Quantum codes from Hermitian dual-containing constacyclic codes over \n \n \n \n $${\\mathbb {F}}_{q^{2}}+{v}{\\mathbb {F}}_{q^{2}}$$\n \n \n \n F\n \n q\n

Abstract

Let $${\\mathbb {R}}$$ R be the finite non-chain ring $${\\mathbb {F}}_{{ q}^{2}}+{v}{\\mathbb {F}}_{{ q}^{2}}$$ F q 2 + v F q 2 , where $${v}^{2}={v}$$ v 2 = v and q is an odd prime power. In this paper, we study quantum codes over $${\\mathbb {F}}_{{ q}}$$ F q from constacyclic codes over $${\\mathbb {R}}$$ R . We define a class of Gray maps, which preserves the Hermitian dual-containing property of linear codes from $${\\mathbb {R}}$$ R to $${\\mathbb {F}}_{{ q}^{2}}$$ F q 2 . We study $${\\alpha }(1-2v)$$ α ( 1 - 2 v ) -constacyclic codes over $${\\mathbb {R}}$$ R , and show that the images of $$\\alpha (1-2v)$$ α ( 1 - 2 v ) -constacyclic codes over $${\\mathbb {R}}$$ R under the special Gray map are $$\\alpha ^{2}$$ α 2 -constacyclic codes over $${\\mathbb {F}}_{{ q}^{2}}$$ F q 2 . Some new non-binary quantum codes are obtained via the Gray map and the Hermitian construction from Hermitian dual-containing $$\\alpha (1-2v)$$ α ( 1 - 2 v ) -constacyclic codes over $${\\mathbb {R}}$$ R .