Quantum bicyclic hyperbolic codes
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Quantum bicyclic hyperbolic codes Sankara Sai Chaithanya Rayudu1
· Pradeep Kiran Sarvepalli1
Received: 7 October 2019 / Accepted: 10 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract 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 narrowsense bicyclic hyperbolic code√ of length n 2 contains its dual if and only if its design distance is lower than n − O( n). We extend the sufficiency condition to the nonprimitive case as well. We also show that over quadratic extension fields, a primitive 2 bicyclic hyperbolic code of length √ n 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. Keywords Quantum error correction · Quantum codes · Bicyclic codes · BCH codes
1 Introduction Cyclic codes are an important class of error-correcting codes. Many popular codes, such as BCH codes and Reed–Solomon codes, are cyclic codes. Cyclic codes with guarantees on the minimum distance of the code are easy to construct. Many subclasses of cyclic codes also have efficient decoders making them suitable for practical applications. For quantum error correction, a classical code can be used to construct quantum code [2,5,6,18] if the code contains its (Euclidean or Hermitian) dual. Using these constructions, many (cyclic) quantum codes have been proposed [10,11,13].
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Sankara Sai Chaithanya Rayudu [email protected] https://www.chaithanyarss.com Pradeep Kiran Sarvepalli [email protected]
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Department of Electrical Engineering, Indian Institute of Technology Madras, Chennai 600036, India 0123456789().: V,-vol
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S. S. C. Rayudu, P. K. Sarvepalli
Grassl et al. gave a simple test for identifying cyclic codes that contain their duals [11]. Steane [19] gave a condition, based on the designed distance, to check whether a primitive binary BCH code contains its Euclidean dual. Subsequently, Aly et al. [1] extended this result to the higher alphabet and non-primitive codes. They proved that a primitive BCH√code of length n contains its dual when its design distance is less than δmax = O( n). These results are based on significant structural results of cyclic codes. However, most of the previous work has been limited to one dimension [7,14–16,21,22], even though classically, cyclic codes have been generalized to higher dimensions. Two-dimensional (2D) cyclic codes, also called bicyclic codes, are a generalizat
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