Consider looking at the substr function, or just convert your string into strings of two elements by doing a match:

my @two_bits = ($bin_num =~ /(..)/g);
[download]

Alternatively, consider creating four binary numbers that check for the four two-bit positions and whether they have the appropriate values:

my %bits = (
  dd => 0b11000000,
  cc => 0b00110000,
  bb => 0b00001100,
  aa => 0b00000011,
);
[download]

Then, check your number wether $v & $bits{ dd } == 0 or $v & $bits{ dd } == $bits{ dd }.