Michel, it is a great honour to be called Luc.
Any name with 平 in it, will somehow reflect reverence for the Heike 平家 (Taira 平) Clan as distant ancestors. As a word on its own, actually it just means 'flat'. Usually it is found with the sound Hei, or in a compound word -pei.
(In the same way many families like to use this character 源 to show they are descended however distantly from the 源氏 Genji or 源 Minamoto. Also 藤 indicates a fondness in the heart for the 藤原 Fujiwara noble family.)
As to the translation, I am deep in a long translation so I do not want to get involved in this at the moment. Also, many people ask for the origins of the 紋/家紋 Mon/Kamon but even after years of searching very few find the exact answer. This is not to discourage you!

But step by step one gets to Rome!
PS I
really liked that Cossack saying, "Everything that is not given is lost".
Thank you for that. I hope I can help you in some way when I have more time.