Quantum codes from linear codes over finite chain rings

Abstract

In this paper, we provide two methods of constructing quantum codes from linear codes over finite chain rings. The first one is derived from the Calderbank-ShorSteane (CSS) construction applied to self-dual codes over finite chain rings. The second construction is derived from the CSS construction applied to Gray images of the linear codes over finite chain ring F p 2m + uF p 2m . The good parameters of quantum codes from cyclic codes over finite chain rings are obtained.

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 [3], Ashraf et al constructed new quantum codes from cyclic codes over F p + vF p with v 2 = v. In [13], Gao gave a construction of quantum codes from cyclic codes over finite non-chain ring F q + vF q + v 2 F q + v 3 F q with v 4 = v. In [4,5], Ashraf et al again obtained new non-binary quantum codes from cyclic codes over F q + uF q + vF q + uvF q with u 2 = u, v 2 = v, uv = vu and F p [u, v]/ u 2 −1, v 3 −v, uv −vu with u 2 = 1, v 3 = v, uv = vu, respectively. In [21], Liu et al obtained good quantum codes from cyclic codes over the finite chain ring F p 2m + uF p 2m , where u 2 = 0. These results illustrate that some good or optimal quantum codes can be constructed by cyclic codes over finite rings.…”

Quantum Codes from Constacyclic Codes over Polynomial Residue Rings

“…We find that some of our obtained quantum codes are better than previously known quantum codes in Refs. [8][9][10][11].…”

“…Recently, Ashraf [10] constructed some non-binary quantum codes from cyclic codes over F q + uF q + vF q + uvF q . Liu et al [11] obtained some nonbinary quantum codes from simple-root cyclic codes over F p 2m + uF p 2m .…”

Explicit Representation and Enumeration of Repeated-Root (δ + αu²)-Constacyclic Codes Over F₂ m [u]/‹u 2λ›

“…The study of error correction coding is based on the practical requirements of modern digital communication technology, and the profound involvement of various mathematical theories promotes the development and prosperity of this area of researches. Now, error-correcting coding theory has been applied to construct authentication codes, quantum error-correcting codes and post-quantum cryptography ( [15], [23], [27], [32], [33], [39], [43], [50], [59]). The problems arising in these fields have put forward various new requirements and subjects for the development of coding theory.…”