Related papers: Arithmetic Operators over Finite Field GF($2^m$) f…
In this paper, we aim to establish a range of numerical radius inequalities. These discoveries will bring us to a recently validated numerical radius inequality and will present numerical radius inequalities that exhibit enhanced precision…
Generalized Reed-Solomon and extended generalized Reed-Solomon (abbreviation to GRS and EGRS) codes are the most well-known family of MDS codes with wide applications in coding theory and practice. Let $\mathbb{F}_q$ be the $q$ elements…
One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment. In past years, many good quantum error-correcting codes had been…
Because of the recent applications to distributed storage systems, researchers have introduced a new class of block codes, i.e., locally recoverable (LRC) codes. LRC codes can recover information from erasure(s) by accessing a small number…
Following recent interest in correctly rounded math library functions (as currently recommended by the IEEE 754 standard), we have designed several SIMD algorithms for one-input single precision functions and integrated them into our CPU…
We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for practical computation in…
Gauss's law-based quantum error correction (GLQEC) offers a promising approach to reducing qubit overhead in lattice gauge theory simulations by leveraging built-in symmetries. For applications of GLQEC to 1+1D lattice quantum…
For the majority of the applications of Reed-Solomon (RS) codes, hard decision decoding is based on syndromes. Recently, there has been renewed interest in decoding RS codes without using syndromes. In this paper, we investigate the…
The aim of this work is to consider multiscale algorithms for solving PDEs with Galerkin methods on bounded domains. We provide results on convergence and condition numbers. We show how to handle PDEs with Dirichlet boundary conditions. We…
Classical hypergeometric functions are well-known to play an important role in arithmetic algebraic geometry. These functions offer solutions to ordinary differential equations, and special cases of such solutions are periods of…
Twisted generalized Reed-Solomon (TGRS) codes were introduced to extend the algebraic capabilities of classical generalized Reed-Solomon (GRS) codes. This extension holds the potential for constructing new non-GRS maximum distance separable…
Graphics Processing Unit (GPU) computing is becoming an alternate computing platform for numerical simulations. However, it is not clear which numerical scheme will provide the highest computational efficiency for different types of…
We study the computational complexity of a very basic problem, namely that of finding solutions to a very large set of random linear equations in a finite Galois Field modulo q. Using tools from statistical mechanics we are able to identify…
Considerable interest has been paid in recent literature to codes combining local and global properties for erasure correction. Applications are in cloud type of implementations, in which fast recovery of a failed storage device is…
BCH codes are an interesting class of cyclic codes due to their efficient encoding and decoding algorithms. In many cases, BCH codes are the best linear codes. However, the dimension and minimum distance of BCH codes have been seldom…
In this paper, we present error-correcting codes which are the results of our research on the sub-exceeding functions. For a short and medium distance data transmission (wifi network, bluetooth, cable, ...), we see that these codes…
Formal proof checkers such as Coq are capable of validating proofs of correction of algorithms for finite field arithmetics but they require extensive training from potential users. The delayed solution of a triangular system over a finite…
In this work, it is revealed that an acyclic multicast network that is scalar linearly solvable over Galois Field of two elements, GF(2), is solvable over all higher finite fields. An algorithm which, given a GF(2) solution for an acyclic…
Significant inaccuracy often occurs during the process of mathematical calculation due to the digit limitation of floating point, which may lead to catastrophic loss. Normally, people believe that adjustment of floating-point precision is…
We give practical algorithms for computing the divisor class group and the gonality of a curve over a finite field, achieving several orders of magnitude speedup over existing methods for sufficiently large genus or residue field. The…