Best CRC Polynomials

Philip Koopman, Carnegie Mellon University

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These are the "Best" general-purpose CRC polynomials with specific Hamming Distance Properties. (See also: Notation and copyright statement). IMPORTANT NOTE: These are "BEST" polynomials under an assumption of a low, constant random independent BERsuch as you'd find in communication networks. If you have a BER that is higher than, say, 1 bit in 100,000, or is non-constant, or is non-random/non-independent, then you need to understand more before using these polynomials.

This data includes work-in-progress results.

CRC Summary Tables

Below is a table of CRC Polynomial performance by Hamming Distance. Click a CRC size for detailed information about CRC polynomials.

(**) means that this is a temporary result which has approximately the longest possible dataword length at the specified HD, but might not be the "best" possible value. (For example, probably there is some as yet unknown result with a slightly longer dataword length at that HD or with lower weights at the same HD.) Ongoing computations will be used to update this value to the "best" value when available. In the meantime, there's nothing wrong with using this polynomial as long as it provides adequate properties for your application.

Additional polynomials (33-64 bits; Special Properties)

• For details about these polynomials, and for CRCs larger than 32 bits, see the CRC Zoo
• For specialized polynomials, see the CRC Selection page. This includes polynomials with combination properties and good polynomials with all zero terms after the lowest byte.

To use these tables: top number in each cell is maximum dataword length at that Hamming Distance. The bottom number in each cell is a "good" polynomial that gives at least that HD up to the indicated dataword length in implicit +1 notation. For example, the polynomial 0x247 is a 10-bit CRC that provides HD=4 (or better) up to 501 bit dataword length (501+10=511 bit codeword length). The corresponding polynomial is: 0x247=x^10 +x^7 +x^3 +x^2 +x +1, and is alternately known as 0x48f in explicit +1 notation. See the Polynomial Zoo for detailed information (or click the CRC size link at the top of each column). Additionally, see the tables below for more nuanced selection criteria.

Notes: Minimum dataword length evaluated for the above table is 4 bits. Grayed-out boxes mean that it has been confirmed that the HD at that row cannot be achived with a dataword length of 4 bits or longer. Color highlighted cells indicated work in progress/missing data.

Notation:

• For details on interpreting the data rows and data file formats, see Hamming Weight Data notes section.
• *p - primitive polynomial. This has optimal length for HD=3, and good HD=2 performance above that length.
• *o - odd bit errors detected. This has a factor of (x+1) and detects all odd bit errors (implying that even number of bit errors have an elevated undetected error rate)
• *op - odd bit errors detected plus primitive. This is a primitive polynomial times (x+1). It has optimal length for HD=4, and detects all odd bit errors.
• ** - HD confirmed, but minimum weight selection at maximum length has been approximated due to long run-time. In other words, this polynomial will perform as advertised, but there is a slight chance a slightly better polynomial exists that was not found due to less than an exhaustive search. This only applies to a few 32-bit and larger polynomials.

This web page and all data files are Copyrighted 2015-2018 by Philip Koopman, Carnegie Mellon University.

This work is licensed under a Creative Commons Attribution 4.0 International License.

Please note that if any data errors or other issues are identified they will be updated at this page, but not necessarily anywhere that has copied these results. Therefore, you should always confirm at this URL: http://users.ece.cmu.edu/~koopman/crc/ that you have the most current version of data before using it.