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Accurate decoding of quantum error-correcting codes is a crucial ingredient in protecting quantum information from decoherence. It requires characterizing the error channels corrupting the logical quantum state and providing this…
We consider decoding of vertically homogeneous interleaved sum-rank-metric codes with high interleaving order $s$, that are constructed by stacking $s$ codewords of a single constituent code. We propose a Metzner--Kapturowski-like decoding…
Twisted generalized Reed-Solomon (TGRS) codes were introduced to extend the algebraic capabilities of classical generalized Reed-Solomon (GRS) codes. This extension holds the potential for constructing new non-GRS maximum distance separable…
The main ideas of quantum error correction are introduced. These are encoding, extraction of syndromes, error operators, and code construction. It is shown that general noise and relaxation of a set of 2-state quantum systems can always be…
We generalize Helberg's number-theoretic construction of multiple insertion-deletion correcting binary codes to non-binary alphabets and describe a linear decoding algorithm for correcting multiple deletions.
Higher order singular value decomposition (HOSVD) is an important tool for analyzing big data in multilinear algebra and machine learning. In this paper, we present two quantum algorithms for HOSVD. Our methods allow one to decompose a…
Quantum codes typically rely on large numbers of degrees of freedom to achieve low error rates. However each additional degree of freedom introduces a new set of error mechanisms. Hence minimizing the degrees of freedom that a quantum code…
We develop novel protocols for generating loss-tolerant quantum codes; these are central for safeguarding information against qubit losses, with most crucial applications in quantum communications. Contrary to current proposals, our method…
Constructing Reed--Solomon (RS) codes that can correct insertions and deletions (insdel errors) has been considered in numerous recent works. For the special case of two-dimensional RS-codes, it is known [CST23] that an $[n,2]_q$ RS-code…
Regenerating codes are a class of codes proposed for providing reliability of data and efficient repair of failed nodes in distributed storage systems. In this paper, we address the fundamental problem of handling errors and erasures during…
This paper presents the first decoding algorithm for Gabidulin codes over Galois rings with provable quadratic complexity. The new method consists of two steps: (1) solving a syndrome-based key equation to obtain the annihilator polynomial…
The interpolation step of Guruswami and Sudan's list decoding of Reed-Solomon codes poses the problem of finding the minimal polynomial of an ideal with respect to a certain monomial order. An efficient algorithm that solves the problem is…
This paper gives some theory and efficient design of binary block systematic codes capable of controlling the deletions of the symbol ``$0$'' (referred to as $0$-deletions) and/or the insertions of the symbol ``$0$'' (referred to as…
A general framework describing the statistical discrimination of an ensemble of quantum channels is given by the name of quantum reading. Several tools can be applied in quantum reading to reduce the error probability in distinguishing the…
Due to the fragility of quantum mechanical effects, real quantum computers are plagued by frequent noise effects that cause errors during computations. Quantum error-correcting codes address this problem by providing means to identify and…
This paper investigates quantum error correction schemes for fully-correlated noise channels on an $n$-qubit system, where error operators take the form $W^{\otimes n}$, with $W$ being an arbitrary $2\times 2$ unitary operator. In previous…
Subspace codes were introduced in order to correct errors and erasures for randomized network coding, in the case where network topology is unknown (the noncoherent case). Subspace codes are indeed collections of subspaces of a certain…
Quantum error correction is expected to be essential in large-scale quantum technologies. However, the substantial overhead of qubits it requires is thought to greatly limit its utility in smaller, near-term devices. Here we introduce a new…
Quantum error correction is a set of methods to protect quantum information--that is, quantum states--from unwanted environmental interactions (decoherence) and other forms of noise. The information is stored in a quantum error-correcting…
Designing quantum error correcting codes that promise a high error threshold, low resource overhead and efficient decoding algorithms is crucial to achieve large-scale fault-tolerant quantum computation. The concatenated quantum Hamming…