Quantum codes with improved minimum distance
Bull. Korean Math. Soc. 2019 Vol. 56, No. 3, 609-619
Published online April 10, 2019
Printed May 31, 2019
Emre Koloto\u{g}lu, Mustafa Sar\i
Yildiz Technical University; Yildiz Technical University
Abstract : The methods for constructing quantum codes is not always sufficient by itself. Also, the constructed quantum codes as in the classical coding theory have to enjoy a quality of its parameters that play a very important role in recovering data efficiently. In a very recent study quantum construction and examples of quantum codes over a finite field of order $q$ are presented by La Garcia in \cite{Guardia}. Being inspired by La Garcia's the paper, here we extend the results over a finite field with $q^2$ elements by studying necessary and sufficient conditions for constructions quantum codes over this field. We determine a criteria for the existence of $q^2$-cyclotomic cosets containing at least three elements and present a construction method for quantum maximum-distance separable (MDS) codes. Moreover, we derive a way to construct quantum codes and show that this construction method leads to quantum codes with better parameters than the ones in \cite{Guardia}.
Keywords : quantum codes, cyclic codes, constacyclic codes, cyclotomic cosets
MSC numbers : 81P45, 81P70, 94B05, 94B15
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