Related papers: Covering sets for limited-magnitude errors

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…

Let $\rho$ f,$\lambda$ be the residual Galois representation attached to a newform f and a prime ideal $\lambda$ in the integer ring of its coefficient field. In this paper, we prove explicit bounds for the residue characteristic of the…

We consider the problem of computing the data-cube marginals of a fixed order $k$ (i.e., all marginals that aggregate over $k$ dimensions), using a single round of MapReduce. The focus is on the relationship between the reducer size (number…

Let $q$ be a fixed odd prime. We show that a finite subset $B$ of integers, not containing any perfect $q^{th}$ power, contains a $q^{th}$ power modulo almost every prime if and only if $B$ corresponds to a blocking set (with respect to…

Let $p$ be a given modulus, let $u$ be prime to $p$, and consider the linear permutation $u\cdot n\pmod p$ of the residue system modulo $p$. Writing $\langle x\rangle_p$ to denote the least nonnegative residue of $x$ modulo $p$, we say that…

For a prime power \( q \) and a positive integer \( n \), a subspace \( U \subseteq \mathbb{F}_q^n \) is called cyclically covering if the union of all its cyclic shifts covers the whole space \( \mathbb{F}_q^n \). Let \( h_q(n) \) denote…

Concatenation of two quantum error correcting codes with complementary sets of transversal gates can provide a means towards universal fault-tolerant computation. We first show that it is generally preferable to choose the inner code with…

In the Metric Capacitated Covering (MCC) problem, given a set of balls $\mathcal{B}$ in a metric space $P$ with metric $d$ and a capacity parameter $U$, the goal is to find a minimum sized subset $\mathcal{B}'\subseteq \mathcal{B}$ and an…

Error-correcting codes resilient to synchronization errors such as insertions and deletions are known as insdel codes. Due to their important applications in DNA storage and computational biology, insdel codes have recently become a focal…

Let $p$ be a large prime number. We prove that any integer $\lambda$ modulo $p$ can be represented in the form $$ m!n! +\sum_{i=1}^{47}n_i!\equiv \lambda \pmod p, $$ with $\max\{m,n,n_1,\ldots,n_{47}\}\ll p^{1300/1301}.$ This improves the…

Let A_{R,q} denote a family of covering codes, in which the covering radius R and the size q of the underlying Galois field are fixed, while the code length tends to infinity. In this paper, infinite sets of families A_{R,q}, where R is…

Sigma-Delta modulation is a popular method for analog-to-digital conversion of bandlimited signals that employs coarse quantization coupled with oversampling. The standard mathematical model for the error analysis of the method measures the…

A set of integers is \emph{primitive} if it does not contain an element dividing another. Denote by $f(n)$ the number of maximum-size primitive subsets of $\{1,\ldots, 2n\}$. We prove that the limit $\alpha=\lim_{n\rightarrow…

Let $Q(x,y)$ be a quadratic form with discriminant $D\neq 0$. We obtain non trivial upper bound estimates for the number of solutions of the congruence $Q(x,y)\equiv\lambda \pmod{p}$, where $p$ is a prime and $x,y$ lie in certain intervals…

Let $\gg$ be a complex reductive Lie algebra and $\kk\subset\gg$ be any reductive in $\gg$ subalgebra. We call a $(\gg,\kk)$-module $M$ bounded if the $\kk$-multiplicities of $M$ are uniformly bounded. In this paper we initiate a general…

We show for each positive integer $a$ that, if $\mathcal{M}$ is a minor-closed class of matroids not containing all rank-$(a+1)$ uniform matroids, then there exists an integer $c$ such that either every rank-$r$ matroid in $\mathcal{M}$ can…

We prove that the covering radius of an $N$-point subset $X_N$ of the unit sphere $S^d \subset R^{d+1}$ is bounded above by a power of the worst-case error for equal weight cubature $\frac{1}{N}\sum_{\mathbf{x} \in X_N}f(\mathbf{x}) \approx…

A $t$-covering array with entries from the alphabet ${\cal Q}=\{0,1,\ldots,q-1\}$ is a $k\times n$ stack, so that for any choice of $t$ (typically non-consecutive) columns, each of the $q^{t}$ possible $t$-letter words over ${\cal Q}$…

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…

An $(m,n,a,b)$-tensor code consists of $m\times n$ matrices whose columns satisfy `$a$' parity checks and rows satisfy `$b$' parity checks (i.e., a tensor code is the tensor product of a column code and row code). Tensor codes are useful in…