AI 中文总结
研究随机线性码自同构群，通过证明当min{k, n - k} ≥ (2 + ε)log_q n时，k维随机码极大概率有平凡自同构群，为基于线性码等价性问题的密码方案安全性分析提供理论支持。
AI 中文摘要
线性码自同构群的研究是编码理论的基础课题。匹配码字框架是分析基于线性码等价性（LCE）问题的密码方案安全性的标准工具，常假定q元随机码有平凡自同构群，但未被文献正式证明。本文证明，当n趋于无穷时，只要min{k, n - k} ≥ (2 + ε)log_q n（对任意ε>0），k维随机码\(\mathcal{C} \subseteq \mathbb{F}_q^n\)极大概率有平凡自同构群。
英文摘要
The matching codewords framework is a key tool in recent algorithms for solving the Linear Code Equivalence (LCE) problem and in security analyses of LCE-based cryptographic schemes such as LESS. These analyses often rely on the assumption that a random $q$-ary linear code has no monomial automorphisms other than scalar multiples of the identity. For binary codes, Lefmann, Phelps, and Rödl established the corresponding rigidity phenomenon in the relevant logarithmic dimension range. For general $q$, Hou established an averaged result over all dimensions, whereas the recent prescribed-dimension result of Di Giusto and Ravagnani applies only in a restricted regime near $n/2$. For every fixed prime power $q$ and every fixed real number $\varepsilon>0$, we prove that a uniformly random $k$-dimensional code $\mathcal{C}\subseteq\mathbb{F}_q^n$ has a trivial monomial automorphism group with probability tending to $1$ as $n\to\infty$, provided that $m:=\min\{k,n-k\}\geq(2+\varepsilon)\log_q n$. Furthermore, when $m \le 2 \log_q n + C$, where $C$ is a constant independent of $n$, we also show that the probability that the automorphism group of $\mathcal{C}$ is nontrivial is at least $\frac{1}{2} - \varepsilon$ for large enough $n$.
