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By employing the residue polynomials, a construction of constant-composition codes is given. This construction generalizes the one proposed by Xing[16]. It turns out that when d=3 this construction gives a lower bound of…
We develop an approach through geometric functional analysis to error correcting codes and to reconstruction of signals from few linear measurements. An error correcting code encodes an n-letter word x into an m-letter word y in such a way…
Locally repairable codes (LRCs) are error correcting codes used in distributed data storage. A traditional approach is to look for codes which simultaneously maximize error tolerance and minimize storage space consumption. However, this…
Given a Fractional Repetition code, finding the reconstruction and repair degree in a distributed storage system is an important problem. In this work, we present algorithms for computing the reconstruction and repair degree of fractional…
Multilevel methods represent a powerful approach in numerical solution of partial differential equations. The multilevel structure can also be used to construct estimates for total and algebraic errors of computed approximations. This paper…
The classical Reed-Muller codes over a finite field $\mathbb{F}_q$ are based on evaluations of $m$-variate polynomials of degree at most $d$ over a product set $U^m$, for some $d$ less than $|U|$. Because of their good distance properties,…
In this paper we propose several strategies for the exact computation of the determinant of a rational matrix. First, we use the Chinese Remaindering Theorem and the rational reconstruction to recover the rational determinant from its…
Higher genus modular invariance of two-dimensional conformal field theories (CFTs) is a largely unexplored area. In this paper, we derive explicit expressions for the higher genus partition functions of a specific class of CFTs: code CFTs,…
Augmenting the balanced residue number system moduli-set $\{m_1=2^n,m_2=2^n-1,m_3=2^n+1\}$, with the co-prime modulo $m_4=2^{2n}+1$, increases the dynamic range (DR) by around 70%. The Mersenne form of product $m_2 m_3 m_4=2^{4n}-1$, in the…
We consider a large-scale matrix multiplication problem where the computation is carried out using a distributed system with a master node and multiple worker nodes, where each worker can store parts of the input matrices. We propose a…
One central theme in quantum error-correction is to construct quantum codes that have a large minimum distance. In this paper, we first present a construction of classical codes based on certain class of polynomials. Through these classical…
Let $g(X)$ be a polynomial over a finite field ${\mathbb F}_q$ with degree $o(q^{1/2})$, and let $\chi$ be the quadratic residue character. We give a polynomial time algorithm to recover $g(X)$ (up to perfect square factors) given the…
We study robust convex quadratic programs where the uncertain problem parameters can contain both continuous and integer components. Under the natural boundedness assumption on the uncertainty set, we show that the generic problems are…
We examine regular and irregular repeat-accumulate (RA) codes with repetition degrees which are all even. For these codes and with a particular choice of an interleaver, we give an upper bound on the decoding error probability of a…
This paper presents and analyzes a novel concatenated coding scheme for enabling error resilience in two distributed storage settings: one being storage using existing regenerating codes and the second being storage using locally repairable…
Quantum error-correcting codes with good parameters can be constructed by evaluating polynomials at the roots of the polynomial trace. In this paper, we propose to evaluate polynomials at the roots of trace-depending polynomials (given by a…
We develop a polynomial reduction procedure that transforms any gauge fixed CHY amplitude integrand for $n$ scattering particles into a $\sigma$-moduli multivariate polynomial of what we call the $\textit{standard form}$. We show that a…
We present a method to increase the dynamical range of a Residue Number System (RNS) by adding virtual RNS layers on top of the original RNS, where the required modular arithmetic for a modulus on any non-bottom layer is implemented by…
We adress the problem of the algebraic decoding of any cyclic code up to the true minimum distance. For this, we use the classical formulation of the problem, which is to find the error locator polynomial in terms of the syndroms of the…
Robust PCA is a widely used statistical procedure to recover a underlying low-rank matrix with grossly corrupted observations. This work considers the problem of robust PCA as a nonconvex optimization problem on the manifold of low-rank…