Problem Statement

Given a series of ones and zeroes in a string, define a repeated string or rep-string as a string which is created by repeating a substring of the first N characters of the string truncated on the right to the length of the input string, and in which the substring appears repeated at least twice in the original. For example, the string 10011001100 is a rep-string as the leftmost four characters of 1001 are repeated three times and truncated on the right to give the original string. Note that the requirement for having the repeat occur two or more times means that the repeating unit is never longer than half the length of the input string.

Let's start with the solution:

# Returns a list of all rep-strings
Reps←(⌊≠÷2˙)((⊣≥≠¨∘⊢)/⊢)(<≡¨≠⥊¨1↓↑)/1↓↑

# Tests
tests←⟨ 
   "1001110011", "1110111011", "0010010010",
   "1010101010", "1111111111", "0100101101",
   "0100100", "101", "11", "00", "1"
⟩

∾´{ 𝕩∾':'∾(•Fmt Reps 𝕩)∾@+10 }¨tests