AVR Freaks

As Jim mentions, the value CRC is based on statistics. The larger the crc, the less likely you will have different sequences generating the same CRC. As well, CRCs are normally applied to serial data - the expectation is that you get bit errors and for a single bit error, the CRC generates a different number. So, it is not foolproof. Where you do have different sequences generate the same CRC value, you'll find the sequences are very different. Apply statistics to the likelihood of the expected error source to cause such a change. 

CRCs are popular because they are easy to generate/check in hardware - they are a shift register with some xor gates.

Don't choose a random CRC polynomial - some work better than others. Best to choose a popular polynomial as it has been proven to work.

If you want something that is unlikely to have a collision, look at using cryptographic hashes.  Unfortunately, these are much more processor intensive unless your processor chip has special hardware to help.

I tried this:

https://crccalc.com

I get different results for 41 and 42 hex. So what is the issue? You say CRC is weak, but where is your evidence? 

Eg: CRC-8 poly

41 hex gives C0 result

42 hex gives C9 result

As well, don't write strange English when it is not your native language - it only makes it harder for us to understand. Keep your language simple and state facts, this will allow us to help you better.

Using the online calculator,

The sequence 3031 hex gives a crc of CD hex using the maxim poly of 31 hex. I can only assume you are calculating the CRC incorrectly.

Since you are using the Maxim polynomial, this suggests you are using 1wire devices. If so, tell us first, don't let us guess.

HD is 'hamming distance'. 

u = You. You can type a couple extra letters to make it easier for us to read.

Give us an example of the problem you are facing. I gave you examples that you can try. Do they not work for you?

//
// Compute a Dallas Semiconductor 8 bit CRC directly.
// this is much slower, but much smaller, than the lookup table.
//
uint8_t crc8(const uint8_t *addr, uint8_t len)
{
	uint8_t crc = 0;

	while (len--) {
		uint8_t inbyte = *addr++;
		for (uint8_t i = 8; i; i--) {
			uint8_t mix = (crc ^ inbyte) & 0x01;
			crc >>= 1;
			if (mix) crc ^= 0x8C;
			inbyte >>= 1;
		}
	}
	return crc;
}

source: https://github.com/adafruit/MAX3...

Just doing an xor is a checksum, not a CRC. CRCs are bit orientated, so you need to shift then xor for each bit.

You still haven't told us precisely what you're trying to achieve - all we've got so far is obscure references to CRC and little evidence. Tell what what you want to achieve and what you've done so far - then we can move forward. At the moment we're going around in circles because you aren't asking a clear question.

Why are you making this so hard? Why do you post a random app note page with no reference?

Just show us your code.