New quantum codes from dual-containing cyclic codes over finite rings

Abstract

Let R = F2m + uF2m + · · · + u k F2m , where F2m is the finite field with 2 m elements, m is a positive integer, and u is an indeterminate with u k+1 = 0. In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over R. A new Gray map over R is defined and a sufficient and necessary condition for the existence of dual-containing cyclic codes over R is given. A new family of 2 m -ary quantum codes is obtained via the Gray map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over R. In particular, a new family of binary quantum codes is obtained via the Gray map, the trace map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over R.

New non-binary quantum codes from constacyclic codes over <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $\end{document}</tex-math></inline-formula>

“…In [9], Dertli et al constructed some new quantum codes from cyclic codes over the ring F 2 + uF 2 + vF 2 + uvF 2 with u 2 = u, v 2 = v, uv = vu. In [26], Tang et al constructed many good quantum codes from dual-containing cyclic codes over the finite chain ring F 2 m + uF 2 m + · · · + u k F 2 m , where u k+1 = 0, m is a positive integer.…”

A Family of Constacyclic Codes over 𝔽 2 m + u 𝔽 2 m and Its Application to Quantum Codes

“…Now we obtain a sufficient and necessary condition for the existence of dual-containing cyclic codes over R by using generator polynomials of (1 + u)-constacyclic codes over R. The following Theorem is improved the main result (Theorem 4.2) in Ref. [30].…”

Quantum Codes from Constacyclic Codes over Polynomial Residue Rings

“…After the realization that non-binary quantum codes are useful for fault-tolerant computations, much attention has been paid to non-binary quantum codes. Tang et al [6] obtained some quaternary quantum codes with simple-root cyclic codes over F 2 m [u]/⟨u k ⟩. Li et al [7] constructed some nonbinary quantum codes and Maximal-distance-separable (MDS) codes from simple-root cyclic codes over F p m + uF p m .…”