Binary Caps and LCD Codes with Large Dimensions
Keita Ishizuka, Yuhi Kamio
TL;DR
This paper links LCD codes to caps in projective space, deriving nonexistence results and determining optimal minimum distances for certain codimensions without computational search.
Contribution
It introduces a new framework connecting LCD codes and caps, enabling nonexistence proofs and optimal distance determination for specific codimensions.
Findings
Nonexistence theorems for LCD codes with min distance ≥ 4
Complete determination of optimal min distances for codimensions 7 and 8
Proofs achieved without exhaustive computation
Abstract
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 , providing computation-free proofs that were previously obtained only through exhaustive search. As an application, we completely determine the optimal minimum distances for codimensions and for the first time.
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