Related papers: Term Coding for Extremal Combinatorics: Dispersion…
For nonnegative integers $n_2, n_3$ and $d$, let $N(n_2,n_3,d)$ denote the maximum cardinality of a code of length $n_2+n_3$, with $n_2$ binary coordinates and $n_3$ ternary coordinates (in this order) and with minimum distance at least…
An algorithm is proposed, analyzed, and tested for solving continuous nonlinear-equality-constrained optimization problems where the objective and constraint functions are defined by expectations or averages over large, finite numbers of…
The distributed index coding problem is studied, whereby multiple messages are stored at different servers to be broadcast to receivers with side information. First, the existing composite coding scheme is enhanced for the centralized…
Linear constraints for a matrix polytope with no fractional vertex are investigated as intersecting research among permutation codes, rank modulations, and linear programming methods. By focusing the discussion to the block structure of…
While most methods for solving mixed-integer optimization problems compute a single optimal solution, a diverse set of near-optimal solutions can often lead to improved outcomes. We present a new method for finding a set of diverse…
We examine an error-correcting coding framework in which each coded symbol is constrained to be a function of a fixed subset of the message symbols. With an eye toward distributed storage applications, we seek to design systematic codes…
We consider several coding discretizations of continuous functions which reflect their variation at some given precision. We study certain statistical and combinatorial properties of the sequence of finite words obtained by coding a typical…
We provide a combinatorial construction for linear codes attaining the maximum possible number of distinct weights. We then introduce the related problem of determining the existence of linear codes with an arbitrary number of distinct…
As far as we know, there is no decoding algorithm of any binary self-dual $[40, 20, 8]$ code except for the syndrome decoding applied to the code directly. This syndrome decoding for a binary self-dual $[40,20,8]$ code is not efficient in…
Assume $D$ is a finite set and $R$ is a finite set of functions from $D$ to the natural numbers. An instance of the minimum $R$-cost homomorphism problem ($MinHom_R$) is a set of variables $V$ subject to specified constraints together with…
In this paper, we use the linear programming approach to find new upper bounds for the moments of isotropic measures. These bounds are then utilized for finding lower packing bounds and energy bounds for projective codes. We also show that…
Let $P = \{p(i)\}$ be a measure of strictly positive probabilities on the set of nonnegative integers. Although the countable number of inputs prevents usage of the Huffman algorithm, there are nontrivial $P$ for which known methods find a…
This paper presents encoding and decoding algorithms for several families of optimal rank metric codes whose codes are in restricted forms of symmetric, alternating and Hermitian matrices. First, we show the evaluation encoding is the right…
Stochastic equations indexed by negative integers and taking values in compact groups are studied. Extremal solutions of the equations are characterized in terms of infinite products of independent random variables. This result is applied…
Polynomial, or Delsarte's, method in coding theory accounts for a variety of structural results on, and bounds on the size of, extremal configurations (codes and designs) in various metric spaces. In recent works of the authors the…
We consider a problem in which a source is encoded into N packets, an unknown number of which are subject to adversarial errors en route to the decoder. We seek code designs for which the decoder is guaranteed to be able to reproduce the…
We study several questions related to diversifying search results. We give improved approximation algorithms in each of the following problems, together with some lower bounds. - We give a polynomial-time approximation scheme (PTAS) for a…
Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in…
It is well known that {\em superposition coding}, namely separately encoding the independent sources, is optimal for symmetric multilevel diversity coding (SMDC) (Yeung-Zhang 1999). However, the characterization of the coding rate region…
In this work the topological order at finite temperature in two-dimensional color code is studied. The topological entropy is used to measure the behavior of the topological order. Topological order in color code arises from the colored…