Related papers: Upper bounds on the length function for covering c…
A binary code Enc$:\{0,1\}^k \to \{0,1\}^n$ is $(0.5-\epsilon,L)$-list decodable if for all $w \in \{0,1\}^n$, the set List$(w)$ of all messages $m \in \{0,1\}^k$ such that the relative Hamming distance between Enc$(m)$ and $w$ is at most…
Let $q\in(1,2)$ and $x\in[0,\frac1{q-1}]$. We say that a sequence $(\epsilon_i)_{i=1}^{\infty}\in\{0,1\}^{\mathbb{N}}$ is an expansion of $x$ in base $q$ (or a $q$-expansion) if x=\sum_{i=1}^{\infty}\epsilon_iq^{-i}. Let…
We give a new construction of algebraic codes which are efficiently list decodable from a fraction $1-R-\eps$ of adversarial errors where $R$ is the rate of the code, for any desired positive constant $\eps$. The worst-case list size output…
Multiple coverings of the farthest-off points ($(R,\mu)$-MCF codes) and the corresponding $(\rho,\mu)$-saturating sets in projective spaces $PG(N,q)$ are considered. We propose and develop some methods which allow us to obtain new small…
We establish connections between the size of circuits and formulas computing monotone Boolean functions and the size of first-order and nonrecursive Datalog rewritings for conjunctive queries over OWL 2 QL ontologies. We use known lower…
In this paper, we revisit the problem of finding the longest systematic-length $k$ for a linear minimum storage regenerating (MSR) code with optimal repair of only systematic part, for a given per-node storage capacity $l$ and an arbitrary…
The study of Armstrong codes is motivated by the problem of understanding complexities of dependencies in relational database systems, where attributes have bounded domains. A $(q,k,n)$-Armstrong code is a $q$-ary code of length $n$ with…
Constructing a Pseudo Random Function (PRF) is a fundamental problem in cryptology. Such a construction, implemented by truncating the last $m$ bits of permutations of $\{0, 1\}^{n}$ was suggested by Hall et al. (1998). They conjectured…
The maximum size of a binary code is studied as a function of its length N, minimum distance D, and minimum codeword weight W. This function B(N,D,W) is first characterized in terms of its exponential growth rate in the limit as N tends to…
An (n,r)-arc in PG(2,q) is a set of n points such that each line contains at most r of the selected points. It is well-known that (n,r)-arcs in PG(2,q) correspond to projective linear codes. Let m_r(2,q) denote the maximal number n of…
In this paper we study the covering numbers of the space of convex and uniformly bounded functions in multi-dimension. We find optimal upper and lower bounds for the $\epsilon$-covering number of $\C([a, b]^d, B)$, in the $L_p$-metric, $1…
In this paper, we propose new lower and upper bounds on the linear extension complexity of regular $n$-gons. Our bounds are based on the equivalence between the computation of (i) an extended formulation of size $r$ of a polytope $P$, and…
We construct new linear codes with high minimum distance d. In at least 12 cases these codes improve the minimum distance of the previously known best linear codes for fixed parameters n,k. Among these new codes there is an optimal ternary…
The explosion in the volumes of data being stored online has resulted in distributed storage systems transitioning to erasure coding based schemes. Local Reconstruction Codes (LRCs) have emerged as the codes of choice for these…
Nearly perfect packing codes are those codes that meet the Johnson upper bound on the size of error-correcting codes. This bound is an improvement to the sphere-packing bound. A related bound for covering codes is known as the van Wee…
A lower bound on the maximum likelihood (ML) decoding error exponent of linear block code ensembles, on the erasure channel, is developed. The lower bound turns to be positive, over an ensemble specific interval of erasure probabilities,…
We determine the minimal bi-degree(s) of an irreducible filling curve over $\mathbb{F}_q$ for $\mathbb{P}^1\times \mathbb{P}^1$. It is $(q+1, q+1)$ if $q\neq 2$, and they are $(4,3)$ and $(3,4)$ if $q=2$.
As an important coding scheme in modern distributed storage systems, locally repairable codes (LRCs) have attracted a lot of attentions from perspectives of both practical applications and theoretical research. As a major topic in the…
Recently, linear codes with a few weights were widely investigated due to their applications in secret sharing schemes and authentication schemes. In this letter, we present a class of $q$-ary linear codes derived from irreducible cyclic…
A fundamental question in the theory of reinforcement learning is: suppose the optimal $Q$-function lies in the linear span of a given $d$ dimensional feature mapping, is sample-efficient reinforcement learning (RL) possible? The recent and…