Asymmetric quantum codes from skew cyclic codes over B1

Abstract

Skew cyclic codes over Bk = 𝔽pr [v1, …, vk]/〈vi2 − vi, vivj – vjvi〉 have been studied by Irwansyah et al in 2018. In this paper we use B1 = 𝔽p2 [v]/〈v2 – v〉 and the Gray map φ : B1→𝔽p22 defined by φ(a + bv) = (a, a + b) for all a + bv ∈ B1 and extend φ to Φ : B1n→𝔽p22n. If C is a skew cyclic code over B1, we can get Φ(C) ⊆ 𝔽p22n a skew cyclic code if n is odd and a skew 2-quasicyclic code if n is even. Then by using the map S introduced by Ezerman et al in 2011, we get a (4n, p2k, 2d) additive code over 𝔽p2 and we use this to construct an additive asymmetric quantum code.