Quantum negacyclic codes

Abstract

In this paper, we construct families of nonbinary quantum codes from classical negacyclic codes. The BCH bound for negacyclic codes is applied to determine the dimension and distance of these quantum codes. The parameters of these quantum codes are good compared with the ones available in the literature. We also construct a family of quantum maximum-distance-separable codes with parameters [[q + 1,q − 2d + 3,d]] q , where q ≡ 1 mod 4 and 2 d (q + 3)/2.

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>

“…Since x 8 − 1 ≡ 0 (mod g i (x)g * i (x)) for i = 1, 2, 3 and x 8 + 1 ≡ 0 (mod g 0 (x)g * 0 (x)), by Theorem 4.4 we get C ⊥ ⊆ C , and Φ(C ) is a [16,12,3] linear Euclidean dualcontaining code over F 17 . Then we obtain a new quantum code with parameters [ [16,8,3]] 17 . This quantum code satisfies n + 2 − k − 2d = 4.…”

“…Afterwards, many good quantum error-correcting codes have been constructed by using classical cyclic codes over finite fields (see Refs. [8,11,17,20,24,27]). Recently, more and more coding scholars have studied the construction of quantum codes from cyclic codes over finite rings.…”

New Binary Quantum Codes Derived From One-Generator Quasi-Cyclic Codes

“…Taking cyclotomic cosets modulo n = 45 with respect to q = 2, one can get minimal polynomials of roots η i (0 ≤ i ≤ 44) which divide the polynomial x 45 − 1. Analogously, let g(x) be a polynomial defined by the following union set T of cyclotomic coset, 2,4,8,16,17,19,23,31, 32, 34, 38}, C (3) = {3, 6, 12, 24}, C (5) = {5, 10, 20, 25, 35, 40}, C (9) = {9, 18, 27, 36} and C (15) = {15, 30}. .…”

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

“…We denote by [[n, l, d]] q a q-ary quantum code for n qubits having q l codewords and minimum distance d. It is well known that quantum codes with parameters [[n, l, d]] q must satisfy the quantum Singleton bound: l ≤ n − 2d + 2 (see Refs. [14][15][16]). A quantum code achieving this bound is called a quantum maximum-distance-separable (MDS) code.…”