A pair of linear codes (C, D) of length n over F-q is called a linear complementary pair (LCP) if their direct sum yields the full space F-q(n). By a result of Carlet et al. (2019), the best security parameters of binary LCPs of codes are left open. Motivated by this, we study binary LCPs of codes. We describe a sufficient condition for binary LCPs of codes which are not optimal. We carry out an exhaustive search to determine the best security parameters for binary LCPs of codes up to length 18. We also obtain results on optimal binary LCPs of codes for infinitely many parameters. For any k >= 2 and length n congruent to 0 or 1 mod (2(k) - 1), we prove that binary [n, k] LCPs of codes are optimal. Binary LCPs of codes of dimensions 2, 3, and 4 are also optimal for all lengths except for two instances, when (n, k) = (4, 3) and (8, 4). We provide explicit constructions of these infinite families of optimal LCPs. Our results also indicate that many security parameters coming from binary LCPs of codes exceed those from binary LCD codes by 1 or 2.