Local list decoding of homomorphisms

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.Includes bibliographical references (leaves 47-49).We investigate the local-list decodability of codes whose codewords are group homomorphisms. The study of such codes was intiated by Goldreich and Levin with the seminal work on decoding the Hadamard code. Many of the recent abstractions of their initial algorithm focus on Locally Decodable Codes (LDC's) over finite fields. We derive our algorithmic approach from the list decoding of the Reed-Muller code over finite fields proposed by Sudan, Trevisan and Vadhan. Given an abelian group G and a fixed abelian group H, we give combinatorial bounds on the number of homomorphisms that have agreement 6 with an oracle-access function f : G --> H. Our bounds are polynomial in , where the degree of the polynomial depends on H. Also, depends on the distance parameter of the code, namely we consider to be slightly greater than 1-minimum distance. Furthermore, we give a local-list decoding algorithm for the homomorphisms that agree on a 3 fraction of the domain with a function f, the running time of which is poly(1/e, log G).by Elena Grigorescu.S.M

Symmetries in algebraic Property Testing

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 94-100).Modern computational tasks often involve large amounts of data, and efficiency is a very desirable feature of such algorithms. Local algorithms are especially attractive, since they can imply global properties by only inspecting a small window into the data. In Property Testing, a local algorithm should perform the task of distinguishing objects satisfying a given property from objects that require many modifications in order to satisfy the property. A special place in Property Testing is held by algebraic properties: they are some of the first properties to be tested, and have been heavily used in the PCP and LTC literature. We focus on conditions under which algebraic properties are testable, following the general goal of providing a more unified treatment of these properties. In particular, we explore the notion of symmetry in relation to testing, a direction initiated by Kaufman and Sudan. We investigate the interplay between local testing, symmetry and dual structure in linear codes, by showing both positive and negative results. On the negative side, we exhibit a counterexample to a conjecture proposed by Alon, Kaufman, Krivelevich, Litsyn, and Ron aimed at providing general sufficient conditions for testing. We show that a single codeword of small weight in the dual family together with the property of being invariant under a 2-transitive group of permutations do not necessarily imply testing. On the positive side, we exhibit a large class of codes whose duals possess a strong structural property ('the single orbit property'). Namely, they can be specified by a single codeword of small weight and the group of invariances of the code. Hence we show that sparsity and invariance under the affine group of permutations are sufficient conditions for a notion of very structured testing. These findings also reveal a new characterization of the extensively studied BCH codes. As a by-product, we obtain a more explicit description of structured tests for the special family of BCH codes of design distance 5.by Elena Grigorescu.Ph.D