相关论文: Perfect single error-correcting codes in the Johns…
We study exact algorithms for Euclidean TSP in $\mathbb{R}^d$. In the early 1990s algorithms with $n^{O(\sqrt{n})}$ running time were presented for the planar case, and some years later an algorithm with $n^{O(n^{1-1/d})}$ running time was…
We investigate the potential of scale-free networks as error-correcting codes. We find that irregular low-density parity-check codes with highest performance known to date have degree distributions well fitted by a power-law function…
A permutationally invariant n-bit code for quantum error correction can be realized as a subspace stabilized by the non-Abelian group S_n. The code corresponds to bases for the trivial representation, and all other irreducible…
The iterated Johnson bound is the best known upper bound on a size of an error-correcting code in the Grassmannian $\mathcal{G}_q(n,k)$. The iterated Sch\"{o}nheim bound is the best known lower bound on the size of a covering code in…
Ternary constant weight codes of length $n=2^m$, weight $n-1$, cardinality $2^n$ and distance $5$ are known to exist for every $m$ for which there exists an APN permutation of order $2^m$, that is, at least for all odd $m \geq 3$ and for…
In 1969 J. Verhoeff provided the first examples of a decimal error detecting code using a single check digit to provide protection against all single, transposition and adjacent twin errors. The three versions of such a code that he…
We shall study three subjects of the Jacobi-Perron Algorithm of dimension 2. First, we study the "ideal convergence". About the approximations (p_n/r_n, q_n/r_n) to (A, B) (where A and B are positive real numbers, r_n, p_n and q_n are…
In an underdetermined system of equations $Ax=y$, where $A$ is an $m\times n$ matrix, only $u$ of the entries of $y$ with $u < m$ are known. Thus $E_jw$, called `measurements', are known for certain $j\in J \subset \{0,1,\ldots,m-1\}$ where…
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,…
The Johnson-type upper bound on the maximum size of a code of length $n$, distance $d=2w-1$ and constant composition ${\overline{w}}$ is $\lfloor\dfrac{n}{w_1}\rfloor$, where $w$ is the total weight and $w_1$ is the largest component of…
Traditional quantum error correction involves the redundant encoding of k quantum bits using n quantum bits to allow the detection and correction of any t bit error. The smallest general t=1 code requires n=5 for k=1. However, the dominant…
We present a construction scheme for quantum error correcting codes. The basic ingredients are a graph and a finite abelian group, from which the code can explicitly be obtained. We prove necessary and sufficient conditions for the graph…
Property testing has been a major area of research in computer science in the last three decades. By property testing we refer to an ensemble of problems, results and algorithms which enable to deduce global information about some data by…
In this survey we concern ourself with the question, wether there exists a fix-free code for a given sequence of codeword lengths. For a given alphabet, we obtain the {\em Kraftsum} of a code, if we divide for every length the number of…
Let $\sigma(n)$ denote the sum of the positive divisors of $n$. We say that $n$ is perfect if $\sigma(n) = 2 n$. Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form…
We show that for every length of form $4^k-1$, there exists a binary $1$-perfect code that does not include any Preparata-like code.
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…
We prove that any completely regular code with minimum eigenvalue in any geometric graph G corresponds to a completely regular code in the clique graph of G. Studying the interrelation of these codes, a complete characterization of the…
We classify all binary error correcting completely regular codes of length $n$ with minimum distance $\delta>n/2$.
In this article we address the computational hardness of optimally decoding a quantum stabilizer code. Much like classical linear codes, errors are detected by measuring certain check operators which yield an error syndrome, and the…