Linear complementary dual (LCD) codes are linear codes that intersect their dual trivially. We study double circulant (Euclidean) LCD codes over the finite chain ring $R=\mathbb F_2+u\mathbb F_2$, where $u^2=0$. Using the Chinese Remainder Theorem and the factorization of $x^n-1$, we derive an explicit enumeration formula, with a separate treatment of the constituent corresponding to $x-1$, which is not obtained by a formal specialization of the previously known counting formula over $\mathbb F_q[u]/(u^k)$. Assuming Artin's primitive root conjecture for $2$, we show that the binary Gray images contain an asymptotically good family of rate $1/2$ whose relative distance can approach $H_2^{-1}(1/4)$ from below. More importantly from the constructive viewpoint, by combining Gray images with a known extension construction, we obtain a binary LCD code with parameters $[29,15,7]$ and construct two explicit permutation-inequivalent binary LCD codes with parameters $[45,23,9]$.

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