### D.10.2 decodegb_lib

Library:
  decodegb.lib
Purpose:
  Decoding and min distance of linear codes with GB
Author:
  Stanislav Bulygin, bulygin@mathematik.uni-kl.de

Overview:
  In this library we generate several systems used for decoding cyclic codes and finding their minimum distance. Namely, we work with the Cooper's philosophy and generalized Newton identities. The origindeal method of quadratic equations is worked out here as well. We also (for comparison) enable to work with the system of Fitzgerald-Lax. We provide some auxiliary functions for further manipulations and decoding. For an overview of the methods mentioned above Decoding codes with Groebner bases. For the vanishing ideal computation the algorithm of Farr and Gao is implemented.

Procedures:

> D.10.2.1 sysCRHT |  | generates the CRHT-ideal as in Cooper's philosophy
> D.10.2.2 sysCRHTMindist |  | CRHT-ideal to find the minimum distance in the binary case
> D.10.2.3 sysNewton |  | generates the ideal with the generalized Newton identities
> D.10.2.4 sysBin |  | generates Bin system using Waring function
> D.10.2.5 encode |  | encodes given message x with the given generator matrix g
> D.10.2.6 syndrome |  | computes a syndrome w.r.t. the given check matrix
> D.10.2.7 sysQE |  | generates the system of quadratic equations for decoding
> D.10.2.8 errorInsert |  | inserts errors in a word
> D.10.2.9 errorRand |  | inserts random errors in a word
> D.10.2.10 randomCheck |  | generates a random check matrix
> D.10.2.11 genMDSMat |  | generates an MDS (actually an RS) matrix
> D.10.2.12 mindist |  | computes the minimum distance of a code
> D.10.2.13 decode |  | decoding of a word using the system of quadratic equations
> D.10.2.14 decodeRandom |  | a procedure for manipulation with random codes
> D.10.2.15 decodeCode |  | a procedure for manipulation with the given code
> D.10.2.16 vanishId |  | computes the vanishing ideal for the given set of points
> D.10.2.17 sysFL |  | generates the Fitzgerald-Lax system
> D.10.2.18 decodeRandomFL |  | manipulation with random codes via Fitzgerald-Lax

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