Some critical applications need to protect a single byte or word of data in memory against corruption. (For example, if you want to be extra-sure about protecting EEPROM values against corruption.) A previous post gives good CRC polynomials for a wide range of circumstances, but it is always nice to know if you can do better for a special case. Below are optimal CRC polynomials for a few special cases involving small data words that might be stored in memory (RAM, EEPROM, or otherwise).
Protecting an 8-bit data word:
8-bit CRC: HD=5; 0x9C = x^8 + x^5 + x^4 + x^3 + 1
16-bit CRC: HD=8; 0xE92F = x^16 + x^15 + x^14 + x^12 + x^9 + x^6 + x^4 + x^3 + x^2 + x + 1
32-bit CRC: HD=16; 0xC563942F = x^32 +x^31 +x^27 +x^25 +x^23 +x^22 +x^18 +x^17 +x^16 +x^13 +x^11 +x^6 +x^4 +x^3 +x^2 +x +1
Protecting a 16-bit data word:
8-bit CRC: HD=4; 0x93 = x^8 + x^5 + x^2 + x + 1
16-bit CRC: HD=7; 0x978A = x^16 + x^13 + x^11 + x^10 + x^9 + x^8 + x^4 + x^2 + 1
32-bit CRC: HD=14; 0xB396786D = x^32 +x^30 +x^29 +x^26 +x^25 +x^24 +x^21 +x^19 +x^18 +x^15 +x^14 +x^13 +x^12 +x^7 +x^6 +x^4 +x^3 +x +1
Protecting a 32-bit data word:
8-bit CRC: HD=4; 0x92 = x^8 + x^5 + x^2 + 1
16-bit CRC: HD=6; 0x8BFC = x^16 + x^12 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + 1
32-bit CRC: HD=12; 0xB527A43B = x^32 +x^30 +x^29 +x^27 +x^25 +x^22 +x^19 +x^18 +x^17 +x^16 +x^14 +x^11 +x^6 +x^5 +x^4 +x^2 +x +1
The hex number has an implicit +1. "HD" means Hamming Distance. Other information about this topic can be found at my previous post on good CRCs.
Note that for this sort of use it is a good idea to "seed" the CRC computation with a value of that is not zero. That avoids a data word value of zero giving a CRC computed value of zero -- which will result in an undetected error if memory is wiped to all zeros for some reason.