arXiv subjects

Lorenz Welter

Publications and source records attributed to Lorenz Welter.

11 recordsLinked to original sources

arXiv2024

Sequential Decoding of Multiple Traces Over the Syndrome Trellis for Synchronization Errors

Standard decoding approaches for convolutional codes, such as the Viterbi and BCJR algorithms, entail significant complexity when correcting synchronization errors. The situation worsens when multiple received sequences should be jointly decoded, as in DNA storage. Previous work has attempted to address this via separate-BCJR decoding, i.e., combining the results of decoding each received sequence separately. Another attempt to reduce complexity adapted sequential decoders for use over channels with insertion and deletion errors. However, these decoding alternatives remain prohibitively expensive for high-rate convolutional codes. To address this, we adapt sequential decoders to decode multiple received sequences jointly over the syndrome trellis. For the short blocklength regime, this decoding strategy can outperform separate-BCJR decoding under certain channel conditions, in addition to reducing decoding complexity. To mitigate the occurrence of a decoding timeout, formally called erasure, we also extend this approach to work bidirectionally, i.e., deploying two independent stack decoders that simultaneously operate in the forward and backward directions.

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arXiv2024

An End-to-End Coding Scheme for DNA-Based Data Storage With Nanopore-Sequenced Reads

We consider error-correcting coding for deoxyribonucleic acid (DNA)-based storage using nanopore sequencing. We model the DNA storage channel as a sampling noise channel where the input data is chunked into $M$ short DNA strands, which are copied a random number of times, and the channel outputs a random selection of $N$ noisy DNA strands. The retrieved DNA reads are prone to strand-dependent insertion, deletion, and substitution (IDS) errors. We construct an index-based concatenated coding scheme consisting of the concatenation of an outer code, an index code, and an inner code. We further propose a low-complexity (linear in $N$) maximum a posteriori probability decoder that takes into account the strand-dependent IDS errors and the randomness of the drawing to infer symbolwise a posteriori probabilities for the outer decoder. We present Monte-Carlo simulations for information-outage probabilities and frame error rates for different channel setups on experimental data. We finally evaluate the overall system performance using the read/write cost trade-off. A powerful combination of tailored channel modeling and soft information processing allows us to achieve excellent performance even with error-prone nanopore-sequenced reads outperforming state-of-the-art schemes.%

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arXiv2023

Index-Based Concatenated Codes for the Multi-Draw DNA Storage Channel

We consider error-correcting coding for DNA-based storage. We model the DNA storage channel as a multi-draw IDS channel where the input data is chunked into $M$ short DNA strands, which are copied a random number of times, and the channel outputs a random selection of $N$ noisy DNA strands. The retrieved DNA strands are prone to insertion, deletion, and substitution (IDS) errors. We propose an index-based concatenated coding scheme consisting of the concatenation of an outer code, an index code, and an inner synchronization code, where the latter two tackle IDS errors. We further propose a mismatched joint index-synchronization code maximum a posteriori probability decoder with optional clustering to infer symbolwise a posterior probabilities for the outer decoder. We compute achievable information rates for the outer code and present Monte-Carlo simulations for information-outage probabilities and frame error rates on synthetic and experimental data, respectively.

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arXiv2022

Concatenated Codes for Multiple Reads of a DNA Sequence

Decoding sequences that stem from multiple transmissions of a codeword over an insertion, deletion, and substitution channel is a critical component of efficient deoxyribonucleic acid (DNA) data storage systems. In this paper, we consider a concatenated coding scheme with an outer nonbinary low-density parity-check code or a polar code and either an inner convolutional code or a time-varying block code. We propose two novel decoding algorithms for inference from multiple received sequences, both combining the inner code and channel to a joint hidden Markov model to infer symbolwise a posteriori probabilities (APPs). The first decoder computes the exact APPs by jointly decoding the received sequences, whereas the second decoder approximates the APPs by combining the results of separately decoded received sequences and has a complexity that is linear with the number of sequences. Using the proposed algorithms, we evaluate the performance of decoding multiple received sequences by means of achievable information rates and Monte-Carlo simulations. We show significant performance gains compared to a single received sequence. In addition, we succeed in improving the performance of the aforementioned coding scheme by optimizing both the inner and outer codes.

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arXiv2022

Equivalence of Insertion/Deletion Correcting Codes for $d$-dimensional Arrays

We consider the problem of correcting insertion and deletion errors in the $d$-dimensional space. This problem is well understood for vectors (one-dimensional space) and was recently studied for arrays (two-dimensional space). For vectors and arrays, the problem is motivated by several practical applications such as DNA-based storage and racetrack memories. From a theoretical perspective, it is interesting to know whether the same properties of insertion/deletion correcting codes generalize to the $d$-dimensional space. In this work, we show that the equivalence between insertion and deletion correcting codes generalizes to the $d$-dimensional space. As a particular result, we show the following missing equivalence for arrays: a code that can correct $t_\mathrm{r}$ and $t_\mathrm{c}$ row/column deletions can correct any combination of $t_\mathrm{r}^{\mathrm{ins}}+t_\mathrm{r}^{\mathrm{del}}=t_\mathrm{r}$ and $t_\mathrm{c}^{\mathrm{ins}}+t_\mathrm{c}^{\mathrm{del}}=t_\mathrm{c}$ row/column insertions and deletions. The fundamental limit on the redundancy and a construction of insertion/deletion correcting codes in the $d$-dimensional space remain open for future work.

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arXiv2022

Feedback Insertion-Deletion Codes

In this paper, a new problem of transmitting information over the adversarial insertion-deletion channel with feedback is introduced. Suppose that the encoder transmits $n$ binary symbols one-by-one over a channel, in which some symbols can be deleted and some additional symbols can be inserted. After each transmission, the encoder is notified about the insertions or deletions that have occurred within the previous transmission and the encoding strategy can be adapted accordingly. The goal is to design an encoder that is able to transmit error-free as much information as possible under the assumption that the total number of deletions and insertions is limited by $τn$, $0<τ<1$. We show how this problem can be reduced to the problem of transmitting messages over the substitution channel. Thereby, the maximal asymptotic rate of feedback insertion-deletion codes is completely established. The maximal asymptotic rate for the adversarial substitution channel has been partially determined by Berlekamp and later finished by Zigangirov. However, the analysis of the lower bound by Zigangirov is quite complicated. We revisit Zigangirov's result and present a more elaborate version of his proof.

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arXiv2021

Multiple Criss-Cross Insertion and Deletion Correcting Codes

This paper investigates the problem of correcting multiple criss-cross insertions and deletions in arrays. More precisely, we study the unique recovery of $n \times n$ arrays affected by $t$-criss-cross deletions defined as any combination of $t_r$ row and $t_c$ column deletions such that $t_r + t_c = t$ for a given $t$. We show an equivalence between correcting $t$-criss-cross deletions and $t$-criss-cross insertions and show that a code correcting $t$-criss-cross insertions/deletions has redundancy at least $tn + t \log n - \log(t!)$. Then, we present an existential construction of $t$-criss-cross insertion/deletion correcting code with redundancy bounded from above by $tn + \mathcal{O}(t^2 \log^2 n)$. The main ingredients of the presented code construction are systematic binary $t$-deletion correcting codes and Gabidulin codes. The first ingredient helps locating the indices of the inserted/deleted rows and columns, thus transforming the insertion/deletion-correction problem into a row/column erasure-correction problem which is then solved using the second ingredient.

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arXiv2021

Criss-Cross Insertion and Deletion Correcting Codes

This paper studies the problem of constructing codes correcting deletions in arrays. Under this model, it is assumed that an $n\times n$ array can experience deletions of rows and columns. These deletion errors are referred to as $(t_r,t_c)$-criss-cross deletions if $t_r$ rows and $t_c$ columns are deleted, while a code correcting these deletion patterns is called a $(t_r,t_c)$-criss-cross deletion correction code. The definitions for criss-cross insertions are similar. It is first shown that when $t_r=t_c$ the problems of correcting criss-cross deletions and criss-cross insertions are equivalent. The focus of this paper lies on the case of $(1,1)$-criss-cross deletions. A non-asymptotic upper bound on the cardinality of $(1,1)$-criss-cross deletion correction codes is shown which assures that the redundancy is at least $2n-3+2\log n$ bits. A code construction with an existential encoding and an explicit decoding algorithm is presented. The redundancy of the construction is at most $2n+4 \log n + 7 +2 \log e$. A construction with explicit encoder and decoder is presented. The explicit encoder adds an extra $5\log n + 5$ bits of redundancy to the construction.

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arXiv2020

Concatenated Codes for Recovery From Multiple Reads of DNA Sequences

Decoding sequences that stem from multiple transmissions of a codeword over an insertion, deletion, and substitution channel is a critical component of efficient deoxyribonucleic acid (DNA) data storage systems. In this paper, we consider a concatenated coding scheme with an outer low-density parity-check code and either an inner convolutional code or a block code. We propose two new decoding algorithms for inference from multiple received sequences, both combining the inner code and channel to a joint hidden Markov model to infer symbolwise a posteriori probabilities (APPs). The first decoder computes the exact APPs by jointly decoding the received sequences, whereas the second decoder approximates the APPs by combining the results of separately decoded received sequences. Using the proposed algorithms, we evaluate the performance of decoding multiple received sequences by means of achievable information rates and Monte-Carlo simulations. We show significant performance gains compared to a single received sequence.

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arXiv2020

Single-Deletion Single-Substitution Correcting Codes

Correcting insertions/deletions as well as substitution errors simultaneously plays an important role in DNA-based storage systems as well as in classical communications. This paper deals with the fundamental task of constructing codes that can correct a single insertion or deletion along with a single substitution. A non-asymptotic upper bound on the size of single-deletion single-substitution correcting codes is derived, showing that the redundancy of such a code of length $n$ has to be at least $2 \log n$. The bound is presented both for binary and non-binary codes while an extension to single deletion and multiple substitutions is presented for binary codes. An explicit construction of single-deletion single-substitution correcting codes with at most $6 \log n + 8$ redundancy bits is derived. Note that the best known construction for this problem has to use 3-deletion correcting codes whose best known redundancy is roughly $24 \log n$.

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arXiv2020

Achievable Rates of Concatenated Codes in DNA Storage under Substitution Errors

In this paper, we study achievable rates of concatenated coding schemes over a deoxyribonucleic acid (DNA) storage channel. Our channel model incorporates the main features of DNA-based data storage. First, information is stored on many, short DNA strands. Second, the strands are stored in an unordered fashion inside the storage medium and each strand is replicated many times. Third, the data is accessed in an uncontrollable manner, i.e., random strands are drawn from the medium and received, possibly with errors. As one of our results, we show that there is a significant gap between the channel capacity and the achievable rate of a standard concatenated code in which one strand corresponds to an inner block. This is in fact surprising as for other channels, such as $q$-ary symmetric channels, concatenated codes are known to achieve the capacity. We further propose a modified concatenated coding scheme by combining several strands into one inner block, which allows to narrow the gap and achieve rates that are close to the capacity.

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