We consider weighted Reed-Muller codes over point ensemble S1 × · · · × Sm where Si needs not be of the same size as Sj. For m = 2 we determine optimal weights and analyze in detail what is the impact of the ratio |S1|/|S2| on the minimum distance. In conclusion the weighted Reed-Muller code construction is much better than its reputation. For a class of affine variety codes that contains the weighted Reed-Muller codes we then present two list decoding algorithms. With a small modification one of these algorithms is able to correct up to 31 errors of the [49,11,28] Joyner code.
We consider multivariate polynomials and investigate how many zeros of multiplicity at least r they can have over a Cartesian product of finite subsets of a field. Here r is any prescribed positive integer and the definition of multiplicity that we use is the one related to Hasse derivatives. As a generalization of material in [2, 5] a general version of the Schwartz-Zippel was presented in [8] which from the leading monomial -with respect to a lexicographic ordering -estimates the sum of zeros when counted with multiplicity. The corresponding corollary on the number of zeros of multiplicity at least r is in general not sharp and therefore in [8] a recursively defined function D was introduced using which one can derive improved information. The recursive function being rather complicated, the only known closed formula consequences of it are for the case of two variables [8]. In the present paper we derive closed formula consequences for arbitrary many variables, but for the powers in the leading monomial being not too large. Our bound can be viewed as a generalization of the footprint bound [10, 6] -the classical footprint bound taking not multiplicity into account.
Abstract. Using tools from algebraic geometry and Gröbner basis theory we solve two problems in network coding. First we present a method to determine the smallest field size for which linear network coding is feasible. Second we derive improved estimates on the success probability of random linear network coding. These estimates take into account which monomials occur in the support of the determinant of the product of Edmonds matrices. Therefore we finally investigate which monomials can occur in the determinant of the Edmonds matrix.
Let S be a finite subset of a field. For multivariate polynomials the generalized Schwartz-Zippel bound [2], [4] estimates the number of zeros over S × • • • × S counted with multiplicity. It does this in terms of the total degree, the number of variables and |S|. In the present work we take into account what is the leading monomial. This allows us to consider more general point ensembles and most importantly it allows us to produce much more detailed information about the number of zeros of multiplicity r than can be deduced from the generalized Schwartz-Zippel bound. We present both upper and lower bounds.
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