無し is a classical style adjective meaning "no", "without", or "not existing" . and this is the predicate form, which will appear in a dictionary or can end a sentence with. Today this form create words , that mean "something without ....", Or " someone who dose not have ..." .
On the other hand , 無き is the participle form of 無し. that means it will be followed by a word like Noun ,for exsample, and modificate the word.
So usage could be
無し: 宿題は無し there is no homework。ゴールは無し there is no goal ( football game)。能無し someone who doesn't have any skill, in other words incompetent.
無き: 国境無き医師団 Doctors without borders、仁義無き戦い battles without Honor and humanity. It is the title of series of yakuza movies.
There might be my misunderstanding or typo, in this case please correct me.
Hope it will help you.