title: Matrix formalism of the Reed—Solomon codes
metadata.name: dc.contributor.author
metadata.value: Marov, Alexei V.
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Uteshev, Alexei Yu.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2017-02-09T08:59:08Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2017-02-09T08:59:08Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2016-12
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Marov A. V., Uteshev A. Yu. Matrix formalism of the Reed—Solomon codes. Vestnik of Saint Petersburg University. Series 10. Applied mathematics. Computer science. Control processes, 2016, issue 4, pp. 4–17.
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: 10.21638/11701/spbu10.2016.401
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/5992
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: The paper is focused onto modification of the involved algorithms of coding and error
  (i. e., failures and silent data corruptions) correction in the Reed—Solomon codes. These modifications involve matrix formalism and are based on an algorithm for the Vandermonde
  matrix inversion. For such a matrix V =
   
  λi−1
  j
   n
  i,j=1
  the suggested algorithm computes the
  columns V
  −1
  [1] , . . . ,V
  −1
  [n−1], V
  −1
  [n] of the matrix V −1 recursively, starting from the last column,
  via the formulas
  V
  −1
  [n] = Ξ0, V
  −1
  [j] = Ξn−j − σ1V
  −1
  [j+1]
  − σ2V
  −1
  [j+2]
  − ·· ·−σn−jV
  −1
  [n] , j = n − 1, n − 2, . . . , 1 .
  Here Ξk =
   
  λk1
  /W (λ1), . . . ,λk
  n/W (λn)
    
  , σk =
   n
  j=1 λn+k−1
  j /W (λj ), k = 1, n, and
  W(x) =
   n
  k=1(x − λk). The obtained result is applied for realization of systematic coding
  of the n-vector of the information blocks with the aid of multiplication (in an appropriate
  Galois field) by the matrix K = [ Wi(aN−j−1)], i = 1,m, j = 0, n − 1. Here  W (x),   = 1,m,
  denote the basic Lagrange interpolation polynomials generated by the powers of a primitive
  element of the field, while m stands for the number of redundancy blocks (syndromes). In
  the framework of this ideology, an error correcting procedure is also realized. The program
  implementation in C demonstrates high performance results (compared to existing software)
  with solid perspectives for parallelization. Refs 17. Fig 1.
metadata.lang: en_GB
metadata.name: dc.language.iso
metadata.value: ru
metadata.lang: en_GB
metadata.name: dc.publisher
metadata.value: St Petersburg State University
metadata.lang: en_GB
metadata.name: dc.relation.ispartofseries
metadata.value: Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes;Issue 4
metadata.lang: -
metadata.name: dc.subject
metadata.value: error-correcting codes
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: Reed—Solomon codes
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: Vandermonde matrix
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: Matrix formalism of the Reed—Solomon codes
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 01-Marov.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/5992/1/01-Marov.pdf
resources.format: Adobe PDF
resources.size: 368 kB