• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.83-98
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• 2014

We reformulate an interpolation-based unique decoding algorithm of AG codes, using the theory of Gr$\ddot{o}$bner bases of modules on the coordinate ring of the base curve. The conceptual description of the reformulated algorithm lets us better understand the majority voting procedure, which is central in the interpolation-based unique decoding. Moreover the smaller Gr$\ddot{o}$bner bases imply smaller space and time complexity of the algorithm.

• Honam Mathematical Journal
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• v.35 no.4
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• pp.793-808
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• 2013

An interpolation-based unique decoding algorithm of Algebraic Geometry codes was recently introduced. The algorithm iteratively computes the sent message through a majority voting procedure using the Gr$\ddot{o}$bner bases of interpolation modules. We now combine the main idea of the Guruswami-Sudan list decoding with the algorithm, and thus obtain a hybrid unique decoding algorithm of plane AG codes, significantly improving the decoding speed.

• Journal of the Korean Data and Information Science Society
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• v.20 no.6
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• pp.1015-1027
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• 2009

Many problems of confounding and identifiability for polynomial models in an experimental design can be solved using methods of algebraic geometry. The theory of $Gr\ddot{o}bner$ basis is used to characterize the design. In addition, a fractional factorial design can be uniquely represented by a polynomial indicator function. $Gr\ddot{o}bner$ bases and indicator functions are powerful computational tools to deal with ideals of fractions based on each different theoretical aspects. The problem posed here is to give how to move from one representation to the other. For a given fractional factorial design, the indicator function can be computed from the generating equations in the $Gr\ddot{o}bner$ basis. The theory is tested using some fractional factorial designs aided by a modern computational algebra package CoCoA.