Effects of parity bit on odd code length regarding its size after alternation
Normally a code's hamming distance is limited to some maximum message length. For example in the case of a 32 bit CRC, for Hamming distance = 3 (2 bit error detection), the maximum length for a message is 2^32-1 bits (including the 32 bits of CRC), depending on the CRC used. With a larger message, a  2 bit error can fail. For a parity check or Hamming distance = 2 (1 bit error detection), there is no maximum length limit.

If the message length is &lt;= maximum message length for that Hamming distance, and k is odd (maximum # of error bits detected is even), then adding a parity check in addition to the CRC should increase Hamming distance to k+1.

For example, CRC16 based on polynomial x^16 + x^12 + x^5 + 1 (0x11021) is the product of 2 irreducible polynomials : (x^15 + x^14 + x^13 + x^12 + x^4 + x^3 + x^2 + x + 1) (0xf01f) and (x + 1) (0x3). For message length (including CRC bits) CRC poly (x^15 + x^14 + x^13 + x^12 + x^4 + x^3 + x^2 + x + 1) Hamming distance is 3 for message length up to 32767 (2^15-1) bits, and the (x + 1) term increases this to Hamming distance 4. As mentioned above, it can detect any odd number of bit errors for any length message.