Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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QUANTUM CODES WITH IMPROVED MINIMUM DISTANCE

  • Kolotoglu, Emre (Department of Mathematics Yildiz Technical University) ;
  • Sari, Mustafa (Department of Mathematics Yildiz Technical University)
  • Received : 2018.04.02
  • Accepted : 2018.09.17
  • Published : 2019.05.31
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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 [14]. 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 [14].

Keywords

Table 1. Some parameters of quantum codes obtained by Theorem 3.2

E1BMAX_2019_v56n3_609_t0001.png 이미지

Table 2. A comparison between our parameters and ones in [15]

E1BMAX_2019_v56n3_609_t0002.png 이미지

Table 3. A comparison of quantum codes of length 32 over F9

E1BMAX_2019_v56n3_609_t0003.png 이미지

Table 4. A comparison of quantum codes of length 35 over F13

E1BMAX_2019_v56n3_609_t0004.png 이미지

Table 5. List of some quantum codes that can not be obtained via the construction given in [15]

E1BMAX_2019_v56n3_609_t0005.png 이미지

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