Universally decodable matrices can be used for coding purposes when transmitting over slow fading channels. These matrices are parameterized by positive integers L and n and a prime power q. Based on Pascal’s triangle we give an explicit construction of universally decodable matrices for any non-zero integers L and n and any prime power q where L ≤ q + 1. This is the largest set of possible par...

and Résumé T challenges in algebraic coding theory are addressed within this dissertation. Œe €rst one is the ecient hardand so‰-decision decoding of Generalized Reed–Solomon codes over €nite €elds in Hamming metric. Œe motivation for this more than 50 years old problem was renewed by the discovery of a polynomial-time interpolation-based decoding principle up to the Johnson radius by Guruswam...

and Résumé T challenges in algebraic coding theory are addressed within this dissertation. Œe €rst one is the ecient hardand so‰-decision decoding of Generalized Reed–Solomon codes over €nite €elds in Hamming metric. Œe motivation for this more than 50 years old problem was renewed by the discovery of a polynomial-time interpolation-based decoding principle up to the Johnson radius by Guruswam...

The interpolation step in the Guruswami-Sudan algorithm has attracted a lot of interest and it is now solved by many algorithms in the literature; they use either structured linear algebra or basis reduction for polynomial lattices. This problem of interpolation with multiplicities has been generalized to multivariate polynomials; in this context, to our knowledge only the approach based on pol...

A family of distance-optimal LRC codes from certain subcodes of q-ary Reed-Solomon codes, proposed by I. Tamo and A. Barg in 2014, assumes that the code length n is a multiple of r + 1. By shortening codes from this family, we show that it is possible to lift this assumption, still obtaining distance-optimal codes.

We present a new method to obtain cyclic subcodes of algebraic geometric codes using their automorphisms. Automorphisms of algebraic geometric codes from F. K. Schmidt curves are proposed. We present an application of this method in designing frequency hopping sequences for spread spectrum systems. Algebraic geometric codes can provide sequences longer (better randomness) than the ones from Ree...

We consider dictionaries of size n over the finite universe u = (0, l)w and introduce a new technique for their implementation: error correcting codes. protected with an error-correcting code (ECC) is presented in this compared with the most recent implementation. treated within industries for their products. ―VLSI Implementation of Decoder Error Correcting in Satellite. Concatenation Code in F...

We determine the dimensions of subfield codes of Reed-Solomon codes and construct certain extensions and lengthenings of these codes. We start from the duals, using the language of orthogonal arrays. As a first result this allows us to obtain a fair number of improvements in the list of binary, ternary and quaternary linear codes with largest known minimal distance.

A closed form formula of the partition weight enumerator of maximum distance separable (MDS) codes is derived for an arbitrary number of partitions. Using this result, some properties of MDS codes are discussed. The results are extended for the average binary image of MDS codes in finite fields of characteristic two. As an application, we study the multiuser error probability of Reed Solomon co...