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，由于自正交嵌入能够产生某些最优自正交码，线性码的自正交嵌入成为研究热点。LCD码具有平凡壳，因此是自正交码的对偶对象。由此自然提出一个问题：能否将线性码嵌入到最优LCD码中？为回答此问题，我们首先确定了需向线性码生成矩阵中添加的列数，以实现将给定码嵌入LCD码。随后，我们刻画了线性码最短LCD嵌入的所有可能形式。作为示例，我们从短长度的二元和三元汉明码出发，获得了最小距离为4的最优LCD码。此外，我们发现了新的三元LCD码，其参数包括$[23, 4, 14]$、$[23, 5, 12]$、$[24, 6, 12]$和$[25, 5, 14]$，以及一个新四元LCD $[21, 10, 8]$码，每个码的最小距离均比已知码大1。这表明我们的最短LCD嵌入方法在各类域上寻找最优LCD码时具有实用价值。