Related papers: Perfect codes over non-prime power alphabets: an a…
Quantum error correction plays an important role in fault-tolerant quantum information processing. It is usually difficult to experimentally realize quantum error correction, as it requires multiple qubits and quantum gates with high…
A major milestone of quantum error correction is to achieve the fault-tolerance threshold beyond which quantum computers can be made arbitrarily accurate. This requires extraordinary resources and engineering efforts. We show that even…
We present two new positive results for reliable computation using formulas over physical alphabets of size $q > 2$. First, we show that for logical alphabets of size $\ell = q$ the threshold for denoising using gates subject to $q$-ary…
Non-overlapping codes are a set of codewords in $\bigcup_{n \ge 2} \mathbb{Z}_q^n$, where $\mathbb{Z}_q = \{0,1,\dots,q-1\}$, such that, the prefix of each codeword is not a suffix of any codeword in the set, including itself; and for…
With the rapid developments in quantum hardware comes a push towards the first practical applications on these devices. While fully fault-tolerant quantum computers may still be years away, one may ask if there exist intermediate forms of…
Cyclic codes are an important subclass of linear codes with wide applications in communication systems and data storage systems. In 2013, Ding and Helleseth presented nine open problems on optimal ternary cyclic codes $\mathcal{C}_{(1,e)}$.…
Large-scale quantum computers will inevitably need quantum error correction (QEC) to protect information against decoherence. Given that the overhead of such error correction is often formidable, autonomous quantum error correction (AQEC)…
A perfect code in a graph is an independent set of the graph such that every vertex outside the set is adjacent to exactly one vertex in the set. A circulant graph is a Cayley graph of a cyclic group. In this paper we study perfect codes in…
In this thesis we present several results in coding theory, concerning error-correcting codes and the Shannon capacity. 1. We give a general symmetry reduction of matrices occuring in semidefinite programs in coding theory. 2. We apply the…
Quantum error correcting codes have a distance parameter, conveying the minimum number of single spin errors that could cause error correction to fail. However, the success thresholds of finite per-qubit error rate that have been proven for…
We study codes with parameters of the ternary Hamming $(n=(3^m-1)/2,3^{n-m},3)$ code, i.e., ternary $1$-perfect codes. The rank of the code is defined to be the dimension of its affine span. We characterize ternary $1$-perfect codes of rank…
A perfect code in a graph $\Gamma = (V, E)$ is a subset $C$ of $V$ that is an independent set such that every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A total perfect code in $\Gamma$ is a subset $C$ of $V$ such…
We present a two-step decoder for the parity code and evaluate its performance in code-capacity and faulty-measurement settings. For noiseless measurements, we find that the decoding problem can be reduced to a series of repetition codes…
Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding…
Quantum error correction allows for faulty quantum systems to behave in an effectively error free manner. One important class of techniques for quantum error correction is the class of quantum subsystem codes, which are relevant both to…
A perfect number is a positive integer $N$ such that the sum of all the positive divisors of $N$ equals $2N$, denoted by $\sigma(N) = 2N$. The question of the existence of odd perfect numbers (OPNs) is one of the longest unsolved problems…
We consider the problem of optimally decoding a quantum error correction code -- that is to find the optimal recovery procedure given the outcomes of partial "check" measurements on the system. In general, this problem is NP-hard. However,…
Quantum error-correcting codes are analyzed from an information-theoretic perspective centered on quantum conditional and mutual entropies. This approach parallels the description of classical error correction in Shannon theory, while…
The maximum operational range of continuous variable quantum key distribution protocols has shown to be improved by employing high-efficiency forward error correction codes. Typically, the secret key rate model for such protocols is…
Mixed codes, which are error-correcting codes in the Cartesian product of different-sized spaces, model degrading storage systems well. While such codes have previously been studied for their algebraic properties (e.g., existence of perfect…