8 research outputs found
List Decoding Algorithm based on Voting in Groebner Bases for General One-Point AG Codes
We generalize the unique decoding algorithm for one-point AG codes over the
Miura-Kamiya Cab curves proposed by Lee, Bras-Amor\'os and O'Sullivan (2012) to
general one-point AG codes, without any assumption. We also extend their unique
decoding algorithm to list decoding, modify it so that it can be used with the
Feng-Rao improved code construction, prove equality between its error
correcting capability and half the minimum distance lower bound by Andersen and
Geil (2008) that has not been done in the original proposal except for
one-point Hermitian codes, remove the unnecessary computational steps so that
it can run faster, and analyze its computational complexity in terms of
multiplications and divisions in the finite field. As a unique decoding
algorithm, the proposed one is empirically and theoretically as fast as the BMS
algorithm for one-point Hermitian codes. As a list decoding algorithm,
extensive experiments suggest that it can be much faster for many moderate
size/usual inputs than the algorithm by Beelen and Brander (2010). It should be
noted that as a list decoding algorithm the proposed method seems to have
exponential worst-case computational complexity while the previous proposals
(Beelen and Brander, 2010; Guruswami and Sudan, 1999) have polynomial ones, and
that the proposed method is expected to be slower than the previous proposals
for very large/special inputs.Comment: Accepted for publication in J. Symbolic Computation. LaTeX2e
article.cls, 42 pages, 4 tables, no figures. Ver. 6 added an illustrative
example of the algorithm executio
δ-Sequences and Evaluation Codes de ned by Plane Valuations at Infinity
We introduce the concept of δ-sequence. A δ-sequence ∆ generates a well-ordered semigroup
S in Z2 or R. We show how to construct (and compute parameters) for the dual code of any evaluation code associated with a weight function defined by ∆ 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
δ-sequences of the diferent existing types, and so this paper provides the tools to know and use a new large set of codes
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum