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Permutation codes in the Ulam metric, which can correct multiple deletions, have been investigated extensively recently owing to their applications. In this work, we are interested in the maximum size of the permutation codes in the Ulam metric and aim to design permutation codes that can correct multiple deletions with efficient decoding algorithms. We first present an improvement on the Gilbert--Varshamov bound of the maximum size of these permutation codes which is the best-known lower bound. Next, we focus on designing permutation codes in the Ulam metric with a decoding algorithm. These constructed codes are the best-known permutation codes that can correct multiple deletions. In particular, the constructed permutation codes can correct $t$ deletions with at most $(3t-1) \log n+o(\log n)$ bits of redundancy where $n$ is the length of the code. Finally, we provide an efficient decoding algorithm for our constructed permutation codes., Comment: 9 pages

Given $d$ defective items in a population of $n$ items with $d \ll n$, in threshold group testing without gap, the outcome of a test on a subset of items is positive if the subset has at least $u$ defective items and negative otherwise, where $1 \leq u \leq d$. The basic goal of threshold group testing is to quickly identify the defective items via a small number of tests. In non-adaptive design, all tests are designed independently and can be performed in parallel. The decoding time in the non-adaptive state-of-the-art work is a polynomial of $(d/u)^u (d/(d-u))^{d - u}, d$, and $\log{n}$. In this work, we present a novel design that significantly reduces the number of tests and the decoding time to polynomials of $\min\{u^u, (d - u)^{d - u}\}, d$, and $\log{n}$. In particular, when $u$ is a constant, the number of tests and the decoding time are $O(d^3 (\log^2{n}) \log{(n/d)} )$ and $O\big(d^3 (\log^2{n}) \log{(n/d)} + d^2 (\log{n}) \log^3{(n/d)} \big)$, respectively. For a special case when $u = 2$, with non-adaptive design, the number of tests and the decoding time are $O(d^3 (\log{n}) \log{(n/d)} )$ and $O(d^2 (\log{n} + \log^2{(n/d)}) )$, respectively. Moreover, with 2-stage design, the number of tests and the decoding time are $O(d^2 \log^2{(n/d)} )$., Comment: 11 pages, 2 figures

An $(m,n,R)$-de Bruijn covering array (dBCA) is a doubly periodic $M \times N$ array over an alphabet of size $q$ such that the set of all its $m \times n$ windows form a covering code with radius $R$. An upper bound of the smallest array area of an $(m,n,R)$-dBCA is provided using a probabilistic technique which is similar to the one that was used for an upper bound on the length of a de Bruijn covering sequence. A folding technique to construct a dBCA from a de Bruijn covering sequence or de Bruijn covering sequences code is presented. Several new constructions that yield shorter de Bruijn covering sequences and $(m,n,R)$-dBCAs with smaller areas are also provided. These constructions are mainly based on sequences derived from cyclic codes, self-dual sequences, primitive polynomials, an interleaving technique, folding, and mutual shifts of sequences with the same covering radius. Finally, constructions of de Bruijn covering sequences codes are also discussed.

A timing and synchronization system based on a de Bruijn sequence has been proposed and studied recently for a channel associated with quantum communication that requires reliable synchronization. To avoid a long period of no-pulse in such a system on-off pulses are used to simulate a zero and on-on pulses are used to simulate a one. However, these sequences have high redundancy. To reduce the redundancy, run-length limited sequences in the de Bruijn graph are proposed for the same purpose. The maximum length of such sequences in the de Bruijn graph is studied and an efficient algorithm to construct a large set of these sequences is presented. A maximum length sequence for which the position of each window can be computed efficiently is constructed. Finally, an enumeration of the number of such sequences is given and some generalizations are discussed.

Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of $\epsilon$ in $\mathrm{poly}(1/\epsilon, \text{input dimension})$ time.

Codes for correcting sticky insertions/deletions and limited-magnitude errors have attracted significant attention due to their applications of flash memories, racetrack memories, and DNA data storage systems. In this paper, we first consider the error type of $t$-sticky deletions with $\ell$-limited-magnitude and propose a non-systematic code for correcting this type of error with redundancy $2t(1-1/p)\cdot\log(n+1)+O(1)$, where $p$ is the smallest prime larger than $\ell+1$. Next, we present a systematic code construction with an efficient encoding and decoding algorithm with redundancy $\frac{\lceil2t(1-1/p)\rceil\cdot\lceil\log p\rceil}{\log p} \log(n+1)+O(\log\log n)$, where $p$ is the smallest prime larger than $\ell+1$., Comment: arXiv admin note: substantial text overlap with arXiv:2301.11680

Codes in the Damerau--Levenshtein metric have been extensively studied recently owing to their applications in DNA-based data storage. In particular, Gabrys, Yaakobi, and Milenkovic (2017) designed a length-$n$ code correcting a single deletion and $s$ adjacent transpositions with at most $(1+2s)\log n$ bits of redundancy. In this work, we consider a new setting where both asymmetric adjacent transpositions (also known as right-shifts or left-shifts) and deletions may occur. We present several constructions of the codes correcting these errors in various cases. In particular, we design a code correcting a single deletion, $s^+$ right-shift, and $s^-$ left-shift errors with at most $(1+s)\log (n+s+1)+1$ bits of redundancy where $s=s^{+}+s^{-}$. In addition, we investigate codes correcting $t$ $0$-deletions, $s^+$ right-shift, and $s^-$ left-shift errors with both uniquely-decoding and list-decoding algorithms. Our main contribution here is the construction of a list-decodable code with list size $O(n^{\min\{s+1,t\}})$ and with at most $(\max \{t,s+1\}) \log n+O(1)$ bits of redundancy, where $s=s^{+}+s^{-}$. Finally, we construct both non-systematic and systematic codes for correcting blocks of $0$-deletions with $\ell$-limited-magnitude and $s$ adjacent transpositions.

\emph{Resistive memories}, such as \emph{phase change memories} and \emph{resistive random access memories} have attracted significant attention in recent years due to their better scalability, speed, rewritability, and yet non-volatility. However, their \emph{limited endurance} is still a major drawback that has to be improved before they can be widely adapted in large-scale systems. In this work, in order to reduce the wear out of the cells, we propose a new coding scheme, called \emph{endurance-limited memories} (\emph{ELM}) codes, that increases the endurance of these memories by limiting the number of cell programming operations. Namely, an \emph{$\ell$-change $t$-write ELM code} is a coding scheme that allows to write $t$ messages into some $n$ binary cells while guaranteeing that each cell is programmed at most $\ell$ times. In case $\ell=1$, these codes coincide with the well-studied \emph{write-once memory} (\emph{WOM}) codes. We study some models of these codes which depend upon whether the encoder knows on each write the number of times each cell was programmed, knows only the memory state, or even does not know anything. For the decoder, we consider these similar three cases. We fully characterize the capacity regions and the maximum sum-rates of three models where the encoder knows on each write the number of times each cell was programmed. In particular, it is shown that in these models the maximum sum-rate is $\log \sum_{i=0}^{\ell} {t \choose i}$. We also study and expose the capacity regions of the models where the decoder is informed with the number of times each cell was programmed. Finally we present the most practical model where the encoder read the memory before encoding new data and the decoder has no information about the previous states of the memory.

The de Bruijn graph, its sequences, and their various generalizations, have found many applications in information theory, including many new ones in the last decade. In this paper, motivated by a coding problem for emerging memory technologies, a set of sequences which generalize sequences in the de Bruijn graph are defined. These sequences can be also defined and viewed as constrained sequences. Hence, they will be called constrained de Bruijn sequences and a set of such sequences will be called a constrained de Bruijn code. Several properties and alternative definitions for such codes are examined and they are analyzed as generalized sequences in the de Bruijn graph (and its generalization) and as constrained sequences. Various enumeration techniques are used to compute the total number of sequences for any given set of parameters. A construction method of such codes from the theory of shift-register sequences is proposed. Finally, we show how these constrained de Bruijn sequences and codes can be applied in constructions of codes for correcting synchronization errors in the $\ell$-symbol read channel and in the racetrack memory channel. For this purpose, these codes are superior in their size on previously known codes.

This year, 2019 we celebrate 30 years of our friendship between Asian and European scientists at the AEW11 in Rotterdam, the Netherlands. Many of the 1989 participants are also present at the 2019 event. This year we have many participants from different parts of Asia and Europe. It shows the importance of this event. It is a good tradition to pay a tribute to a special lecturer in our community. This year we selected Hiroyoshi Morita, who is a well known information theorist with many original contributions.

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 attention. In this work we design codes which combat shift errors in racetrack memory, called position errors. Namely, shifting the domains is not an error-free operation and the domains may be over-shifted or are not shifted, which can be modeled as deletions and sticky insertions. While it is possible to use conventional deletion and insertion-correcting codes, we tackle this problem with the special structure of racetrack memory, where the domains can be read by multiple heads. Each head outputs a noisy version of the stored data and the multiple outputs are combined in order to reconstruct the data. Under this paradigm, we will show that it is possible to correct, with at most a single bit of redundancy, $d$ deletions with $d+1$ heads if the heads are well-separated. Similar results are provided for burst of deletions, sticky insertions and combinations of both deletions and sticky insertions.