The Importance of Special Position while Working with Algebraic-Geometry Codes

Abstract

Feng-Rao majority voting can be used to decode functionally decoded AG codes in special form by predicting syndrome values not directly produced from the received word.

So we first recast special form in terms of valuations, pointing out that at least theoretically AG codes not in special position can be mapped into subcodes of codes in special position. This process can be computationally expensive.

We can then apply a slightly modified version of majority voting to decode this image, hence the original code as well. The modification varies based on the choice of valuation used in special position. We include the Macaulay2 code written to do the examples.

Then we note that a designed minimum distance based on the genus can sometimes be improved by counting the actual number of votes on each back-diagonal rather than just giving a lower bound on that number. This count is based on the gap sequence at the special valuation chosen. For different choices of that valuation, there may be a better gap sequence that gives a better minimum designed distance for a particular code.