取编辑信道的短块长度高码率场景，提出结合标记分隔码与LDPC码的级联编码方案，提升了纠错效率。
Deep Learning-Based Detection for Marker Codes over Insertion and Deletion Channels
针对标记码检测需完美信道状态信息（CSI）的问题，提出两种基于深度学习的CSI无关算法，模型驱动方法将CSI转化为可学习权重，数据驱动方法构建端到端系统，仿真显示其错误性能显著优于原始算法及未知信道模型下的其他方法。
Tail-Erasure-Correcting Codes
针对分子数据存储等二维阵列存储范式的尾部擦除与跨行任意删除错误，本研究提出近乎最优的编码方案及编解码器，可有效缓解错误，推动新兴存储平台应用。
The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a communication scenario where the sender transmits a codeword from some codebook and the receiver obtains multiple noisy reads of the codeword. The common setup assumes the codebook to be the entire space and the problem is to determine the minimum number of distinct reads that is required to reconstruct the transmitted codeword. Motivated by modern storage devices, we study a variant of the problem where the number of noisy reads $N$ is fixed. Specifically, we design reconstruction codes that reconstruct a code…
This paper starts by considering the minimization of the Renyi divergence subject to a constraint on the total variation distance. Based on the solution of this optimization problem, the exact locus of the points $\bigl( D(Q\|P_1), D(Q\|P_2) \bigr)$ is determined when $P_1, P_2, Q$ are arbitrary probability measures which are mutually absolutely continuous, and the total variation distance between $P_1$ and $P_2$ is not below a given value. It is further shown that all the points of this convex region are attained by probability measures which are defined on a binary alphabet. This characteriz…
An End-to-End Coding Scheme for DNA-Based Data Storage With Nanopore-Sequenced Reads
针对DNA存储中纳米孔测序读段的链相关IDS错误等问题，提出基于索引的级联编码方案与低复杂度最大后验概率解码器，经仿真验证性能优于现有方案。
In this paper, we present a novel communication channel, called the absorption channel, inspired by information transmission in neurons. Our motivation comes from in-vivo nano-machines, emerging medical applications, and brain-machine interfaces that communicate over the nervous system. Another motivation comes from viewing our model as a specific deletion channel, which may provide a new perspective and ideas to study the general deletion channel. For any given finite alphabet, we give codes that can correct absorption errors. For the binary alphabet, the problem is relatively trivial and we …
High Information Density and Low Coverage Data Storage in DNA with Efficient Channel Coding Schemes
本研究提出一种集成高效信道编码方案的DNA数据存储架构，通过内外编码结合纠错码与约束码，在实验中实现1.731和1.815的信息密度，以4.5和6.0的低覆盖度无错恢复数据，达到现有标准DNA存储的最高净信息密度和最低覆盖度。
Transformer-Based Decoding in Concatenated Coding Schemes Under Synchronization Errors
针对DNA数据存储中等多副本受插入、删除和替换破坏的码字重建问题，提出基于Transformer的内层解码器BCJRFormer及交叉注意力架构ConvBCJRFormer，将复杂度降至二次方并实现端到端高效解码。
We consider the problem of determining Gröbner bases of binomial ideals associated with linear error correcting codes. Computation of Gröbner bases of linear codes have become a topic of interest to many researchers in coding theory because of its several applications in decoding and error corrections. In this paper, Gröbner bases of linear codes associated to Grassmann varieties and Schubert varieties over a binary field have been obtained. We also use them to study the decoding of binary Schubert codes.
Motivated by average-case trace reconstruction and coding for portable DNA-based storage systems, we initiate the study of \emph{coded trace reconstruction}, the design and analysis of high-rate efficiently encodable codes that can be efficiently decoded with high probability from few reads (also called \emph{traces}) corrupted by edit errors. Codes used in current portable DNA-based storage systems with nanopore sequencers are largely based on heuristics, and have no provable robustness or performance guarantees even for an error model with i.i.d.\ deletions and constant deletion probability.…
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