Commutative Algebra Methods in Coding Theory
by
Horacio Tapia-Recillas
Universidad Autónoma Metropolitana-Iztapalapa
Coauthors: Carlos Rentería

Let K be a finite field with q elements (q a power of a prime p), let P_m(K) be the m-projective space over the field K and let S = { P₁, ... , P_s} be a subset of P_m(K). If L is a finite dimensional K-vector space of functions over S with values in K, consider the following evaluation map:

ev : L --> Ks,    ev(f) = (f(P1), ..., f(Ps))

Then ev(L) = C_S is a code of length s=|S| over K and it would be interesting to determine its parameters, its dual code, a generating matrix, and some of its properties. For an arbitrary subset S and any subspace of functions L it is not easy to describe the code C_S. By using some techniques from commutative algebra such as the Hilbert function, free (graded) resolutions, Gröbner bases, the canonical module, and some invariants of ideals, several results about the code C_S  such as the dimension, a bound on the minimal distance, the a-invariant of the vanishing ideal  I_S of the set S, and a set of generators of the ideal I_S will be presented where S is the affine space A_m(K), the projective space  P_m(K), a complete intersection in P_m(K), and the K-rational points of the Veronese variety. If

A = K[ X0, ... , Xm] = \oplusj >= 0Aj

is the ring of polynomials in the variables X₀, ... , X_m over the field K (with the natural graduation), the space of functions L will be taken as \oplus_j=0^tA_j, i.e., the polynomials of degree at most t, or the homogeneous polynomials of degree d (t >= 1, d >= 1). Several examples will be given in order to illustrate the main ideas.

Date received: January 29, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabw-61.