Quantum Codes from Trace Codes

For an odd prime p and a positive integer m, let 𝔽_p^m be the finite field with p^m elements. For D_1={d∈𝔽_p^m^* : Tr_e^m(d^2)=0}={d_1,d_2,… ,d_n} (say) and D_2={d∈𝔽_p^m : d^k=1} ( k=p^l-1 for a divisor l of m), first we define classical linear codes by 𝒞_D_1= {(Tr_e^m(ad_1), Tr_e^m(ad_2),… ,Tr_e^m(ad_n)): a∈𝔽_p^m}; 𝒞_D_1= {( u+Tr_e^m(ad_1),u+Tr_e^m(ad_2),… ,u+Tr_e^m(ad_n)): a∈𝔽_p^m, u∈𝔽_p^e}; 𝒞_D_2= {(Tr_e^m(ad_1),… ,Tr_e^m(ad_k), u+Tr_e^m(ad_1),… ,u+Tr_e^m(ad_k)): a∈𝔽_p^m, u∈𝔽_p^e}, where Tr^m_e denotes the trace function from 𝔽_p^m onto 𝔽_p^e and e is a divisor of m. Then we determine weight distribution of the code 𝒞_D_1∖𝒞_D_1 and construct quantum codes from the codes 𝒞_D_1 and 𝒞_D_1 based on CSS code construction. Finally, we construct quantum code from the code 𝒞_D_2 and show that the code obtained from 𝒞_D_2 is MDS if and only if s=1, where s=m/e .