In the recent years, there has been active research on self-orthogonal embeddings of linear codes since they yielded some optimal self-orthogonal codes. LCD codes have a trivial hull so they are counterparts of self-orthogonal codes. So it is a natural question whether one can embed linear codes into optimal LCD codes. To answer it, we first determine the number of columns to be added to a generator matrix of a linear code in order to embed the given code into an LCD code. Then we characterize all possible forms of shortest LCD embeddings of a linear code. As examples, we start from binary and ternary Hamming codes of small lengths and obtain optimal LCD codes with minimum distance 4. Furthermore, we find new ternary LCD codes with parameters including $[23, 4, 14]$, $[23, 5, 12]$, $[24, 6, 12]$, and $[25, 5, 14]$ and a new quaternary LCD $[21, 10, 8]$ code, each of which has minimum distance one greater than those of known codes. This shows that our shortest LCD embedding method is useful in finding optimal LCD codes over various fields.

翻译：近年来，线性码的自正交嵌入研究十分活跃，因为它们产生了一些最优的自正交码。LCD码具有平凡核，因此它们是自正交码的对偶。一个自然的问题是：能否将线性码嵌入到最优的LCD码中？为回答此问题，我们首先确定了为使给定码嵌入到一个LCD码中，需要在其生成矩阵中添加的列数。接着，我们刻画了线性码所有可能形式的最短LCD嵌入。作为示例，我们从短长度的二元和三元汉明码出发，得到了最小距离为4的最优LCD码。此外，我们发现了参数为$[23, 4, 14]$、$[23, 5, 12]$、$[24, 6, 12]$和$[25, 5, 14]$的新三元LCD码，以及一个参数为$[21, 10, 8]$的新四元LCD码，其中每个码的最小距离均比已知码的最小距离大1。这表明我们的最短LCD嵌入方法对于在不同域上寻找最优LCD码是有效的。