Is there any bit level error detection algorithm that use minimum extra bits?

Question

I have a 32-bit number that is created by encoding some data, I want to be more confident that the data (a max 32-bit number) is not changed when decoding it, so I am going to add some error detection bits.

I need to keep the data as short as possible, so I can only add a few bits for error detection, in some cases just 1 bit.

I'm looking for an algorithm that detects more bit changes and needs fewer extra bits.

I was thinking of calculating a checksum or CRC and just dropping extra bits or maybe xor the result to make it shorter but I'm not sure if the error detection remains good enough.

Thanks in advance for any help.

Answer 1

A 1-bit CRC, with polynomial x+1 would simply be the parity of your 32 message bits. That will detect any one-bit error in the resulting 32 bits. For a 2-bit CRC, you can use x²+1. You can define a CRC of any length. See Koopman's list for good CRC polynomials for CRCs of degree 3 and higher.