$δ$-sequences and Evaluation Codes defined by Plane Valuations at Infinity

Details $δ$-sequences and Evaluation Codes defined by Plane Valuations at Infinity $δ$-sequences and Evaluation Codes defined by Plane Valuations at Infinity

2007-04-26

arxiv.org/abs/0704.3536v2

Computer Science

Information Theory

Scientific paper

We introduce the concept of $\delta$-sequence. A $\delta$-sequence $\Delta$ generates a well-ordered semigroup $S$ in $\mathbb{Z}^2$ or $\mathbb{R}$. We show how to construct (and compute parameters) for the dual code of any evaluation code associated with a weight function defined by $\Delta$ from the polynomial ring in two indeterminates to a semigroup $S$ as above. We prove that this is a simple procedure which can be understood by considering a particular class of valuations of function fields of surfaces, called plane valuations at infinity. We also give algorithms to construct an unlimited number of $\delta$-sequences of the different existing types, and so this paper provides the tools to know and use a new large set of codes.

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