Some Bounds on Binary LCD Codes

Abstract

A linear code with a complementary dual (or LCD code) is defined to be a linear code $C$ whose dual code $C^{\perp }$ satisfies $C\cap C^{\perp }$= $\left\{0\right\}$. Let $LCD[n,k]$ denote the maximum of possible values of $d$ among $[n,k,d]$ binary LCD codes. We give exact values of $LCD[n,k]$ for $1\le k\le n\le 12$. We also show that $LCD[n,n-i]=2$ for any $i\ge 2$ and $n\ge 2^i$. Furthermore, we show that $LCD[n,k]\le LCD[n,k-1]$ for $k$ odd and $LCD[n,k]\le LCD[n,k-2]$ for $k$ even.

9 pages, 2 tables

References in corpus (3)

• Complementary Dual Algebraic Geometry Codes
• LCD Cyclic Codes over Finite Fields
• Erasure Techniques in MRD codes

Cited by in corpus (2)

• Linear codes over Fq which are equivalent to LCD codes
• Euclidean and Hermitian LCD MDS codes