Re: Triple-parity raid6

[David Brown <david.brown@hesbynett.no>]

On 09/06/11 02:01, David Brown wrote:
> Has anyone considered triple-parity raid6 ? As far as I can see, it
> should not be significantly harder than normal raid6 - either to
> implement, or for the processor at run-time. Once you have the GF(2⁸)
> field arithmetic in place for raid6, it's just a matter of making
> another parity block in the same way but using a different generator:
>
> P = D_0 + D_1 + D_2 + .. + D_(n.1)
> Q = D_0 + g.D_1 + g².D_2 + .. + g^(n-1).D_(n.1)
> R = D_0 + h.D_1 + h².D_2 + .. + h^(n-1).D_(n.1)
>
> The raid6 implementation in mdraid uses g = 0x02 to generate the second
> parity (based on "The mathematics of RAID-6" - I haven't checked the
> source code). You can make a third parity using h = 0x04 and then get a
> redundancy of 3 disks. (Note - I haven't yet confirmed that this is
> valid for more than 100 data disks - I need to make my checker program
> more efficient first.)
>
> Rebuilding a disk, or running in degraded mode, is just an obvious
> extension to the current raid6 algorithms. If you are missing three data
> blocks, the maths looks hard to start with - but if you express the
> equations as a set of linear equations and use standard matrix inversion
> techniques, it should not be hard to implement. You only need to do this
> inversion once when you find that one or more disks have failed - then
> you pre-compute the multiplication tables in the same way as is done for
> raid6 today.
>
> In normal use, calculating the R parity is no more demanding than
> calculating the Q parity. And most rebuilds or degraded situations will
> only involve a single disk, and the data can thus be re-constructed
> using the P parity just like raid5 or two-parity raid6.
>
>
> I'm sure there are situations where triple-parity raid6 would be
> appealing - it has already been implemented in ZFS, and it is only a
> matter of time before two-parity raid6 has a real probability of hitting
> an unrecoverable read error during a rebuild.
>
>
> And of course, there is no particular reason to stop at three parity
> blocks - the maths can easily be generalised. 1, 2, 4 and 8 can be used
> as generators for quad-parity (checked up to 60 disks), and adding 16
> gives you quintuple parity (checked up to 30 disks) - but that's maybe
> getting a bit paranoid.
>
>
> ref.:
>
> <http://kernel.org/pub/linux/kernel/people/hpa/raid6.pdf>
> <http://blogs.oracle.com/ahl/entry/acm_triple_parity_raid>
> <http://queue.acm.org/detail.cfm?id=1670144>
> <http://blogs.oracle.com/ahl/entry/triple_parity_raid_z>
>
>
> mvh.,
>
> David
>

Just to follow up on my numbers here - I've now checked the validity of 
triple-parity using generators 1, 2 and 4 for up to 254 data disks 
(i.e., 257 disks altogether).  I've checked the validity of quad-parity 
up to 120 disks - checking the full 253 disks will probably take the 
machine most of the night.  I'm sure there is some mathematical way to 
prove this, and it could certainly be checked more efficiently than with 
a Python program - but my computer has more spare time than me!


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