Gröbner Bases and Abelian Codes
We investigate the links between Gröbner Bases and Abelian Codes. Abelian codes are multivariate generalization of cyclic codes, and our purpose it to show how Gröbner bases can replace the generating polynomial of cyclic codes.
We use these bases to properly define the information symbols of an abelian code in polynomial representation. In particular, we show that Gröbner bases are related with parity check matrices of the code. From this, we are able to generalize permutation decoding to abelian codes. This generalization introduces some choice which allows us to decode more errors than in the cyclic case.
Finally, we examine the connections between Gröbner Basis for an abelian code and those of its reciprocal, annihilator and orthogonal code.
KeywordsParity Check Cyclic Code Generate Polynomial Information Symbol Parity Check Matrix
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