Let $R_2 = \mathbb{F}_{p^m} + u\mathbb{F}_{p^m}$ ($u^2 = 0$), where $p$ is an odd prime and $m, s \in \mathbb{N}$. For $p \equiv 3, 5 \pmod 7$ with $\gcd(m, 6) = 1$, the cyclotomic polynomial $Φ_7(x)$ is irreducible over $\mathbb{F}_{p^m}$. This yields a direct sum decomposition $C = C_1 \oplus C_2$ for any cyclic code $C$ of length $7p^s$ over $R_2$, where $C_1$ has length $p^s$ and $C_2$ is a $7$-cyclotomic code of length $6p^s$. We classify $7$-cyclotomic codes into four disjoint generator-based types and calculate exact cardinalities using residue and torsion subcodes. Furthermore, explicit generators for the Euclidean dual codes $C^\perp$ are determined. As operational applications of these classified codes, we construct new families of quantum stabilizer codes via the CSS framework and establish parameter criteria for linear codes with complementary duals (LCD codes).

翻译：暂无翻译