Recent advances in DNA sequencing technology and DNA storage systems have rekindled the interest in deletion channels. Multiple recent works have looked at variants of sequence reconstruction over a single and over multiple deletion channels, a notoriously difficult problem due to its highly combinatorial nature. Although works in theoretical computer science have provided algorithms which guarantee perfect reconstruction with multiple independent observations from the deletion channel, they are only applicable in the large blocklength regime and more restrictively, when the number of observations is also large. Indeed, with only a few observations, perfect reconstruction of the input sequence may not even be possible in most cases. In such situations, maximum likelihood (ML) and maximum aposteriori (MAP) estimates for the deletion channels are natural questions that arise and these have remained open to the best of our knowledge. In this work, we take steps to answer the two aforementioned questions. Specifically: 1. We show that solving for the ML estimate over the single deletion channel (which can be cast as a discrete optimization problem) is equivalent to solving its relaxation, a continuous optimization problem; 2. We exactly compute the symbolwise posterior distributions (under some assumptions on the priors) for both the single as well as multiple deletion channels. As part of our contributions, we also introduce tools to visualize and analyze error events, which we believe could be useful in other related problems concerning deletion channels.
Optimal Reconstruction Codes for Deletion Channels
Levenshtein在2001年提出的序列重构问题考虑了一种通信场景，其中发送方从某个码本发送码字，接收方获得码字的多个噪声读取。受现代存储设备的启发，我们引入了问题的一种变体，其中噪声读取$N$的数量是固定的（Kiah等人，2020）。重要的是，对于单删除通道，使用$\log_2\log_2 n +O(1)$冗余位，我们设计了长度为$n$的重建码，其从两个不同的噪声读取重建码字。在这项工作中，我们表明，$\log_2\log_2 n -O(1)$冗余位是必要的，这样的重建代码，从而证明了我们以前的建设的最优性。此外，我们表明，这些重建代码可以用于$t$删除通道（与$t\ge 2$），以唯一地重建码字从$n^{t-1}+O\left(n^{t-2}\right)$不同的噪声读取。
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…
Improved Capacity Upper Bounds for the Deletion Channel using a Parallelized Blahut-Arimoto Algorithm
Instituto Superior Técnico, Universidade de Lisboa（里斯本技术学院，里斯本大学）；Instituto de Telecomunicações and Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa（电信研究所与数学系，里斯本技术学院，里斯本大学）
This paper presents the first proof of polarization for the deletion channel with a constant deletion rate and a regular hidden-Markov input distribution. A key part of this work involves representing the deletion channel using a trellis and describing the plus and minus polar-decoding operations on that trellis. In particular, the plus and minus operations can be seen as combining adjacent trellis stages to yield a new trellis with half as many stages. Using this viewpoint, we prove a weak polarization theorem for standard polar codes on the deletion channel. To achieve strong polarization, w…
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…
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…
The channel output entropy of a transmitted word is the entropy of the possible channel outputs and similarly, the input entropy of a received word is the entropy of all possible transmitted words. The goal of this work is to study these entropy values for the k-deletion, k-insertion channel, where exactly k symbols are deleted, and inserted in the transmitted word, respectively. If all possible words are transmitted with the same probability then studying the input and output entropies is equivalent. For both the 1-insertion and 1-deletion channels, it is proved that among all words with a fi…
Two of the most common models for channels with synchronisation errors are the Binary Deletion Channel with parameter $p$ ($\text{BDC}_p$) -- a channel where every bit of the codeword is deleted i.i.d with probability $p$, and the Poisson Repeat Channel with parameter $λ$ ($\text{PRC}_λ$) -- a channel where every bit of the codeword is repeated $\text{Poisson}(λ)$ times. Previous constructions based on synchronisation strings yielded codes with rates far lower than the capacities of these channels [CS19, GL18], and the only efficient construction to achieve capacity on the BDC at the time of w…
In this paper, we consider a class of symmetry groups associated to communication channels, which can informally be viewed as the transformations of the set of inputs that ``commute'' with the action of the channel. These groups were first studied by Polyanskiy in (IEEEToIT 2013). We show the simple result that the input distribution that attains the maximum mutual information for a given channel is a ``fixed point'' of its group. We conjecture (and give empirical evidence) that the channel group of the deletion channel is extremely small (it contains a number of elements constant in the block…
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…
Transmit a codeword $x$, that belongs to an $(\ell-1)$-deletion-correcting code of length $n$, over a $t$-deletion channel for some $1\le \ell\le t<n$. Levenshtein, in 2001, proposed the problem of determining $N(n,\ell,t)+1$, the minimum number of distinct channel outputs required to uniquely reconstruct $x$. Prior to this work, $N(n,\ell,t)$ is known only when $\ell\in\{1,2\}$. Here, we provide an asymptotically exact solution for all values of $\ell$ and $t$. Specifically, we show that $N(n,\ell,t)=\binom{2\ell}{\ell}/(t-\ell)! n^{t-\ell} - O(n^{t-\ell-1})$ and in the special instance where…
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…
