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Recent progress in quantum cryptography and quantum computers has given hope to their imminent practical realization. An essential element at the heart of the application of these quantum systems is a quantum error correction scheme. We…
We study perfect error-correcting codes in $\mathbb{Z}^n$ for the symmetric limited-magnitude error channel, where at most $e$ coordinates of an integer vector may be altered by a value whose magnitude is at most $s$. Geometrically, such…
In this paper, the problem of correction of a single error in $q$-ary symmetric channel with noiseless feedback is considered. We propose an algorithm to construct codes with feedback inductively. For all prime power $q$ we prove that two…
Standard approaches to quantum error correction for fault-tolerant quantum computing are based on encoding a single logical qubit into many physical ones, resulting in asymptotically zero encoding rates and therefore huge resource…
In this paper, we investigate the error correction capability of column-weight-three LDPC codes when decoded using the Gallager A algorithm. We prove that the necessary condition for a code to correct $k \geq 5$ errors is to avoid cycles of…
We show that errors in data transmitted through linear codes can be thought of as codewords of minimum weight of new linear codes. To determine errors we can then use methods specific to finding such special codewords. One of these methods…
This paper studies \emph{linear} and \emph{affine} error-correcting codes for correcting synchronization errors such as insertions and deletions. We call such codes linear/affine insdel codes. Linear codes that can correct even a single…
Error-correcting codes are a method for representing data, so that one can recover the original information even if some parts of it were corrupted. The basic idea, which dates back to the revolutionary work of Shannon and Hamming about a…
We study uniquely decodable codes and list decodable codes in the high-noise regime, specifically codes that are uniquely decodable from $\frac{1-\varepsilon}{2}$ fraction of errors and list decodable from $1-\varepsilon$ fraction of…
Realizing the potential of quantum computing will require achieving sufficiently low logical error rates. Many applications call for error rates in the $10^{-15}$ regime, but state-of-the-art quantum platforms typically have physical error…
We present a method of concatenated quantum error correction in which improved classical processing is used with existing quantum codes and fault-tolerant circuits to more reliably correct errors. Rather than correcting each level of a…
In this paper, we construct systematic $q$-ary two-deletion correcting codes and burst-deletion correcting codes, where $q\geq 2$ is an even integer. For two-deletion codes, our construction has redundancy $5\log n+O(\log q\log\log n)$ and…
Determining the maximum number of unit vectors in $\mathbb{R}^r$ with no pairwise inner product exceeding $\alpha$ is a fundamental problem in geometry and coding theory. In 1955, Rankin resolved this problem for all $\alpha \leq 0$ and in…
Non-overlapping codes are block codes that have arisen in diverse contexts of computer science and biology. Applications typically require finding non-overlapping codes with large cardinalities, but the maximum size of non-overlapping codes…
Intersecting codes are a classical object in coding theory whose rank-metric analogue has recently been introduced. Although the definition formally parallels the Hamming-metric case, the structure and parameter constraints of rank-metric…
In quantum error correction using imperfect primitives, errors of high weight arising from a few faults are major concerns since they might not be correctable by the quantum error correcting code. Fortunately, some errors of different…
Motivated by limitations and capabilities of neutral atom qubits, we examine whether measurement-free error correction can produce practical error thresholds. We show that this can be achieved by extracting redundant syndrome information,…
Quantum computers hold the promise of solving computational problems which are intractable using conventional methods. For fault-tolerant operation quantum computers must correct errors occurring due to unavoidable decoherence and limited…
In the last 60 years coding theory has been studied a lot over finite fields $\mathbb{F}_q$ or commutative rings $\mathcal{R}$ with unity. Although in $1993$, a study on the classification of the rings (not necessarily commutative or ring…
In this paper, we investigate the optimal nonadditive quantum error-detecting codes with distance two. The the numerical simulation shows that, with n being can be 5, 6, 7, 8, 10 and 12, such the n-qubit quantum error-detecting codes with…