Cyclic Codes of Length 7_p^s over F_p^m + uF_p^m : Characterization, Duals, and Applications to Quantum and LCD Codes

2026-09-07

AI Summary

研究通过分解和分类方法解决了在特定条件下构造长度为7p^s的循环码问题，并应用于量子稳定码和LCD码。

Abstract

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 $\Phi_7(x)$ 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).

Problem

• cyclic codes
• quantum codes
• LCD codes
• cyclotomic codes

Innovation

• cyclic codes
• cyclotomic codes
• quantum stabilizer codes
• LCD codes