Improved local testing for multiplicity codes

Multiplicity codes are a generalization of Reed-Muller codes which include derivatives as well as the values of low degree polynomials, evaluated in every point in F mp . Similarly to Reed-Muller codes, multiplicity codes have a local nature that allows for local correction and local testing. Recently, [6] showed that the plane test , which tests the degree of the codeword on a random plane, is a good local tester for small enough degrees . In this work we simplify and extend the analysis of local testing for multiplicity codes, giving a more general and tight analysis. In particular, we show that multiplicity codes MRM p ( m, d, s ) over prime fields with arbitrary d are locally testable by an appropriate k -flat test , which tests the degree of the codeword on a random k -dimensional affine subspace. The relationship between the degree parameter d and the required dimension k is shown to be nearly optimal, and improves on [6] in the case of planes. Our analysis relies on a generalization of the technique of canonincal monomials introduced in [5]. Generalizing canonical monomials to the multiplicity case requires substantially different proofs which exploit the algebraic structure of multiplicity codes.