Related papers: Perfect Codes Correcting a Single Burst of Limited…
Perfect codes in the $n$-dimensio\-nal grid $\Lambda_n$ of the lattice $\mathbb{Z}^n$ ($0<n\in\mathbb{Z}$) and its quotient toroidal grids were obtained via the truncated distance in $\mathbb{Z}^n$ given between $u=(u_1,\cdots,u_n)$ and…
MDS codes are erasure-correcting codes that can correct the maximum number of erasures for a given number of redundancy or parity symbols. If an MDS code has $r$ parities and no more than $r$ erasures occur, then by transmitting all the…
In this paper, we introduce the notion of perfect space-time block codes (STBC). These codes have full rate, full diversity, non-vanishing constant minimum determinant for increasing spectral efficiency, uniform average transmitted energy…
The high repair cost of (n,k) Maximum Distance Separable (MDS) erasure codes has recently motivated a new class of codes, called Regenerating Codes, that optimally trade off storage cost for repair bandwidth. On one end of this spectrum of…
The known quantum error-correcting codes are typically built on approximative open-quantum-system models such as Born--Markov master equations. However, it is an open question how such codes perform in actual physical systems that, to some…
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…
Motivated by DNA storage in living organisms, and by known biological mutation processes, we study the reverse-complement string-duplication system. We fully classify the conditions under which the system has full expressiveness, for all…
We present a quantum error correction code which protects a qubit of information against general one qubit errors which maybe caused by the interaction with the environment. To accomplish this, we encode the original state by distributing…
The perfect space-time block codes (STBCs) are based on four design criteria - full-rateness, non-vanishing determinant, cubic shaping and uniform average transmitted energy per antenna per time slot. Cubic shaping and transmission at…
Motivated by DNA storage systems and 3D fingerprinting, this work studies the adversarial torn paper channel with edit errors. This channel first applies at most $t_e$ edit errors (i.e., insertions, deletions, and substitutions) to the…
Random quantum circuits have played a central role in establishing the computational advantages of near-term quantum computers over their conventional counterparts. Here, we use ensembles of low-depth random circuits with local connectivity…
Recently, network error correction coding (NEC) has been studied extensively. Several bounds in classical coding theory have been extended to network error correction coding, especially the Singleton bound. In this paper, following the…
The minimum distance graph of a code has the codewords as vertices and edges exactly when the Hamming distance between two codewords equals the minimum distance of the code. A constructive proof for reconstructibility of an extended perfect…
Many quantum technologies are now reaching a high level of maturity and control, and it is likely that the first demonstrations of suppression of naturally occurring quantum noise using small topological error correcting codes will soon be…
To ensure resilience against the unavoidable noise in quantum computers, quantum information needs to be encoded using an error-correcting code, and circuits must have a particular structure to be fault-tolerant. Compilation of…
Surface codes are versatile quantum error-correcting codes known for their planar geometry, making them ideal for practical implementations. While the original proposal used Pauli $X$ or Pauli $Z$ operators in a square structure, these…
This is an expository article aiming to introduce the reader to the underlying mathematics and geometry of quantum error correction. Information stored on quantum particles is subject to noise and interference from the environment. Quantum…
The construction of topological error correction codes requires the ability to fabricate a lattice of physical qubits embedded on a manifold with a non-trivial topology such that the quantum information is encoded in the global degrees of…
Quantum error correction protects quantum information against environmental noise. When using qubits, a measure of quality of a code is the maximum number of errors that it is able to correct. We show that a suitable notion of ``number of…
In [1], K\"otter and Kschischang presented a new model for error correcting codes in network coding. The alphabet in this model is the subspace lattice of a given vector space, a code is a subset of this lattice and the used metric on this…