1 Basics of Forward Error Correction (FEC) .............2 2 Theory ..................................................................... 3 2.1 Galois Fields ........................................................... 3 2.2 Reed-Solomon Codes ..............................................6 2.3 Error-Correcting Performance of Reed-Solomon Codes ..............................................9 3 SC...

In the last lecture, we studied codes based on univarite polynomials, also known as Reed-Solomon codes, and on multivariate polynomials, also known as Reed-Muller codes. The Reed-Solomon codes had rate R = 1 − δ, which is optimal (given by the Singleton bound), but required alphabets of large size (|Σ| ≥ n). Multivariate polynomials in a constant number of variables, however, gave codes of posi...

A projection of binary Reed–Muller codes ( ) onto GF (4) is presented. For an ( ) code, this operation yields a linear quaternary code with the same length, dimension, and minimum distance as the Reed–Muller ( 1 2) code. Based upon this projection, multilevel construction is given for ( ), where the constituent codes applied to the different levels are themselves the Reed–Muller codes ( 2 2) an...

A formula, which relates the state dimensions of a minimal trellis of a Reed-Muller code to those of another Reed-Muller code with lower order and shorter length, is derived in this correspondence. State dimension at every position of a minimal trellis of any Reed-Muller code can be obtained by a recursive application of this formula.

a Cardiovascular Disease Service, Department of Medicine, Walter Reed National Military Medical Center, Bethesda, MD, United States b Pulmonary Disease and Critical Care Service, Department of Medicine, Walter Reed National Military Medical Center, Bethesda, MD, United States c Internal Medicine Service, Department of Medicine, Walter Reed National Military Medical Center, Bethesda, MD, United ...

Gopalan, Klivans, and Zuckerman proposed a list-decoding algorithm for Reed-Muller codes. Their algorithm works up to a given list-decoding radius. Dumer, Kabatiansky, and Tavernier improved the complexity of the algorithm for binary Reed-Muller codes by using wellknown Plotkin construction. In this study, we propose a list-decoding algorithm for non-binary Reed-Muller codes as a natural genera...

OFDM, power, Reed-Muller code, decoding, algorithms, error correction Recently a class of generalised Reed-Muller codes has been suggested for use in power-controlled OFDM modulation. A number of approaches to decoding these codes have already been developed. Here we present low complexity alternatives which are inspired by the classical Reed decoding algorithm for binary Reed-Muller codes. We ...

Aquatic reed is an important indicator for the ecological assessment of freshwater lakes. Monitoring is essential to document its expansion or deterioration and decline. The applicability of Green-LiDAR data for the status assessment of aquatic reed beds of Bavarian freshwater lakes was investigated. The study focused on mapping diagnostic structural parameters of aquatic reed beds by exploring...

Journal of the Society for Industrial and Applied Mathematics. In June of 1960 the paper was published: five pages under the rather unpretentious title "Polynomial Codes over Certain Finite Fields" [17]. This paper described a new class of error-correcting codes that are now called Reed-Solomon codes. In the decades since their discovery, Reed-Solomon codes have enjoyed countless applications, ...