# How many parity syndromes are there?

In RAID 6, there is a parity scheme that allows 2 concurrent disk failures. This requires 2 ‘syndromes’, one of which is simply XOR, as used for RAID 5's only parity. In trying to find out whether RAID 6 can support 3 or more parities, it has been disputed that there even exist any other parity syndromes.
I don't understand the maths for computing RAID 6 parity or ‘syndrome decoding’ but I sort of make out that the 2 syndromes used in RAID 6 are just 2 of a very large group and that XOR is simply a special-case. Presumably the number of syndromes is something along the lines of an exponential function of the number of bits in a chunk (or other rapidly increasing combinatorial function such as factorial). I see something there about the bits being coëfficients of a polynomial, but this is somewhat over my head at the moment.
If I'm correct that there are more than 2 parity syndromes, I'd like to know how to calculate the number of different syndromes for a given chunk size (or whatever parameters are required).

• What's your question? I can't tell what your specific confusion/question is. Please edit to clarify, and ask just one question per question. Is your question "Can RAID 6 support more than 3 or more parities?" If so, edit to ask that question (and remove the stuff about how to number of syndromes etc.; that should be asked separately). Be warned that you need to phrase the question carefully to be on-topic; questions about products or standard usage are off-topic. – D.W. Mar 11 '15 at 16:27
• Also, have you looked at en.wikipedia.org/wiki/RAID#Concept and en.wikipedia.org/wiki/Reed%E2%80%93Solomon_error_correction? It seems clear that it would be possible to construct a RAID-6-like scheme with any number of "parities", using a suitable error-correcting code. Is that what you are asking? If not, what specifically is the technical question? (I don't know whether that would technically be called RAID-6, but that's probably not on-topic here. I don't know whether any products implement it, but that's off-topic here, too.) – D.W. Mar 11 '15 at 16:28
• Syndrome decoding is a particular (efficient) technique for decoding Linear Block codes. RAID 6 is defined to be a particular linear block code with 2 parity bits. You could define your own linear block code with an arbitrary number of parity bits. For correcting more than 1 error you would want something like a Hadamard code. (It wouldn't be RAID 6 anymore, of course.) – Wandering Logic Mar 11 '15 at 22:30