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Skew Cyclic codes over \F_q+u\F_q+v\F_q+uv\F_q
In this paper, we study skew cyclic codes over the ring
R=\F_q+u\F_q+v\F_q+uv\F_q, where , and
is an odd prime. We investigate the structural properties of skew cyclic codes
over through a decomposition theorem. Furthermore, we give a formula for
the number of skew cyclic codes of length over $R.
From Type-II Triply Degenerate Nodal Points and Three-Band Nodal Rings to Type-II Dirac Points in Centrosymmetric Zirconium Oxide
Using first-principles calculations, we report that ZrO is a topological
material with the coexistence of three pairs of type-II triply degenerate nodal
points (TNPs) and three nodal rings (NRs), when spin-orbit coupling (SOC) is
ignored. Noticeably, the TNPs reside around Fermi energy with large linear
energy range along tilt direction (> 1 eV) and the NRs are formed by three
strongly entangled bands. Under symmetry-preserving strain, each NR would
evolve into four droplet-shaped NRs before fading away, producing distinct
evolution compared with that in usual two-band NR. When SOC is included, TNPs
would transform into type-II Dirac points while all the NRs have gaped.
Remarkably, the type-II Dirac points inherit the advantages of TNPs: residing
around Fermi energy and exhibiting large linear energy range. Both features
facilitate the observation of interesting phenomena induced by type-II
dispersion. The symmetry protections and low-energy Hamiltonian for the
nontrivial band crossings are discussed.Comment: 7 pages, 5 figures, J. Phys. Chem. Lett. 201
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