QUANTUM CODES FROM CYCLIC CODES OVER Fq+uFq+vFq

HU Peng,LI Hui,LIU Xiu-sheng

(School of Mathematics and Physics,Hubei Polytechnic University,Huangshi 435003,China)

Abstract:In this paper,we investigate the method of constructing quantum codes from cyclic codes over the ring R=Fq+uFq+vFq(u2=u,v2=v,uv=vu=0).By means of the generator polynomials and some decomposition of cyclic codes and their dual over R,we give a necessary and sufficient condition of the cyclic codes over R to be construct the quantum codes.Also as an application,we obtain new non binary quantum codes from those classes of cyclic codes.

Keywords:cyclic codes;quantum codes;dual codes

1 Introduction

Quantum error-correcting codes play an important role in quantum communications and quantum computations.After the pioneering work in[1–4],the theory of quantum codes has developed rapidly in recent years.As we know,the approach of constructing new quantum codes which have good parameters is an interesting research field.However,obtaining the parameters of the new quantum codes,especially the new good quantum codes,is a difficult problem.Recently,a lot of new quantum codes have been constructed by classical linear codes with Hermitian dual containing,which can be found in[5–10].

Cyclic codes over finite rings are an important class of codes from both a theoretical and a practical viewpoint.It has been shown that certain good quantum codes could be found as images of linear codes over some special rings under the Gray map(see[11]).In[12],Kai and Zhu established a construction for quantum codes from cyclic codes of odd length over finite chain ring F4+uF4,whereu2=0.Qian et al.in[13]gave a new method of constructing quantum codes from cyclic codes of odd length over finite ring F2+vF2,wherev2=v.Motivated by two papers above,we study quantum codes from cyclic codes over Fq+uFq+vFqwhereu2=u,v2=v,uv=vu=0,andq=ptfor some primepand positive integert.investigate some results about linear codes overR.In Section 3,we address the relation of Hermitian dual-containing codes betweenRand Fq.In light of the relation,we get quantum codes with new parameters over Fq.

2 Linear Codes over R

3 Quantum Codes from Cyclic Codes over R

In Table 1,we list some quantum codes obtained from Theorem 3.10.The table shows that our quantum codes have new parameters compared with the previous quantum codes available(see[18]).

Table 1:Quantum codes comparison

4 Conclusion

We have developed a new method of constructing quantum codes from cyclic codes over finite ringR.Using this method,we have constructed new quantum codes.We believe that cyclic codes over finite ringRwill be a good source for constructing new quantum codes.In a future work,we will use the computer algebra system MAGMA to find more new quantum codes.