萩原研究室を修了した中山君（博士前期課程）とハワイで会いました。

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お祝いを兼ねて、中山君と奥様と一緒に食事へ行きました。

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# 月: 2022年11月

## 結婚おめでとう

## 削除符号論文メモ：2022/11/26

## 削除符号論文メモ：2022/11/25

萩原研究室を修了した中山君（博士前期課程）とハワイで会いました。

結婚式をハワイで挙げ、その前後の数週間もハワイで過ごすとのこと。

お祝いを兼ねて、中山君と奥様と一緒に食事へ行きました。

おめでとう！

末永くお幸せに。

タイトル：On Prefixed Varshamov-Tenengolts Codes for Segmented Edit Channels 著者名：Xiaopeng Jiao; Haiyang Liu; Jianjun Mu; Hui Han; Yu-Cheng He 雑誌名：IEEE Transactions on Communications 号数：Volume 70, Issue 3 DOI:10.1109/TCOMM.2022.3146285 Abstract：The prefixed Varshamov-Tenengolts (VT) codes, which are subsets of VT codes with predetermined prefixes, can be used for error correction over segmented edit channels. In this paper, we investigate the construction and analysis of this class of codes. First, we derive upper bounds on the size of zero-error codes for segmented edit channels with segment-by-segment decoding. Second, we establish a one-to-one correspondence between prefixed VT codes and Levenshtein codes. Based on this relation, we can obtain explicit formulas on the size of prefixed VT codes via the existing results on the size of Levenshtein codes. Third, we construct a new zero-error prefixed VT code and show that the size of the constructed code is strictly larger than that of the existing prefixed VT code for the segmented deletion channel. Finally, an efficient systematic encoding method of prefixed VT codes is proposed for the segmented edit channels.

タイトル : A Single-Round File Synchronization Scheme from Insertions and Deletions with Low Redundancy 著者名 : Guochen Ma; Xiaopeng Jiao; Jianjun Mu; Hui Han; Yu-Cheng He 雑誌名 : IEEE Communications Letters 号数 : - DOI: 10.1109/LCOMM.2022.3193809 Abstract : Consider a file synchronization problem at two remote nodes A and B connected by a noiseless link. The node A has a binary file X, and node B has a binary file Y which is an edited version of X with random insertions and deletions (indels). The goal is to reconstruct X at node B from Y with minimal exchange of information between A and B. In this letter, we design a single-round, thus low latency, file synchronization scheme with low redundancy. In particular, a series of redundancy bits, including markers, check symbols of Reed-Solomon (RS) codes for hashes and Varshamov-Tenengolts (VT) syndromes, are constructed from X at node A and send to node B. According to the received redundancy and Y, we develop a recovery algorithm at node B to reconstruct X. The amount of exchanged information of the proposed scheme is significantly reduced when compared with the existing single-round synchronization scheme under random indels.

タイトル： Error Probability Bounds for Coded-Index DNA Storage Systems 著者名： Nir Weinberger 雑誌名： IEEE Transactions on Information Theory 号数： Volume: 68, Issue: 11 DOI: 10.1109/TIT.2022.3176917 Abstract： The DNA storage channel is considered, in which a codeword is comprised of M unordered DNA molecules. At reading time, N molecules are sampled with replacement, and then each molecule is sequenced. A coded-index concatenated-coding scheme is considered, in which the m th molecule of the codeword is restricted to a subset of all possible molecules (an inner code), which is unique for each m . The decoder has low-complexity, and is based on first decoding each molecule separately (the inner code), and then decoding the sequence of molecules (an outer code). Only mild assumptions are made on the sequencing channel, in the form of the existence of an inner code and decoder with vanishing error. The error probability of a random code as well as an expurgated code is analyzed and shown to decay exponentially with N . This establishes the importance of increasing the coverage depth N/M in order to obtain low error probability.

タイトル： Correcting Deletions With Multiple Reads 著者名： Johan Chrisnata; Han Mao Kiah; Eitan Yaakobi 雑誌名： IEEE Transactions on Information Theory 号数： Volume: 68, Issue: 11 DOI: 10.1109/TIT.2022.3184868 Abstract： 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. Motivated by modern storage devices, we introduced a variant of the problem where the number of noisy reads N is fixed. Of significance, for the single-deletion channel, using log_2log_2(n)+O(1) redundant bits, we designed a reconstruction code of length n that reconstructs codewords from two distinct noisy reads (Cai et al. , 2021). In this work, we show that log_2log_2(n)−O(1) redundant bits are necessary for such reconstruction codes, thereby, demonstrating the optimality of the construction. Furthermore, we show that these reconstruction codes can be used in t -deletion channels (with t⩾2) to uniquely reconstruct codewords from n^(t−1)/(t−1)!+O(n^(t−2)) distinct noisy reads. For the two-deletion channel, using higher order VT syndromes and certain runlength constraints, we designed the class of higher order constrained shifted VT code with 2log_2(n)+o(log_2(n)) redundancy bits that can reconstruct any codeword from any N⩾5 of its length- (n−2) subsequences.

タイトル： On the Reverse-Complement String-Duplication System 著者名： Eyar Ben-Tolila; Moshe Schwartz 雑誌名： IEEE Transactions on Information Theory 号数： Volume: 68, Issue: 11 DOI: 10.1109/TIT.2022.3182873 Abstract： Motivated by DNA storage in living organisms, and by known biological mutation processes, we study the reverse-complement string-duplication system. We fully classify the conditions under which the system has full expressiveness, for all alphabets and all fixed duplication lengths. We then focus on binary systems with duplication length 2 and prove that they have full capacity, yet surprisingly, have zero entropy-rate. Finally, by using binary single burst-insertion correcting codes, we construct codes that correct a single reverse- complement duplication of odd length, over any alphabet. The redundancy (in bits) of the constructed code does not depend on the alphabet size.

タイトル： Weight Enumerators and Cardinalities for Number-Theoretic Codes 著者名： Takayuki Nozaki 雑誌名： IEEE Transactions on Information Theory 号数： Volume: 68, Issue: 11 DOI: 10.1109/TIT.2022.3184776 Abstract： The number-theoretic code is a class of codes defined by single or multiple congruences. These codes are mainly used for correcting insertion and deletion errors, and for correcting asymmetric errors. This paper presents a formula for a generalization of the complete weight enumerator for the number-theoretic codes. This formula allows us to derive the weight enumerators and cardinalities for the number-theoretic codes. As a special case, this paper provides the Hamming weight enumerators and cardinalities of the non-binary Tenengolts’ codes, correcting single insertion or deletion. Moreover, we show that the formula deduces the MacWilliams identity for the linear codes over the ring of integers modulo r.