@article{Liu2024, 
author = {Yang Liu and Ruihu Li and Qiang Fu and Hao Song},
title = {On the minimum distances of binary optimal LCD codes with dimension 5},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {7},
pages = {19137-19153},
keywords = {optimal code, LCD code, hull dimension, defining vector, reduced code},
url = {https://www.sciopen.com/article/10.3934/math.2024933},
doi = {10.3934/math.2024933},
abstract = {Let        d          a        (  n  ,  5  ) and        d          l        (  n  ,  5  ) be the minimum weights of optimal binary    [  n  ,  5  ] linear codes and linear complementary dual (LCD) codes, respectively. This article aims to investigate        d          l        (  n  ,  5  ) of some families of binary    [  n  ,  5  ] LCD codes when    n  =  31  s  +  t  ≥  14 with    s integer and    t  ∈    {  2  ,  8  ,  10  ,  12  ,  14  ,  16  ,  18  }. By determining the defining vectors of optimal linear codes and discussing their reduced codes, we classify optimal linear codes and calculate their hull dimensions. Thus, the non-existence of these classes of binary    [  n  ,  5  ,      d          a        (  n  ,  5  )  ] LCD codes is verified, and we further derive that        d          l        (  n  ,  5  )  =      d          a        (  n  ,  5  )  −  1 for    t  ≠  16 and        d          l        (  n  ,  5  )  =  16  s  +  6  =      d          a        (  n  ,  5  )  −  2 for    t  =  16. Combining them with known results on optimal LCD codes,        d          l        (  n  ,  5  ) of all    [  n  ,  5  ] LCD codes are completely determined.}
}