Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields

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• 作者：Carlos Renterí ; a-Má ; rquez ; Aron Simis ; Rafael H. Villarreal
• 关键词：Evaluation codes ; Parameterized codes ; Binomial and lattice ideals ; Parameters of a code ; Grö ; bner bases ; Projective variety ; Degree ; Index of regularity ; Hilbert function ; Minimum distance
• 刊名：Finite Fields and Their Applications
• 出版年：2011
• 出版时间：January 2011
• 年：2011
• 卷：17
• 期：1
• 页码：81-104
• 全文大小：317 K

文摘

Let be a finite field with q elements and let X be a subset of a projective space , over the field K, parameterized by Laurent monomials. Let I(X) be the vanishing ideal of X. Some of the main contributions of this paper are in determining the structure of I(X) to compute some of its invariants. It is shown that I(X) is a lattice ideal. We introduce the notion of a parameterized code arising from X and present algebraic methods to compute and study its dimension, length and minimum distance. For a parameterized code, arising from a connected graph, we are able to compute its length and to make our results more precise. If the graph is non-bipartite, we show an upper bound for the minimum distance.