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 is structured like a tape, 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 paper, 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. This setup is a special case of the reconstruction problem studied by Levenshtein, however, in our case, the position errors from different heads are correlated. We will show how to take advantage of this special feature of racetrack memories in order to construct codes correcting deletions and sticky insertions. In particular, 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.

An indel refers to a single insertion or deletion, while an edit refers to a single insertion, deletion or substitution. In this article, we investigate codes that correct either a single indel or a single edit and provide linear-time algorithms that encode binary messages into these codes of length n. Over the quaternary alphabet, we provide two linear-time encoders. One corrects a single edit with ⌈log n⌉+ O(loglog n) redundancy bits, while the other corrects a single indel with ⌈log n⌉+2 redundant bits. These two encoders are order-optimal. The former encoder is the first known order-optimal encoder that corrects a single edit, while the latter encoder (that corrects a single indel) reduces the redundancy of the best known encoder of Tenengolts (1984) by at least four bits. Over the DNA alphabet, we impose an additional constraint: the GC-balanced constraint and require that exactly half of the symbols of any DNA codeword to be either C or G. In particular, via a modification of Knuth's balancing technique, we provide a linear-time map that translates binary messages into GC-balanced codewords and the resulting codebook is able to correct a single indel or a single edit. These are the first known constructions of GC-balanced codes that correct a single indel or a single edit.

We propose coding techniques that simultaneously limit the length of homopolymers runs, ensure the GC-content constraint, and are capable of correcting a single edit error in strands of nucleotides in DNA-based data storage systems. In particular, for given l , ∈ > 0, we propose simple and efficient encoders/decoders that transform binary sequences into DNA base sequences (codewords), namely sequences of the symbols A, T, C and G, that satisfy all of the following properties: 1) runlength constraint: the maximum homopolymer run in each codeword is at most l ; 2) GC-content constraint: the GC-content of each codeword is within [0.5-∈,0.5+∈]; 3) error-correction: each codeword is capable of correcting a single deletion, or single insertion, or single substitution error. While various combinations of these properties have been considered in the literature, this work provides generalizations of codes constructions that satisfy all the properties with arbitrary parameters of l and ∈. Furthermore, for practical values of l and ∈, we show that our encoders achieve higher rates than existing results in the literature and approach capacity. Our methods have low encoding/decoding complexity and limited error propagation.

Despite their exceptional error-correcting properties, Reed-Solomon codes have been overlooked in distributed storage applications due to the common belief that they have poor repair bandwidth: A naive repair approach would require the whole file to be reconstructed in order to recover a single erased codeword symbol. In a recent work, Guruswami and Wootters (STOC’16) proposed a single-erasure repair method for Reed-Solomon codes that achieves the optimal repair bandwidth amongst all linear encoding schemes. Their key idea is to recover the erased symbol by collecting a sufficiently large number of its traces, each of which can be constructed from a number of traces of other symbols. We extend the trace collection technique to cope with two and three erasures.

We introduce the notion of weakly mutually uncorrelated (WMU) sequences, motivated by applications in DNAbased data storage systems and synchronization between communication devices. WMU sequences are characterized by the property that no sufficiently long suffix of one sequence is the prefix of the same or another sequence. WMU sequences used for primer design in DNA-based data storage systems are also required to be at large mutual Hamming distance from each other, have balanced compositions of symbols, and avoid primer-dimer byproducts. We derive bounds on the size of WMU and various constrained WMU codes and present a number of constructions for balanced, error-correcting, primer-dimer free WMU codes using Dyck paths, prefix-synchronized and cyclic codes.

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 codeword from $N$ distinct noisy reads. We focus on channels that introduce a single edit error (i.e. a single substitution, insertion, or deletion) and their variants, and design reconstruction codes for all values of $N$ . In particular, for the case of a single edit, we show that as the number of noisy reads increases, the number of redundant symbols required can be gracefully reduced from $\log _{q} n+O(1)$ to $\log _{q} \log _{q} n+O(1)$ , and then to $O(1)$ , where $n$ denotes the length of a codeword. We also show that these reconstruction codes are asymptotically optimal. Finally, via computer simulations, we demonstrate that in certain cases, reconstruction codes can achieve similar performance as classical error-correcting codes with less redundant symbols.