Binary LCD Codes from ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$

Linear complementary dual (LCD) codes over finite fields are linear codes satisfying $C\cap C^{\perp }=\{0\}$. We generalize the LCD codes over finite fields to ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-LCD codes over the ring ${\mathbf{Z}}_2\times ({\mathbf{Z}}_2+u{\mathbf{Z}}_2)$. Under suitable conditions, ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-linear codes that are ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-LCD codes are characterized. We then prove that the binary image of a ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-LCD code is a binary LCD code. Finally, by means of these conditions, we construct new binary LCD codes using ${\mathbb{Z}}_2{\mathbb{Z}}_2[u]$-LCD codes, most of which have better parameters than current binary LCD codes available.