quantum codes with improved minimum distance
Bull. Korean Math. Soc.
Published online 2019 Apr 10
Emre Kolotoğlu and Mustafa Sarı
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 [La Guardia, G. G.: Quantum codes derived from cyclic codes. Int. J. Theor. Phys., \textbf{56}(8), p. 2479–2484 (2017)]. 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 existing in the literature.
Keywords : Quantum codes, cyclic codes, constacyclic codes, cyclotomic cosets
MSC numbers : 81P45, 81P70, 94B05, 94B15
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