Binary Caps and LCD Codes with Large Dimensions
Provides theoretical proofs and new optimality results for LCD codes, benefiting coding theory researchers.
The paper links LCD codes to caps in projective space, using maximal cap theory to prove nonexistence of LCD codes with minimum distance ≥4 without exhaustive search, and determines optimal minimum distances for codimensions 7 and 8 for the first time.
We establish a connection between linear complementary dual (LCD) codes and caps in projective space. Using this framework and the structure theory of maximal caps, we derive nonexistence theorems for LCD codes with minimum distance at least $4$, providing computation-free proofs that were previously obtained only through exhaustive search. As an application, we completely determine the optimal minimum distances for codimensions $7$ and $8$ for the first time.