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    Publication
    Quantum bicyclic hyperbolic codes
    (01-08-2020)
    Rayudu, Sankara Sai Chaithanya
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    Bicyclic codes are a generalization of the one-dimensional (1D) cyclic codes to two dimensions (2D). Similar to the 1D case, in some cases, 2D cyclic codes can also be constructed to guarantee a specified minimum distance. Many aspects of these codes are yet unexplored. Motivated by the problem of constructing quantum codes, we study some structural properties of certain bicyclic codes. We show that a primitive narrow-sense bicyclic hyperbolic code of length n2 contains its dual if and only if its design distance is lower than n-O(n). We extend the sufficiency condition to the non-primitive case as well. We also show that over quadratic extension fields, a primitive bicyclic hyperbolic code of length n2 contains Hermitian dual if and only if its design distance is lower than n-O(n). Our results are analogous to some structural results known for BCH and Reed–Solomon codes. They further our understanding of bicyclic codes. We also give an application of these results by showing that we can construct two classes of quantum bicyclic codes based on our results.
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    Publication
    Decoding Toric Codes on Three Dimensional Simplical Complexes
    (01-02-2021)
    Aloshious, Arun B.
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    Three dimensional (3D) toric codes are a class of stabilizer codes with local checks and come under the umbrella of topological codes. While decoding algorithms have been proposed for the 3D toric code on a cubic lattice, there have been very few studies on the decoding of 3D toric codes over arbitrary lattices. Color codes in 3D can be mapped to toric codes. However, the resulting toric codes are not defined on cubic lattice. They are arbitrary lattices with triangular faces. Decoding toric codes over an arbitrary lattice will help in studying the performance of color codes. Furthermore, gauge color codes can also be decoded via 3D toric codes. Motivated by this, we propose an efficient algorithm to decode 3D toric codes on arbitrary lattices (with and without boundaries).