The following is a list of some of the smaller error correcting generator polynomials. There are thousands more.

generator            bits      bits      total   hpo2
polynomial            in        in       bits
                   message   checksum
decimal     binary
    7        111       1        2          3       4
   11       1011       4        3          7       8
   13       1101       4        3          7       8
   19      10011      11        4         15      16
   25      11001      11        4         15      16
   37     100101      26        5         31      32
   41     101001      26        5         31      32
   47     101111      26        5         31      32
   55     110111      26        5         31      32
   59     111011      26        5         31      32
   61     111101      26        5         31      32
   67    1000011      57        6         63      64
   91    1011011      57        6         63      64
   97    1100001      57        6         63      64
  103    1100111      57        6         63      64
  109    1101101      57        6         63      64
  115    1110011      57        6         63      64