2,946 research outputs found
Concatenated structure and construction of certain code families
In this thesis, we consider concatenated codes and their generalizations as the main tool for two different purposes. Our first aim is to extend the concatenated structure of quasi-cyclic codes to its two generalizations: generalized quasi-cyclic codes and quasi-abelian codes. Concatenated structure have consequences such as a general minimum distance bound. Hence, we obtain minimum distance bounds, which are analogous to Jensen's bound for quasi-cyclic codes, for generalized quasicyclic and quasi-abelian codes. We also prove that linear complementary dual quasi-abelian codes are asymptotically good, using the concatenated structure. Moreover, for generalized quasi-cyclic and quasi-abelian codes, we prove, as in the quasi-cyclic codes, that their concatenated decomposition and the Chinese Remainder decomposition are equivalent. The second purpose of the thesis is to construct a linear complementary pair of codes using concatenations. This class of codes have been of interest recently due to their applications in cryptography. This extends the recent result of Carlet et al. on the concatenated construction of linear complementary dual codes
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
ACD codes over skew-symmetric dualities
The applications of additive codes mainly lie in quantum error correction and
quantum computing. Due to their applications in quantum codes, additive codes
have grown in importance. In addition to this, additive codes allow the
implementation of a variety of dualities. The article begins by developing the
properties of Additive Complementary Dual (ACD) codes with respect to arbitrary
dualities over finite abelian groups. Further, we introduce a subclass of
non-symmetric dualities referred to as skew-symmetric dualities. Then, we
precisely count symmetric and skew-symmetric dualities over finite fields. Two
conditions have been obtained: one is a necessary and sufficient condition, and
the other is a necessary condition. The necessary and sufficient condition is
for an additive code to be an ACD code over arbitrary dualities. The necessary
condition is on the generator matrix of an ACD code over skew-symmetric
dualities. We provide bounds for the highest possible minimum distance of ACD
codes over skew-symmetric dualities. Finally, we find some new quaternary ACD
codes over non-symmetric dualities with better parameters than the symmetric
ones
List decoding group homomorphisms between supersolvable groups
We show that the set of homomorphisms between two supersolvable groups can be
locally list decoded up to the minimum distance of the code, extending the
results of Dinur et al who studied the case where the groups are abelian.
Moreover, when specialized to the abelian case, our proof is more streamlined
and gives a better constant in the exponent of the list size. The constant is
improved from about 3.5 million to 105.Comment: 11 page
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