Shortest LCD embeddings of binary, ternary and quaternary linear codes

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.

💡 Research Summary

The paper investigates the problem of embedding an arbitrary linear code into a linear complementary‑dual (LCD) code with the smallest possible increase in length. An LCD code is defined as a linear code whose hull (the intersection of the code with its dual) is trivial, i.e., {0}. While much recent work has focused on self‑orthogonal embeddings (which produce codes with large hulls), the authors turn the attention to the opposite extreme: codes with a zero hull.

The central theoretical contribution is a precise formula for the length of a shortest LCD embedding. Let C be an