Let $C$ be a linear code of length $n$ and dimension $k$ over the finite field $\mathbb{F}_{q^m}$. The trace code $\mathrm{Tr}(C)$ is a linear code of the same length $n$ over the subfield $\mathbb{F}_q$. The obvious upper bound for the dimension of the trace code over $\mathbb{F}_q$ is $mk$. If equality holds, then we say that $C$ has maximum trace dimension. The problem of finding the true dimension of trace codes and their duals is relevant for the size of the public key of various code-based cryptographic protocols. Let $C_{\mathbf{a}}$ denote the code obtained from $C$ and a multiplier vector $\mathbf{a}\in (\mathbb{F}_{q^m})^n$. In this paper, we give a lower bound for the probability that a random multiplier vector produces a code $C_{\mathbf{a}}$ of maximum trace dimension. We give an interpretation of the bound for the class of algebraic geometry codes in terms of the degree of the defining divisor. The bound explains the experimental fact that random alternant codes have minimal dimension. Our bound holds whenever $n\geq m(k+h)$, where $h\geq 0$ is the Singleton defect of $C$. For the extremal case $n=m(h+k)$, numerical experiments reveal a closed connection between the probability of having maximum trace dimension and the probability that a random matrix has full rank.

翻译：设$C$为有限域$\mathbb{F}_{q^m}$上长度为$n$、维数为$k$的线性码。迹码$\mathrm{Tr}(C)$是子域$\mathbb{F}_q$上相同长度$n$的线性码。迹码在$\mathbb{F}_q$上的维数的显然上界为$mk$。若等式成立，则称$C$具有最大迹维。迹码及其对偶码真实维数的求解问题，与多种基于编码的密码协议的公钥尺寸密切相关。令$C_{\mathbf{a}}$表示由$C$与乘子向量$\mathbf{a}\in (\mathbb{F}_{q^m})^n$导出的码。本文给出了随机乘子向量产生具有最大迹维的码$C_{\mathbf{a}}$的概率下界。针对代数几何码类，我们利用定义除子的次数对该下界进行了解释。该下界解释了随机交替码具有最小维数的实验事实。当$n\geq m(k+h)$时（其中$h\geq 0$为$C$的Singleton缺陷），该下界成立。对于极端情形$n=m(h+k)$，数值实验揭示了具有最大迹维的概率与随机矩阵满秩概率之间的密切联系。