Quantum codes from $ \sigma $-dual-containing constacyclic codes over $ \mathfrak{R}_{l, k} $

Abstract

Let $ \mathfrak{R}_{l, k} = {\mathbb F}_{p^m}[u_1, u_2, \cdots, u_k]/ \langle u_{i}^{l} = u_{i}, u_iu_j = u_ju_i = 0 \rangle $, where $ p $ is a prime, $ l $ is a positive integer, $ (l-1)\mid(p-1) $ and $ 1\leq i, j\leq k $. First, we define a Gray map $ \phi_{l, k} $ from $ \mathfrak{R}_{l, k}^n $ to $ {\mathbb F}_{p^m}^{((l-1)k+1)n} $, and study its Gray image. Further, we study the algebraic structure of $ \sigma $-self-orthogonal and $ \sigma $-dual-containing constacyclic codes over $ \mathfrak{R}_{l, k} $, and give the necessary and sufficient conditions for $ \lambda $-constacyclic codes over $ \mathfrak{R}_{l, k} $ to satisfy $ \sigma $-self-orthogonal and $ \sigma $-dual-containing. Finally, we construct quantum codes from $ \sigma $-dual-containing constacyclic codes over $ \mathfrak{R}_{l, k} $ using the CSS construction or Hermitian construction and compare new codes our obtained better than the existing codes in some recent references.

Keywords

• quantum codes
• $ \sigma $-self-orthogonal
• $ \sigma $-dual-containing
• constacyclic codes
• css construction
• hermitian construction