John Little

Ideals, Varieties, and Algorithms

(here, > is the Maple prompt). Once the Groebner package is loaded, you can perform the division algorithm, compute Groebner bases, and carry out a variety of other commands described below. In Maple, a monomial ordering is called a monomial order. The monomial orderings lex, grlex, and grevlex from Chapter 2 are easy to use in Maple. Lex order is called plex (for “pure lexicographic”), grlex order is called grlex, and grevlex order is called tdeg (for “total degree”). Be careful not to confuse tdeg with grlex. Since a monomial order depends also on how the variables are ordered, Maple needs to know both the monomial order you want (plex, grlex or tdeg) and a list of variables. For example, to tell Maple to use lex order with variables x > y > z, you would need to input plex(x,y,z). The Groebner package also knows some elimination orders, as defined in Exercise 5 of Chapter 3, §1. To eliminate the first k variables from x1, . . . , xn, one can use the monomial order lexdeg([x 1,. . .,x k],[x {k+1},. . . ,x n]) (remember that Maple encloses a list inside brackets [. . .]). This order is the elimination order of Bayer and Stillman described in Exercise 6 of Chapter 3, §1. The Maple documentation for the Groebner package also describes how to use certain weighted orders, and we will explain below how matrix orders give us many more monomial orderings. The most commonly used commands in the Groebner package are NormalForm, for doing the division algorithm, and Basis, for computing a Groebner basis. NormalForm has the following syntax:

Systematic encoding via Grobner bases for a class of algebraic-geometric Goppa codes

Any linear code with a nontrivial automorphism has the structure of a module over a polynomial ring. The theory of Grobner bases for modules gives a compact description and implementation of a systematic encoder. We present examples of algebraic-geometric Goppa codes that can be encoded by these methods, including the one-point Hermitian codes.

Toric Surface Codes and Minkowski Sums

Toric codes are evaluation codes obtained from an integral convex polytope

On toric codes and multivariate Vandermonde matrices

P \subset {\mathbb R}^n

On the structure of Hermitian codes

and finite field

On webs of maximum rank

{\mathbb F}_q

Remarks on generalized toric codes

. They are, in a sense, a natural extension of Reed-Solomon codes, and have been studied recently in [V. Diaz, C. Guevara, and M. Vath, Proceedings of Simu Summer Institute, 2001], [J. Hansen, Appl. Algebra Engrg. Comm. Comput., 13 (2002), pp. 289-300; Coding Theory, Cryptography and Related Areas (Guanajuato, 1998), Springer, Berlin, pp. 132-142], and [D. Joyner, Appl. Algebra Engrg. Comm. Comput., 15 (2004), pp. 63-79]. In this paper, we obtain upper and lower bounds on the minimum distance of a toric code constructed from a polygon

Automorphisms and Encoding of AG and Order Domain Codes

P \subset {\mathbb R}^2

Some remarks on Fitzpatrick and Flynn's Gröbner basis technique for Padé approximation

by examining Minkowski sum decompositions of subpolygons of

Computation in Local Rings

P