Physics-based three-dimensional numerical simulations are becoming more predictive and have already become essential. As an example in geophysics, simulations at scale with a very fine resolution, including uncertainty quantification procedures are crucial to provide the relevant physical parameters for forward modeling of seismic wave propagation. Consequently, the diversity of HPC architectures available (heterogeneity, high core counts or depth of the memory hierarchy) leads to increasing concerns regarding the portability of applicative performances. In this paper, we discuss the implementation of the classical spectral finite-elements method using the Arm C Language Extensions for both Neon and SVE SIMD units.

Modular integer arithmetic occurs in many algorithms for computer algebra, cryptography, and error correcting codes. Although recent microprocessors typically offer a wide range of highly optimized arithmetic functions, modular integer operations still require dedicated implementations. In this article, we survey existing algorithms for modular integer arithmetic and present detailed vectorized counterparts. We also describe several applications, such as fast modular Fourier transforms and multiplication of integer polynomials and matrices. The vectorized algorithms have been implemented in C++ inside the free computer algebra and analysis system M athemagix . The performance of our implementation is illustrated by various benchmarks.

In this paper, we study generalized Reed-Solomon codes (GRS codes) over commutative and noncommutative rings, we show that the classical Welch-Berlekamp and Guruswami-Sudan decoding algorithms still hold in this context, and we investigate their complexities. Under some hypothesis, the study of noncommutative GRS codes over finite rings leads to the fact that GRS codes over commutative rings have better parameters than their noncommutative counterparts. Also, GRS codes over finite fields have better parameters than their commutative rings counterparts. But we also show that given a unique decoding algorithm for a GRS code over a finite field, there exists a unique decoding algorithm for a GRS code over a truncated power series ring with a better asymptotic complexity. Moreover, we generalize a lifting decoding scheme to obtain new unique and list decoding algorithms designed to work when the base ring is, for example, a Galois ring or a truncated power series ring or the ring of square matrices over the latter ring.

We present an algorithm for list decoding codewords of algebraic number field codes in polynomial time. This is the first explicit procedure for decoding number field codes whose construction were previously described by Lenstra [1] and Guruswami [2]. We rely on a new algorithm for computing the Hermite normal form of the basis of an OK-module due to Biasse and Fieker [3] where OKis the ring of integers of a number field K.

In this paper we design a decoding algorithm based on a lifting decoding scheme. This leads to a unique decoding algorithm with complexity quasi linear in all the parameters for Reed-Solomon codes over Galois rings and a list decoding algorithm. We show that, using erasures in our algorithms, allows one to decode more errors than half the minimum distance with a high probability. Finally we apply these techniques to interleaved linear codes over a finite field and obtain a decoding algorithm that can recover more errors than half the minimum distance.