Many good quantum error-correcting codes were constructed from cyclic codes. However, it is a difficult problem to determine the true minimum distance of quantum cyclic codes for large length n. In this work, we construct nonbinary quantum cyclic codes and asymmetric quantum cyclic codes that are derived from repeated-root cyclic codes for an arbitrary length ps, and determine the true minimum distance of all those codes. Some proposed quantum cyclic codes are optimal. Additionally, some proposed asymmetric quantum cyclic codes have better parameters than the ones available in the literature.Many good quantum error-correcting codes were constructed from cyclic codes. However, it is a difficult problem to determine the true minimum distance of quantum cyclic codes for large length n. In this work, we construct nonbinary quantum cyclic codes and asymmetric quantum cyclic codes that are derived from repeated-root cyclic codes for an arbitrary length ps, and determine the true minimum distance of all those codes. Some proposed quantum cyclic codes are optimal. Additionally, some proposed asymmetric quantum cyclic codes have better parameters than the ones available in the literature.

Modern Physics Letters BMar 4, 20137 citations

Quantum Codes Over an Extension of ${\mathbb {Z}_4}$ Quantum Codes Over an Extension of $${\mathbb {Z}_4}$$

Mohammad Ashraf, Naim Khan, Washiqur Rehman +1 more

Springer proceedings in mathematics & statisticsJan 1, 20240 citations

Quantum Codes from Cyclic Codes over $F_2+vF_2$ Quantum Codes from Cyclic Codes over $F_2+vF_2$

Jianfa Qian

Journal of Information and Computational ScienceApr 10, 201357 citations

Quantum Cyclic Code

Sagarmoy Dutta, Piyush P. Kurur

In this paper, we define and study \emph{quantum cyclic codes}, a generalisation of cyclic codes to the quantum setting. Previously studied examples of quantum cyclic codes were all quantum codes obtained from classical cyclic codes via the CSS construction. However, the codes that we study are much more general. In particular, we construct cyclic stabiliser codes with parameters $[[5,1,3]]$ , $[[17,1,7]]$ and $[[17,9,3]]$ , all of which are \emph{not} CSS. The $[[5,1,3]]$ code is the well known Laflamme code and to the best of our knowledge the other two are new examples. Our definition of cyclicity applies to non-stabiliser codes as well; in fact we show that the $((5,6,2))$ nonstabiliser first constructed by Rains\etal~ cite{rains97nonadditive} and latter by Arvind \etal~\cite{arvind:2004:nonstabilizer} is cyclic. We also study stabiliser codes of length $4^m +1$ over $\mathbb{F}_2$ for which we define a notation of BCH distance. Much like the Berlekamp decoding algorithm for classical BCH codes, we give efficient quantum algorithms to correct up to $\floor{\frac{d-1}{2}}$ errors when the BCH distance is $d$ .In this paper, we define and study \emph{quantum cyclic codes}, a generalisation of cyclic codes to the quantum setting. Previously studied examples of quantum cyclic codes were all quantum codes obtained from classical cyclic codes via the CSS construction. However, the codes that we study are much more general. In particular, we construct cyclic stabiliser codes with parameters $[[5,1,3]]$, $[[17,1,7]]$ and $[[17,9,3]]$, all of which are \emph{not} CSS. The $[[5,1,3]]$ code is the well known Laflamme code and to the best of our knowledge the other two are new examples. Our definition of cyclicity applies to non-stabiliser codes as well; in fact we show that the $((5,6,2))$ nonstabiliser first constructed by Rains\etal~ cite{rains97nonadditive} and latter by Arvind \etal~\cite{arvind:2004:nonstabilizer} is cyclic. We also study stabiliser codes of length $4^m +1$ over $\mathbb{F}_2$ for which we define a notation of BCH distance. Much like the Berlekamp decoding algorithm for classical BCH codes, we give efficient quantum algorithms to correct up to $\floor{\frac{d-1}{2}}$ errors when the BCH distance is $d$.

arXiv:1007.1697Jul 10, 20103 citations

Linear Quantum Codes Constructed from Quaternary Cyclic Codes

LI Rui-hu

A method of describing quaternary cyclic codes with 4-cyclotomic cosets of modulo odd n and generator polynomials is presented.A necessary and sufficient condition under which a quaternary cyclic code containing its dual code is given,thus generalizing the related known results on self-orthogonal simple root cyclic codes and on BCH codes to al quaternary cyclic codes.Using the relation of simple root cyclic codes and repeat root cyclic codes,the linear quantum codes that can be constructed from quaternary cyclic codes of short lengths are determined.A method of describing quaternary cyclic codes with 4-cyclotomic cosets of modulo odd n and generator polynomials is presented.A necessary and sufficient condition under which a quaternary cyclic code containing its dual code is given,thus generalizing the related known results on self-orthogonal simple root cyclic codes and on BCH codes to al quaternary cyclic codes.Using the relation of simple root cyclic codes and repeat root cyclic codes,the linear quantum codes that can be constructed from quaternary cyclic codes of short lengths are determined.

Journal of Air Force Engineering UniversityJan 1, 20070 citations

Constacyclic codes over $\mathbb{F}_q R_r$ and new quantum codesConstacyclic codes over $$\mathbb{F}_q R_r$$ and new quantum codes

Yu Qian, Yu Wang, L Wang

Journal of Applied Mathematics and ComputingApr 21, 20260 citations

Quantum Cyclic Codes Over ℤ m

{\mathbb{Z}}_m

Quantum Codes Over an Extension of

{\mathbb {Z}_4}

Quantum Codes from Cyclic Codes over

F_2+vF_2

[[5,1,3]]

[[17,1,7]]

and

[[17,9,3]]

, all of which are \emph{not} CSS. The

[[5,1,3]]

code is the well known Laflamme code and to the best of our knowledge the other two are new examples. Our definition of cyclicity applies to non-stabiliser codes as well; in fact we show that the

((5,6,2))

nonstabiliser first constructed by Rains\etal~ cite{rains97nonadditive} and latter by Arvind \etal~\cite{arvind:2004:nonstabilizer} is cyclic. We also study stabiliser codes of length

4^m +1

over

\mathbb{F}_2

for which we define a notation of BCH distance. Much like the Berlekamp decoding algorithm for classical BCH codes, we give efficient quantum algorithms to correct up to $\floor{\frac{d-1}{2}}$ errors when the BCH distance is

d

Constacyclic codes over

\mathbb{F}_q R_r

and new quantum codes