There are 2 reasons an antibiotic would affect birth control. 1. The medications interact with each other (the antibiotic decreases hormone levels in the body) and 2. The antibiotic kills bacteria in the GI track that are involved with the body's recycling of estrogen - which could "theoretically" have an effect on birth control effectiveness. This is why we tell women to use a back-up method even if there is no known "interaction".
Flucoxacillin is not known to interact with Implanon, but it's advisable to use a non-medicinal backup method of contraception (condom, diaphragm, etc.) while taking antibiotics.