AI 中文总结
本文确定F_q+uF_q上线性码的最短自正交与LCD嵌入的精确长度，利用Witt理论构造最短自正交嵌入，刻画最短LCD嵌入，给出的部分嵌入的Gray映射像为F_q上的最优码。
AI 中文摘要
本文确定了F_q+uF_q上线性码的最短自正交与LCD嵌入的精确长度。通过将F_q+uF_q上的Gram矩阵分解为有限域F_q上的对称矩阵对，嵌入问题被约化为有限域上对称矩阵与交错矩阵的同余分类问题。得到了最短自正交嵌入长度的完整公式，在偶特征与奇特征下各产生两种不同情形。还证明了每个具有非零自由秩的F_q+uF_q上自正交码都可视为另一码的最短自正交嵌入，利用Witt理论构造了所有最短自正交嵌入。还建立了最短LCD嵌入的完整刻画，涉及在生成矩阵后附加指定大小的可逆矩阵与任意矩阵。给出了所考虑码的具有最大最小距离的自正交与LCD嵌入的例子，其中一些的Gray映射像是F_q上的最优码。
英文摘要
This paper determines the exact lengths of shortest self-orthogonal and LCD embeddings of linear codes over $\mathbb{F}_q+u\mathbb{F}_q$. By decomposing Gram matrices over $\mathbb{F}_q+u\mathbb{F}_q$ into pairs of symmetric matrices over the finite field $\mathbb{F}_q$, the embedding problems are reduced to the congruence classification of symmetric and alternate matrices over finite fields. Complete formulas for the shortest self-orthogonal embedding length are obtained, with two distinct cases arising in both even and odd characteristic. We also show that every self-orthogonal code over $\mathbb{F}_q+u\mathbb{F}_q$ with nonzero free rank can be viewed as a shortest self-orthogonal embedding of another code. We use Witt theory to construct all shortest self-orthogonal embeddings. A complete characterization of shortest LCD embeddings is also established in terms of invertible and arbitrary matrices of prescribed sizes appended to a generator matrix. Examples of self-orthogonal and LCD embeddings with the largest minimum distance for the code considered are also presented, some of whose Gray images are optimal codes over $\mathbb{F}_q$.
