A new family of locally correctable codes based on degree-lifted algebraic geometry codes

We describe new constructions of error correcting codes, obtained by "degree-lifting" a short algebraic geometry base-code of block-length q to a lifted-code of block-length qm, for arbitrary integer m. The construction generalizes the way degree-d, univariate polynomials evaluated over the q-element field (also known as Reed-Solomon codes) are "lifted" to degree-d, m-variate polynomials (Reed-Muller codes). A number of properties are established: The rate of the degree-lifted code is approximately a 1/m!-fraction of the rate of the base-code. The relative distance of the degree-lifted code is at least as large as that of the base-code. This is proved using a generalization of the Schwartz-Zippel Lemma to degree-lifted Algebraic-Geometry codes. [Local correction] If the base code is invariant under a group that is "close" to being doubly-transitive (in a precise manner defined later then the degree-lifted code is locally correctable with query complexity at most q2. The automorphisms of the base-code are crucially used to generate query-sets, abstracting the use of affine-lines in the local correction procedure of Reed-Muller codes. Taking a concrete illustrating example, we show that degree-lifted Hermitian codes form a family of locally correctable codes over an alphabet that is significantly smaller than that obtained by Reed-Muller codes of similar constant rate, message length, and distance.

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