Toric codes and lattice ideals


ŞAHİN M.

FINITE FIELDS AND THEIR APPLICATIONS, cilt.52, ss.243-260, 2018 (SCI İndekslerine Giren Dergi) identifier identifier

  • Cilt numarası: 52
  • Basım Tarihi: 2018
  • Doi Numarası: 10.1016/j.ffa.2018.04.007
  • Dergi Adı: FINITE FIELDS AND THEIR APPLICATIONS
  • Sayfa Sayıları: ss.243-260

Özet

Let X be a complete simplicial toric variety over a finite field F-q with homogeneous coordinate ring S = F-q [x(1),., x(r)] and split torus T-X congruent to (F*(q))(n). We prove that submonoids of T-X are exactly those subsets that are parameterized by Laurents monomials. We give an algorithm for determining this parametrization if the submonoid is the zero locus of a lattice ideal in the torus. We also show that vanishing ideals of submonoids of T-X are radical homogeneous lattice ideals of dimension r - n. We identify the lattice corresponding to a degenerate torus in X and completely characterize when its lattice ideal is a complete intersection. We compute dimension and length of some generalized toric codes defined on these degenerate tori. (C) 2018 Elsevier Inc. All rights reserved.