Quantum Codes Derived from Cyclic Codes

Abstract

In this note, we present a construction of new nonbinary quantum codes with good parameters. These codes are obtained by applying the Calderbank-Shor-Steane (CSS) construction. In order to do this, we show the existence of (classical) cyclic codes whose defining set consists of only one cyclotomic coset containing at least two consecutive integers.

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>

“…11 . This quantum code has the same minimum distance as the known quantum code with parameters [[63, 51, 3]] 11 appeared in [19], but our code has the larger code rate than that code. = {5}, by Theorem 4.9 we get C ⊥ H ⊆ C , and Φ(C ) is a [12,8,4] linear Hermitian dualcontaining code over F 13 2 .…”

New Non-Binary Quantum Codes Derived From a Class of Linear Codes

“…Example 3: Let C be an [11,6,5] linear code over [11,1,4]] 3 appeared in [19], but our code has the larger minimum distance than that code.…”

“…In recent years, a lot of good quantum codes are constructed by self-orthogonal codes over finite fields because the self-orthogonal codes satisfy the conditions of CSS quantum codes construction method, and the self-orthogonal codes can be constructed effectively by algebraic codes such as cyclic codes, constacyclic codes, AG codes and so on. For example, Thangaraj and McLaughlin [10] provided a construction for quantum codes from Hermitian self-orthogonal cyclic codes over F 4 m , Guardia [11] constructed some new quantum codes from a class of special cyclic codes whose defining set consists of only one cyclotomic coset containing at least two consecutive integers; Xiaoyan [12] constructed quantum code from some classes of constacyclic codes, Koroglu [13] constructed eight new classes of entanglementassisted quantum MDS codes by virtue of a decomposition of the defining set of constacyclic codes; Feng et al [14] gave an asymptotic bounds on quantum codes from algebraic geometry codes, Chen [15] constructed asymptotically good family of quantum codes from algebraic geometric codes, Munuera et al [16] studies quantum codes construction from algebraic geometric codes of Castle type.…”

Some New Quantum BCH Codes over Finite Fields

“…Aly et al extended Steane's results [8] to narrow-sense (not necessarily primitive) BCH codes with certain distances over GF(q) [5]. Nonbinary quantum codes with better code parameters were obtained by CSS construction [9]. Steane's enlargement construction was generalized from binary quantum codes to q-ary quantum codes [10].…”

“…Good nonbinary quantum codes were constructed by corresponding cyclotomic cosets with given parameters [16]. The designed quantum BCH codes were obtained with given code lengths [9,[16][17][18][19][20].…”