Related papers: Fast Xor-based Erasure Coding based on Polynomial …
We propose a generic transformation that can convert any nonbinary $(n=k+r,k)$ maximum distance separable (MDS) code into another $(n,k)$ MDS code over the same field such that 1) some arbitrarily chosen $r$ nodes have the optimal repair…
Recent advances in erasure coding for distributed storage systems have demonstrated that adapting redundancy to varying disk failure rates can lead to substantial storage savings. Such adaptation requires code conversion, wherein data…
Normal basis is used in many applications because of the efficiency of the implementation. However, most space complexity reduction techniques for binary field multiplier are applicable for only optimal normal basis or Gaussian normal basis…
We explore a new form of DFT, which we call the Polynomial Transform. It functions over finite fields, and a size $n$ transform takes $O(n)$ operations. In the multitape Turing machine model, it allows us to multiply two $n$ bit numbers in…
A method of fast linear transform algorithm synthesis for an arbitrary tensor, matrix, or vector is proposed. The method is based on factorization of a tensor and using the factors for building computational structures performing fast…
Transpose convolution has shown prominence in many deep learning applications. However, transpose convolution layers are computationally intensive due to the increased feature map size due to adding zeros after each element in each row and…
We consider the Fast Fourier Transform (FFT) based numerical method for thin film magnetization problems [Vestg{\aa}rden and Johansen, SuST, 25 (2012) 104001], compare it with the finite element methods, and evaluate its accuracy. Proposed…
We introduce two efficient algorithms for computing the partial Fourier transforms in one and two dimensions. Our study is motivated by the wave extrapolation procedure in reflection seismology. In both algorithms, the main idea is to…
We propose a new way for speeding up the search of the maximal solution $X_+$ of $X + A^\top X^{-1} A = Q$. It is known that the speed of convergence of traditional approaches for solving this problem depends highly on the spectral radius…
In this paper, we give algorithms and methods of construction of self-dual codes over finite fields using orthogonal matrices. Randomization in the orthogonal group, and code extension are the main tools. Some optimal, almost MDS, and MDS…
This paper deals with circulant matrices. It is shown that a circulant matrix can be multiplied by a vector in time O(n log(n)) in a ring with roots of unity without making use of an FFT algorithm. With our algorithm we achieve a speedup of…
Example-based mesh deformation methods are powerful tools for realistic shape editing. However, existing techniques typically combine all the example deformation modes, which can lead to overfitting, i.e. using a overly complicated model to…
This paper shows how to decode errors and erasures with Gabidulin codes in sub-quadratic time in the code length, improving previous algorithms which had at least quadratic complexity. The complexity reduction is achieved by accelerating…
In this paper we briefly discuss \Rings --- an efficient lightweight library for commutative algebra. Polynomial arithmetic, GCDs, polynomial factorization and Gr\"obner bases are implemented with the use of modern asymptotically fast…
Discrete transforms such as the discrete Fourier transform (DFT) and the discrete Hartley transform (DHT) are important tools in numerical analysis. The successful application of transform techniques relies on the existence of efficient…
This paper examines finite field trigonometry as a tool to construct trigonometric digital transforms. In particular, by using properties of the k-cosine function over GF(p), the Finite Field Discrete Cosine Transform (FFDCT) is introduced.…
We analyze the Double Fourier Sphere (DFS) method on the rotation group $\mathcal{SO}(3)$ in the frequency domain and demonstrate its central role in fast algorithms. Fast Fourier algorithms on $\mathcal{SO}(3)$ are commonly formulated as a…
This tutorial aims to establish connections between polynomial modular multiplication over a ring to circular convolution and discrete Fourier transform (DFT). The main goal is to extend the well-known theory of DFT in signal processing…
This paper shows that, for matrix multiplications and convolutions, it is possible to asymptotically replace each real multiplication with a single squaring operation. Similarly, a single complex multiplication can be replaced with 3…
In large-scale distributed storage systems, erasure codes are used to achieve fault tolerance in the face of node failures. Tuning code parameters to observed failure rates has been shown to significantly reduce storage cost. Such tuning of…