Some Bounds on Binary LCD Codes

A linear code with a complementary dual (or An LCD code) is defined to be a linear code C whose dual code C⊥ satisfies C ∩ C⊥= 0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\left \{ \mathbf {0}\right \} $\end{document}. Let LD (n, k) denote the maximum of possible values of d among [n, k, d] binary LCD codes. We give the exact values of LD (n, k) for k = 2 for all n and some bounds on LD (n, k) for other cases. From our results and some direct search we obtain a complete table for the exact values of LD (n, k) for 1 ≤ k ≤ n ≤ 12. As a consequence, we also derive bounds on the dimensions of LCD codes with fixed lengths and minimum distances.