A "strong reputation" means that someone or something is highly regarded and respected by others because of their good qualities, achievements, or behavior. Full definition
But as the craft expansion continues at a phenomenal clip, other cities are starting to build strong reputations of their own. (fortune.com)
You can rest assured that we are one of the leading players in a tightly defined market area, offering well known corporate lawyers with strong reputations. (mooreblatch.com)
Strong reputation among colleagues and peers for exceptional communication, project, and people management. (members.climber.com)