Extended footprint [algebraic geometric codes

Abstract

Many important problems in error-correcting codes can be related to a system of polynomial equations. For instance, a lower bound on the generalized Hamming weights of algebraic geometric (AG) codes can be obtained by the number of common zeros of a set of polynomials. There are many results on the number of roots or even the roots of a set of polynomial equations, such as the method of Grobner bases and the generalized Bezout's theorem. Even though these methods can, in principle, solve any polynomial equation, they have high computational complexity. Moreover, the procedure becomes more intractable when the coefficients of the non-leading terms of each polynomial can have arbitrary value, which is common in AG codes applications. We give some upper bounds on the number of common roots, for special types of equation sets in coding applications. They can be proved by the generalized Bezout's theorem.