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Binary Caps and LCD Codes with Large Dimensions

Keita Ishizuka, Yuhi Kamio

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 $4$, 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.

Binary Caps and LCD Codes with Large Dimensions

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.

Paper Structure

This paper contains 15 sections, 18 theorems, 25 equations, 1 table.

Table of Contents

  1. Introduction
  2. Our Contribution.
  3. Organization.
  4. Preliminaries
  5. Linear Codes and LCD Codes
  6. Caps in Projective Space
  7. Connection Between Caps and LCD Codes
  8. The Gram Matrix and Large Caps
  9. Properties of the Gram Matrix
  10. Structure Theorem
  11. Determination of $d_2^E(n,n-m)$
  12. Nonexistence Theorems
  13. Complete Results for Codimensions $6$ and $7$
  14. Complete Results for Codimension $8$
  15. Conclusion

Key Result

Lemma 1

Let $C$ be an $[n,k]$ code with generator matrix $G$. Then In particular, $C$ is LCD if and only if $GG^T$ is nonsingular, and self-orthogonal if and only if $GG^T=0$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (34)

  • Lemma 1: guenda2018constructions
  • Lemma 2: carlet2018hull
  • Definition 1
  • Theorem 1: bruen1999long; see also DavydovTombak
  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Definition 2
  • ...and 24 more