Related papers: Codes over a subset of Octonion Integers
Given a matrix over a skew field fixing the column (1,...,1)^t, we give formulas for a row vector fixed by this matrix. The same techniques are applied to give noncommutative extensions of probabilistic properties of codes.
When digital data are transmitted over a noisy channel, it is important to have a mechanism allowing recovery against a limited number of errors. Normally, a user string of 0's and 1's, called bits, is encoded by adding a number of…
This paper provides a new instance of quantum deletion error-correcting codes. This code can correct any single quantum deletion error, while our code is only of length 4. This paper also provides an example of an encoding quantum circuit…
We present a general theory to obtain good linear network codes utilizing the osculating nature of algebraic varieties. In particular, we obtain from the osculating spaces of Veronese varieties explicit families of equidimensional vector…
We define multi-block interleaved codes as codes that allow reading information from either a small sub-block or from a larger full block. The former offers faster access, while the latter provides better reliability. We specify the…
The connection between index coding and matroid theory have been well studied in the recent past. El Rouayheb et al. established a connection between multilinear representation of matroids and wireless index coding. Muralidharan and Rajan…
We consider $t$-Lee-error-correcting codes of length $n$ over the residue ring $\mathbb{Z}_m := \mathbb{Z}/m\mathbb{Z}$ and determine upper and lower bounds on the number of $t$-Lee-error-correcting codes. We use two different methods,…
Color codes are a class of topological quantum codes with a high error threshold and large set of transversal encoded gates, and are thus suitable for fault tolerant quantum computation in two-dimensional architectures. Recently,…
Extended variants of the recently introduced spread unary coding are described. These schemes, in which the length of the code word is fixed, allow representation of approximately n^2 numbers for n bits, rather than the n numbers of the…
This paper presents conditions for constructing permutation-invariant quantum codes for deletion errors and provides a method for constructing them. Our codes give the first example of quantum codes that can correct two or more deletion…
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…
This paper introduces the notion of multiset codes as relevant to the problem of reliable information transmission over permutation channels. The motivation for studying permutation channels comes from the effect of out of order delivery of…
We define multilevel codes on bipartite graphs that have properties analogous to multilevel serial concatenations. A decoding algorithm is described that corrects a proportion of errors equal to half the Blokh-Zyablov bound on the minimum…
Matrix Code gives imperative programming a mathematical semantics and heuristic power comparable in quality to functional and logic programming. A program in Matrix Code is developed incrementally from a specification in pre/post-condition…
We demonstrate a construction of error-correcting codes from graphs by means of $k$-resolving sets, and present a decoding algorithm which makes use of covering designs. Along the way, we determine the $k$-metric dimension of grid graphs…
This paper addresses network code design for robust transmission of sources over an orthogonal two-hop wireless network with a broadcasting relay. The network consists of multiple sources and destinations in which each destination,…
An error correcting code using a tree-like multilayer perceptron is proposed. An original message $\mbi{s}^0$ is encoded into a codeword $\boldmath{y}_0$ using a tree-like committee machine (committee tree) or a tree-like parity machine…
A binary 1-error-correcting code can always be embedded in a 1-perfect code of some larger length
We study the largest possible length $B$ of $(B-1)$-dimensional linear codes over $\mathbb{F}_q$ which can correct up to $t$ errors taken from a restricted set $\mathcal{A}\subseteq \mathbb{F}_q^*$. Such codes can be applied to multilevel…
A permutation-invariant code on m qubits is a subspace of the symmetric subspace of the m qubits. We derive permutation-invariant codes that can encode an increasing amount of quantum information while suppressing leading order spontaneous…