Hi,
Yes, being a generator for GF(2^8) is a requirement for a parity
generator (sorry for the confusing terminology here - if anyone has a
better suggestion, please say) to be part of a 255 data disk system.
However, being a GF generator is necessary but not sufficient - using
parity generators (1, 2, 4, 16) will /not/ give quad parity for 255 data
disks, even though individually each of 1, 2, 4 and 16 are generators
for GF.
255 data disks is the theoretical limit for GF(2⁸). But it is a
theoretical limit of the algorithms - I don't know whether Linux md raid
actually supports that many disks. I certainly doubt if it is useful.
It might well be that a 21 data disk limit quad parity is useful - or at
least, as useful as quad parity ever would be. It would fit well within
a typical large chassis with 24 disk slots. And then it doesn't matter
that 8 is not a generator for GF(2⁸) - it becomes the best choice
because of the easiest implementation.
On 20/04/12 05:32, Alex wrote:
I understand we need a generator to facilitate a 255 data disks
array, but 255 sounds like a theoretic limit to me. ZFS now only
supports an array of only 9 disks(6 of them are data disks), so
having, say, a quad-parity array of 48 disks(theoretically) doesn't
sound that bad, does it?
Cheers, Alex
On Fri, Apr 20, 2012 at 11:00 AM, H. Peter Anvin<hpa@xxxxxxxxx>
wrote:
Being a generator is a requirement for that.
Alex<creamyfish@xxxxxxxxx> wrote:
I think when David says 'generator', he doesn't mean the
generator of the order 8 Galois field, he means an arbitrary set
of number in it which can render the system of equations solvable
to up to a certain number of data disks(not necessarily 255). He
uses a brute-force method with the help of a Python program to
actually figure that out. It looks pretty cool to me since I have
known the system of 4 equations generally fails to render a
solution for a while, but now I know exactly how many ways it may
fail...
Cheers, Alex
On Fri, Apr 20, 2012 at 2:16 AM, H. Peter Anvin<hpa@xxxxxxxxx>
wrote:
On 04/17/2012 01:18 PM, David Brown wrote:
For quad parity, we can try g3 = 8 as the obvious next choice
in the pattern. Unfortunately, we start hitting conflicts.
To recover
missing
data, we have to solve multiple simultaneous equations over
G(2⁸),
whose
coefficients depend on the index numbers of the missing
disks. With parity generators (1, 2, 4, 8), some of these
combinations of
missing
disk indexes lead to insoluble equations when you have more
that 21
disks.
That is because 255 = 3*5*17... this means {02}^3 = {08} is not
a
generator.
-hpa
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