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We present error-correcting codes that achieve the information-theoretically best possible trade-off between the rate and error-correction radius. Specifically, for every $0 < R < 1$ and $\eps> 0$, we present an explicit construction of…
Entanglement-assisted quantum error-correcting codes (EAQECCs) to desired rate, error-correcting capability and maximum shared entanglement are constructed. Thus for a required rate $R$, required error-correcting capability to correct $t$…
A key principle in string processing is local consistency: using short contexts to handle matching fragments of a string consistently. String synchronizing sets [Kempa, Kociumaka; STOC 2019] are an influential instantiation of this…
Limited magnitude asymmetric error model is well suited for flash memory. In this paper, we consider the construction of asymmetric codes correcting single error over $\mathbb{Z}_{2^{k}r}$ and which are based on so called $B_{1}[4](2^{k}r)$…
We present a semidefinite program optimization approach to quantum error correction that yields codes and recovery procedures that are robust against significant variations in the noise channel. Our approach allows us to optimize the…
We prove that every $1$-error-correcting code over a finite field can be embedded in a $1$-perfect code of some larger length. Embedding in this context means that the original code is a subcode of the resulting $1$-perfect code and can be…
Series of maximum distance quantum error-correcting codes are developed and analysed. For a given rate and given error-correction capability, quantum error-correcting codes with these specifications are constructed. The codes are explicit…
Autonomous quantum memories are a way to passively protect quantum information using engineered dissipation that creates an ``always-on'' decoder. We analyze Markovian autonomous decoders that can be implemented with a wide range of qubit…
We explore the design of quantum error-correcting codes for cases where the decoherence events of qubits are correlated. In particular, we consider the case where only spatially contiguous qubits decohere, which is analogous to the case of…
We study 4 problems in string matching, namely, regular expression matching, approximate regular expression matching, string edit distance, and subsequence indexing, on a standard word RAM model of computation that allows logarithmic-sized…
In the framework of write-once memory (WOM) codes, it is important to distinguish between codes that can be decoded directly and those that require that the decoder knows the current generation to successfully decode the state of the…
Traditional error-correcting codes (ECCs) assume a fixed message length, but many scenarios involve ongoing or indefinite transmissions where the message length is not known in advance. For example, when streaming a video, the user should…
Quantum error correction is capable of digitizing quantum noise and increasing the robustness of qubits. Typically, error correction is designed with the target of eliminating all errors - making an error so unlikely it can be assumed that…
We consider rate R = k/n causal linear codes that map a sequence of k-dimensional binary vectors {b_t} to a sequence of n-dimensional binary vectors {c_t}, such that each c_t is a function of {b_1,b_2,...,b_t}. Such a code is called anytime…
We describe a simple quantum error correcting code built out of a time-dependent transverse field Ising model. The code is similar to a repetition code, but has two advantages: an $N$-qubit code can be implemented with a finite-depth…
Indexing highly repetitive texts --- such as genomic databases, software repositories and versioned text collections --- has become an important problem since the turn of the millennium. A relevant compressibility measure for repetitive…
A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (2-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not…
We consider the approximate pattern matching problem under edit distance. In this problem we are given a pattern $P$ of length $w$ and a text $T$ of length $n$ over some alphabet $\Sigma$, and a positive integer $k$. The goal is to find all…
Quantum error correction is widely believed to be essential for large-scale quantum computation, but the required qubit overhead remains a central challenge. Quantum low-density parity-check codes can substantially reduce this overhead…
Many problems that can be solved in quadratic time have bit-parallel speed-ups with factor $w$, where $w$ is the computer word size. A classic example is computing the edit distance of two strings of length $n$, which can be solved in…