TY - JOUR
T1 - Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields
AU - Rentería-Márquez, Carlos
AU - Simis, Aron
AU - Villarreal, Rafael H.
N1 - Funding Information:
✩ The first author was partially supported by COFAA-IPN and SNI. The second author was partially supported by a grant of CNPq. The third author was partially supported by CONACyT grant 49251-F and SNI. * Corresponding author. E-mail addresses: renteri@esfm.ipn.mx (C. Rentería-Márquez), aron@dmat.ufpe.br (A. Simis), vila@math.cinvestav.mx (R.H. Villarreal).
PY - 2011/1
Y1 - 2011/1
N2 - Let K=Fq be a finite field with q elements and let X be a subset of a projective space Ps-1, 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.
AB - Let K=Fq be a finite field with q elements and let X be a subset of a projective space Ps-1, 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.
KW - Binomial and lattice ideals
KW - Degree
KW - Evaluation codes
KW - Gröbner bases
KW - Hilbert function
KW - Index of regularity
KW - Minimum distance
KW - Parameterized codes
KW - Parameters of a code
KW - Projective variety
UR - http://www.scopus.com/inward/record.url?scp=78649515591&partnerID=8YFLogxK
U2 - 10.1016/j.ffa.2010.09.007
DO - 10.1016/j.ffa.2010.09.007
M3 - Artículo
SN - 1071-5797
VL - 17
SP - 81
EP - 104
JO - Finite Fields and their Applications
JF - Finite Fields and their Applications
IS - 1
ER -