Polynomial encoding of discrete probability using Gröbner bases

Efficient encoding via Gröbner bases and discrete Fourier transforms for several kinds of algebraic codes

Abstract—Novel encoding scheme for algebraic codes, such as codes on algebraic curves, multidimensional cyclic codes, and hyperbolic cascaded Reed–Solomon codes, is proposed with numerical examples. We make use of the 2-dimensional inverse discrete Fourier transform, which generalizes the Mattson–Solomon polynomial for Reed–Solomon codes. We also generalize the workings of the generator polynom...

Generalized Discrete Comprehensive Gröbner Bases

We showed special types of comprehensive Gröbner bases can be defined and calculated as the applications of Gröbner bases in polynomial rings over commutative Von Neumann regular rings in [5] and [6]. We called them discrete comprehensive Gröbner bases, since there is a strict restriction on specialization of parameters, that is parameters can take values only 0 and 1. In this paper, we show th...

Projective modules over polynomial rings and dynamical Gröbner bases

Approximate Gröbner Bases and Overdetermined Algebraic Systems

We discuss computation of Gröbner bases using approximate arithmetic for coefficients. We show how certain considerations of tolerance, corresponding roughly to accuracy and precision from numeric computation, allow us to obtain good approximate solutions to problems that are overdetermined. We provide examples of solving overdetermined systems of polynomial equations. As a secondary feature we...

XIDEAL Gröbner Bases for Exterior Algebra

The method of Gröbner bases in commutative polynomial rings introduced by Buchberger (e.g. [1]) is a well-known and very important tool in polynomial ideal theory, for example in solving the ideal membership problem. XIDEAL extends the method to exterior algebras using algorithms from [2] and [3]. There are two main departures from the commutative polynomial case. First, owing to the non-commut...