\(\mathbb{Z}_{p}\mathbb{Z}_{p}[u]\)-additive cyclic codes
Summary: Additive cyclic codes of length \((\alpha,\beta)\) over \(\mathbb Z_p\mathbb Z_p[u]\) can be viewed as \(\mathbb Z_p[u][x]\)-submodules of \(\mathbb Z_p[x]/(x^\alpha - 1) \times \mathbb Z_p[u][x]/(x^\beta - 1)\), where \(\mathbb Z_p[u] = \mathbb Z_p + u\mathbb Z_p\), \(u^2 = 0\). In this paper, we determine the generator polynomials and the minimal generating sets of this family of codes as \(\mathbb Z_p[u]\)-submodules of \(\mathbb Z_p[x] / (x^\alpha-1) \times \mathbb Z_p[u][x] / (x^\beta-1)\). Further, we also determine the generator polynomials of the dual codes of \(\mathbb Z_p\mathbb Z_p[u]\)-additive cyclic codes. Moreover, some binary quantum codes are constructed by additive cyclic codes over \(\mathbb Z_2\mathbb Z_2[u]\).
scientific article; zbMATH DE number 6973112