### Re: Triple parity and beyond

On 28/11/13 08:16, Stan Hoeppner wrote: Late reply. This one got lost in the flurry of activity... On 11/22/2013 7:24 AM, David Brown wrote: On 22/11/13 09:38, Stan Hoeppner wrote: On 11/21/2013 3:07 AM, David Brown wrote: For example, with 20 disks at 1 TB each, you can have: ...

### Re: Triple parity and beyond

Late reply. This one got lost in the flurry of activity... On 11/22/2013 7:24 AM, David Brown wrote: On 22/11/13 09:38, Stan Hoeppner wrote: On 11/21/2013 3:07 AM, David Brown wrote: For example, with 20 disks at 1 TB each, you can have: ... Maximum: RAID 10 = 10 disk redundancy RAID 15

### Re: Triple parity and beyond

On Thu, 28 Nov 2013, Stan Hoeppner s...@hardwarefreak.com wrote: We must follow different definitions of redundancy. I view redundancy as the number of drives that can fail without taking down the array. In the case of the above 20 drive RAID15 that maximum is clearly 11 drives-- one of

### Re: Triple parity and beyond

On 21/11/2013 21:31, David Brown wrote: On 21/11/13 21:05, Piergiorgio Sartor wrote: Having a multi parity RAID allows to check even which disk. This would provide the user with a more comprehensive (I forgot the spelling) information. Of course, since we are there, we can also give the option

### Re: Triple parity and beyond

This is great stuff. Now, how can we get this into btrfs and md? On Wed, Nov 20, 2013 at 1:23 PM, Andrea Mazzoleni amadva...@gmail.com wrote: Hi, First, create a 3 by 6 cauchy matrix, using x_i = 2^-i, and y_i = 0 for i=0, and y_i = 2^i for other i. In this case: x = { 1, 142, 71, 173,

### Re: Triple parity and beyond

On 24/11/13 22:13, Stan Hoeppner wrote: On 11/23/2013 11:14 PM, John Williams wrote: On Sat, Nov 23, 2013 at 8:03 PM, Stan Hoeppner s...@hardwarefreak.com wrote: Parity array rebuilds are read-modify-write operations. The main difference from normal operation RMWs is that the write is

### Re: Triple parity and beyond

On 25/11/13 03:14, Russell Coker wrote: On Mon, 25 Nov 2013, Stan Hoeppner s...@hardwarefreak.com wrote: If that is the problem then the solution would be to just enable read-ahead. Don't we already have that in both the OS and the disk hardware? The hard- drive read-ahead buffer should at

### Re: Triple parity and beyond

On Fri, Nov 22, 2013 at 07:12:39PM +1100, Russell Coker wrote: On Thu, 21 Nov 2013 18:30:49 Stan Hoeppner wrote: I suggest that anyone in the future needing fast random write IOPS is going to move those workloads to SSD, which is steadily increasing in capacity. And I suggest anyone

### Re: Triple parity and beyond

Hi Piergiorgio, Checking better I found that for PAR3 it was also evaluated a Cauchy matrix, but they preferred to use a RS with FFT in GF(2^16 +1). http://sourceforge.net/mailarchive/forum.php?forum_name=parchive-develmax_rows=25style=nestedviewmonth=201006 Note that using a Cauchy matrix for

### Re: Triple parity and beyond

On 11/23/2013 11:14 PM, John Williams wrote: On Sat, Nov 23, 2013 at 8:03 PM, Stan Hoeppner s...@hardwarefreak.com wrote: Parity array rebuilds are read-modify-write operations. The main difference from normal operation RMWs is that the write is always to the same disk. As long as the

### Re: Triple parity and beyond

On 11/23/2013 11:19 PM, Russell Coker wrote: On Sun, 24 Nov 2013, Stan Hoeppner s...@hardwarefreak.com wrote: I have always surmised that the culprit is rotational latency, because we're not able to get a real sector-by-sector streaming read from each drive. If even only one disk in the array

### Re: Triple parity and beyond

On Sun, Nov 24, 2013 at 1:44 PM, Stan Hoeppner s...@hardwarefreak.com wrote: SNIP Are you suggesting that it would be a common case that people just write data to an array and never read it or do an array scrub? I hope that it will become standard practice to have a cron job scrubbing all

### Re: Triple parity and beyond

On 24-11-13 22:13, Stan Hoeppner wrote: I freely admit I may have drawn an incorrect conclusion about md parity rebuild performance based on incomplete data. I simply don't recall anyone stating here in ~3 years that their parity rebuilds were speedy, but quite the opposite. I guess it's

### Re: Triple parity and beyond

On 11/24/2013 5:53 PM, Alex Elsayed wrote: Stan Hoeppner wrote: On 11/23/2013 11:14 PM, John Williams wrote: On Sat, Nov 23, 2013 at 8:03 PM, Stan Hoeppner s...@hardwarefreak.com wrote: snip But I, and a number of other people I have talked to or corresponded with, have had mdadm RAID 5

### Re: Triple parity and beyond

On Mon, 25 Nov 2013, Stan Hoeppner s...@hardwarefreak.com wrote: If that is the problem then the solution would be to just enable read-ahead. Don't we already have that in both the OS and the disk hardware? The hard- drive read-ahead buffer should at least cover the case where a seek

### Re: Triple parity and beyond

On 22/11/13 23:59, NeilBrown wrote: On Fri, 22 Nov 2013 10:07:09 -0600 Stan Hoeppner s...@hardwarefreak.com wrote: snip In the event of a double drive failure in one mirror, the RAID 1 code will need to be modified in such a way as to allow the RAID 5 code to rebuild the first replacement

### Re: Triple parity and beyond

Hi Andrea, On Sat, Nov 23, 2013 at 08:55:08AM +0100, Andrea Mazzoleni wrote: Hi Piergiorgio, How about par2? How does this work? I checked the matrix they use, and sometimes it contains some singular square submatrix. It seems that in GF(2^16) these cases are just less common. Maybe they

### Re: Triple parity and beyond

On 11/23/2013 1:12 AM, NeilBrown wrote: On Fri, 22 Nov 2013 21:34:41 -0800 John Williams jwilliams4...@gmail.com Even a single 8x PCIe 3.0 card has potentially over 7GB/s of bandwidth. Bottom line is that IO bandwidth is not a problem for a system with prudently chosen hardware. Quite

### Re: Triple parity and beyond

On Sat, Nov 23, 2013 at 8:03 PM, Stan Hoeppner s...@hardwarefreak.com wrote: Parity array rebuilds are read-modify-write operations. The main difference from normal operation RMWs is that the write is always to the same disk. As long as the stripe reads and chunk reconstruction outrun the

### Re: Triple parity and beyond

On Sun, 24 Nov 2013, Stan Hoeppner s...@hardwarefreak.com wrote: I have always surmised that the culprit is rotational latency, because we're not able to get a real sector-by-sector streaming read from each drive. If even only one disk in the array has to wait for the platter to come round

### Re: Triple parity and beyond

On Thu, 21 Nov 2013 20:54:54 Adam Goryachev wrote: On a pure storage server, the CPU would normally have nothing to do, except a little interrupt handling, it is just shuffling bytes around. Of course, if you need RAID7.5 then you probably have a dedicated storage server, so I don't see the

### Re: Triple parity and beyond

Hi David, On 11/21/2013 3:07 AM, David Brown wrote: On 21/11/13 02:28, Stan Hoeppner wrote: ... WRT rebuild times, once drives hit 20TB we're looking at 18 hours just to mirror a drive at full streaming bandwidth, assuming 300MB/s average--and that is probably being kind to the drive makers.

### Re: Triple parity and beyond

On 11/21/2013 3:07 AM, David Brown wrote: For example, with 20 disks at 1 TB each, you can have: All correct, and these are maximum redundancies. Maximum: raid5 = 19TB, 1 disk redundancy raid6 = 18TB, 2 disk redundancy raid6.3 = 17TB, 3 disk redundancy raid6.4 = 16TB, 4 disk redundancy

### Re: Triple parity and beyond

On 11/21/2013 5:38 PM, John Williams wrote: On Thu, Nov 21, 2013 at 2:57 PM, Stan Hoeppner s...@hardwarefreak.com wrote: He wrote that article in late 2009. It seems pretty clear he wasn't looking 10 years forward to 20TB drives, where the minimum mirror rebuild time will be ~18 hours, and

### Re: Triple parity and beyond

On 22/11/13 09:13, Stan Hoeppner wrote: Hi David, On 11/21/2013 3:07 AM, David Brown wrote: On 21/11/13 02:28, Stan Hoeppner wrote: ... WRT rebuild times, once drives hit 20TB we're looking at 18 hours just to mirror a drive at full streaming bandwidth, assuming 300MB/s average--and that

### Re: Triple parity and beyond

On 22/11/13 09:38, Stan Hoeppner wrote: On 11/21/2013 3:07 AM, David Brown wrote: For example, with 20 disks at 1 TB each, you can have: All correct, and these are maximum redundancies. Maximum: raid5 = 19TB, 1 disk redundancy raid6 = 18TB, 2 disk redundancy raid6.3 = 17TB, 3 disk

### Re: Triple parity and beyond

On Fri, Nov 22, 2013 at 1:35 AM, Stan Hoeppner s...@hardwarefreak.com wrote: Only one graph goes to 2019, the rest are 2010 or less. That being the case, his 2019 graph deals with projected reliability of single, double, and triple parity. The whole article goes to 2019 (or longer). He shows

### Re: Triple parity and beyond

On 11/22/2013 2:13 AM, Stan Hoeppner wrote: Hi David, On 11/21/2013 3:07 AM, David Brown wrote: ... I don't see that there needs to be any changes to the existing md code to make raid15 work - it is merely a raid 5 made from a set of raid1 pairs. The sole purpose of the parity layer of

### Re: Triple parity and beyond

On Fri, Nov 22, 2013 at 12:13 AM, Stan Hoeppner s...@hardwarefreak.com wrote: Hi David, On 11/21/2013 3:07 AM, David Brown wrote: SNIP Shouldn't we be talking about RAID 15 here, rather than RAID 51 ? I interpret RAID 15 to be like RAID 10 - a raid5 set of raid1 mirrors, while RAID 51 would

### Re: Triple parity and beyond

Mark Knecht posted on Fri, 22 Nov 2013 08:50:32 -0800 as excerpted: On Fri, Nov 22, 2013 at 12:13 AM, Stan Hoeppner s...@hardwarefreak.com wrote: Now that you mention it, yes, RAID 15 would fit much better with convention. Not sure why I thought 51. So it's RAID 15 from here. SNIP For

### Re: Triple parity and beyond

Hi David, On Fri, Nov 22, 2013 at 01:32:09AM +0100, David Brown wrote: One typical case is when many errors are found, belonging to the same disk. This case clearly shows the disk is to be replaced or the interface checked... But, again, the user is the master, not the machine... :-)

### Re: Triple parity and beyond

On 11/22/2013 9:01 AM, John Williams wrote: snip I see no advantage of RAID 15, and several disadvantages. Of course not, just as I sated previously. On 11/22/2013 2:13 AM, Stan Hoeppner wrote: Parity users who currently shun RAID 10 for this reason will also shun this RAID 15. With that

### Re: Triple parity and beyond

On Fri, 22 Nov 2013 10:07:09 -0600 Stan Hoeppner s...@hardwarefreak.com wrote: On 11/22/2013 2:13 AM, Stan Hoeppner wrote: Hi David, On 11/21/2013 3:07 AM, David Brown wrote: ... I don't see that there needs to be any changes to the existing md code to make raid15 work - it is merely

### Re: Triple parity and beyond

On Thu, 21 Nov 2013 16:57:48 -0600 Stan Hoeppner s...@hardwarefreak.com wrote: On 11/21/2013 1:05 AM, John Williams wrote: On Wed, Nov 20, 2013 at 10:52 PM, Stan Hoeppner s...@hardwarefreak.com wrote: On 11/20/2013 8:46 PM, John Williams wrote: For myself or any machines I managed for

### Re: Triple parity and beyond

On 11/22/2013 5:07 PM, NeilBrown wrote: On Thu, 21 Nov 2013 16:57:48 -0600 Stan Hoeppner s...@hardwarefreak.com wrote: On 11/21/2013 1:05 AM, John Williams wrote: On Wed, Nov 20, 2013 at 10:52 PM, Stan Hoeppner s...@hardwarefreak.com wrote: On 11/20/2013 8:46 PM, John Williams wrote: For

### Re: Triple parity and beyond

On Fri, 22 Nov 2013 21:46:50 -0600 Stan Hoeppner s...@hardwarefreak.com wrote: On 11/22/2013 5:07 PM, NeilBrown wrote: On Thu, 21 Nov 2013 16:57:48 -0600 Stan Hoeppner s...@hardwarefreak.com wrote: On 11/21/2013 1:05 AM, John Williams wrote: On Wed, Nov 20, 2013 at 10:52 PM, Stan

### Re: Triple parity and beyond

On Fri, Nov 22, 2013 at 9:04 PM, NeilBrown ne...@suse.de wrote: I guess with that many drives you could hit PCI bus throughput limits. A 16-lane PCIe 4.0 could just about give 100MB/s to each of 16 devices. So you would really need top-end hardware to keep all of 16 drives busy in a

### Re: Triple parity and beyond

On Fri, 22 Nov 2013 21:34:41 -0800 John Williams jwilliams4...@gmail.com wrote: On Fri, Nov 22, 2013 at 9:04 PM, NeilBrown ne...@suse.de wrote: I guess with that many drives you could hit PCI bus throughput limits. A 16-lane PCIe 4.0 could just about give 100MB/s to each of 16 devices.

### Re: Triple parity and beyond

Hi Piergiorgio, How about par2? How does this work? I checked the matrix they use, and sometimes it contains some singular square submatrix. It seems that in GF(2^16) these cases are just less common. Maybe they were just unnoticed. Anyway, this seems to be an already known problem for PAR2,

### Re: Triple parity and beyond

On 21/11/2013 02:28, Stan Hoeppner wrote: On 11/20/2013 10:16 AM, James Plank wrote: Hi all -- no real comments, except as I mentioned to Ric, my tutorial in FAST last February presents Reed-Solomon coding with Cauchy matrices, and then makes special note of the common pitfall of assuming that

### Re: Triple parity and beyond

On 20/11/13 19:09, John Williams wrote: On Wed, Nov 20, 2013 at 2:31 AM, David Brown david.br...@hesbynett.no wrote: That's certainly a reasonable way to look at it. We should not limit the possibilities for high-end systems because of the limitations of low-end systems that are unlikely to

### Re: Triple parity and beyond

On 20/11/13 19:34, Andrea Mazzoleni wrote: Hi David, The choice of ZFS to use powers of 4 was likely not optimal, because to multiply by 4, it has to do two multiplications by 2. I can agree with that. I didn't copy ZFS's choice here David, it was not my intention to suggest that you

### Re: Triple parity and beyond

On 21/11/13 02:28, Stan Hoeppner wrote: On 11/20/2013 10:16 AM, James Plank wrote: Hi all -- no real comments, except as I mentioned to Ric, my tutorial in FAST last February presents Reed-Solomon coding with Cauchy matrices, and then makes special note of the common pitfall of assuming that

### Re: Triple parity and beyond

On 21/11/13 20:07, David Brown wrote: I can see plenty of reasons why raid15 might be a good idea, and even raid16 for 5 disk redundancy, compared to multi-parity sets. However, it costs a lot in disk space. For example, with 20 disks at 1 TB each, you can have: raid5 = 19TB, 1 disk

### Re: Triple parity and beyond

On 20/11/13 22:59, Piergiorgio Sartor wrote: On Wed, Nov 20, 2013 at 11:44:39AM +0100, David Brown wrote: [...] In RAID-6 (as per raid6check) there is an easy way to verify where an HDD has incorrect data. I think the way to do that is just to generate the parity blocks from the data

### Re: Triple parity and beyond

On 21/11/13 10:54, Adam Goryachev wrote: On 21/11/13 20:07, David Brown wrote: I can see plenty of reasons why raid15 might be a good idea, and even raid16 for 5 disk redundancy, compared to multi-parity sets. However, it costs a lot in disk space. For example, with 20 disks at 1 TB each,

### Re: Triple parity and beyond

Hi David, On Thu, Nov 21, 2013 at 09:31:46PM +0100, David Brown wrote: [...] If this can all be done to give the user an informed choice, then it sounds good. that would be my target. To _offer_ more options to the (advanced) user. It _must_ always be under user control. One issue here is

### Re: Triple parity and beyond

On 11/21/2013 1:05 AM, John Williams wrote: On Wed, Nov 20, 2013 at 10:52 PM, Stan Hoeppner s...@hardwarefreak.com wrote: On 11/20/2013 8:46 PM, John Williams wrote: For myself or any machines I managed for work that do not need high IOPS, I would definitely choose triple- or quad-parity

### Re: Triple parity and beyond

On Thu, Nov 21, 2013 at 11:13:29AM +0100, David Brown wrote: [...] Ah, you are trying to find which disk has incorrect data so that you can change just that one disk? There are dangers with that... Hi David, http://neil.brown.name/blog/20100211050355 I think we already did the exercise,

### Re: Triple parity and beyond

On 11/21/2013 2:08 AM, joystick wrote: On 21/11/2013 02:28, Stan Hoeppner wrote: ... WRT rebuild times, once drives hit 20TB we're looking at 18 hours just to mirror a drive at full streaming bandwidth, assuming 300MB/s average--and that is probably being kind to the drive makers. With 6 or

### Re: Triple parity and beyond

On 21/11/13 21:52, Piergiorgio Sartor wrote: Hi David, On Thu, Nov 21, 2013 at 09:31:46PM +0100, David Brown wrote: [...] If this can all be done to give the user an informed choice, then it sounds good. that would be my target. To _offer_ more options to the (advanced) user. It _must_

### Re: Triple parity and beyond

On 11/21/2013 04:30 PM, Stan Hoeppner wrote: The rebuild time of a parity array normally has little to do with CPU overhead. Unless you have to fall back to table driven code. Anyway, this looks like a great concept. Now we just need to implement it ;) -hpa -- To unsubscribe from

### Re: Triple parity and beyond

On 22/11/13 01:30, Stan Hoeppner wrote: I don't like it either. It's a compromise. But as RAID1/10 will soon be unusable due to URE probability during rebuild, I think it's a relatively good compromise for some users, some workloads. An alternative is to move to 3-way raid1 mirrors rather

### Re: Triple parity and beyond

On Wed, Nov 20, 2013 at 07:28:37PM -0600, Stan Hoeppner wrote: [...] It's always perilous to follow a Ph.D., so I guess I'm feeling suicidal today. ;) I'm not attempting to marginalize Andrea's work here, but I can't help but ponder what the real value of triple parity RAID is, or quad, or

### Re: Triple parity and beyond

On 19/11/13 18:36, Andrea Mazzoleni wrote: Hi David, Just to say that I know your good past work, and it helped me a lot. Thanks for that! If we end up with your Cauchy matrix implementation going into the kernel and btrfs (and you've persuaded me, anyway), then perhaps I should make a new

### Re: Triple parity and beyond

On 20/11/13 02:23, John Williams wrote: On Tue, Nov 19, 2013 at 4:54 PM, Chris Murphy li...@colorremedies.com wrote: If anything, I'd like to see two implementations of RAID 6 dual parity. The existing implementation in the md driver and btrfs could remain the default, but users could opt

### Re: Triple parity and beyond

On 19/11/13 19:12, Piergiorgio Sartor wrote: On Mon, Nov 18, 2013 at 11:08:59PM +0100, Andrea Mazzoleni wrote: snip for brevity Hi Andrea, great job, this was exactly what I was looking for. Do you know if there is a fast way not to correct errors, but to find them? In RAID-6 (as

### Re: Triple parity and beyond

Hi all -- no real comments, except as I mentioned to Ric, my tutorial in FAST last February presents Reed-Solomon coding with Cauchy matrices, and then makes special note of the common pitfall of assuming that you can append a Vandermonde matrix to an identity matrix. Please see

### Re: Triple parity and beyond

On Wed, Nov 20, 2013 at 2:31 AM, David Brown david.br...@hesbynett.no wrote: That's certainly a reasonable way to look at it. We should not limit the possibilities for high-end systems because of the limitations of low-end systems that are unlikely to use 3+ parity anyway. I've also looked

### Re: Triple parity and beyond

Hi David, The choice of ZFS to use powers of 4 was likely not optimal, because to multiply by 4, it has to do two multiplications by 2. I can agree with that. I didn't copy ZFS's choice here David, it was not my intention to suggest that you copied from ZFS. Sorry to have expressed myself

### Re: Triple parity and beyond

It is also possible to quickly multiply by 2^-1 which makes for an interesting R parity. Andrea Mazzoleni amadva...@gmail.com wrote: Hi David, The choice of ZFS to use powers of 4 was likely not optimal, because to multiply by 4, it has to do two multiplications by 2. I can agree with that.

### Re: Triple parity and beyond

Hi John, Yes. There are still AMD CPUs sold without SSSE3. Most notably Athlon. Instead, Intel is providing SSSE3 from the Core 2 Duo. A detailed list is available at: http://en.wikipedia.org/wiki/SSSE3 Ciao, Andrea On Wed, Nov 20, 2013 at 7:09 PM, John Williams jwilliams4...@gmail.com wrote:

### Re: Triple parity and beyond

Hi, Yep. At present to multiply for 2^-1 I'm using in C: static inline uint64_t d2_64(uint64_t v) { uint64_t mask = v 0x0101010101010101U; mask = (mask 8) - mask; v = (v 1) 0x7f7f7f7f7f7f7f7fU; v ^= mask 0x8e8e8e8e8e8e8e8eU; return v; } and for SSE2:

### Re: Triple parity and beyond

On 11/20/2013 10:56 AM, Andrea Mazzoleni wrote: Hi, Yep. At present to multiply for 2^-1 I'm using in C: static inline uint64_t d2_64(uint64_t v) { uint64_t mask = v 0x0101010101010101U; mask = (mask 8) - mask; v = (v 1) 0x7f7f7f7f7f7f7f7fU; v ^=

### Re: Triple parity and beyond

On 11/20/2013 10:56 AM, Andrea Mazzoleni wrote: Hi, Yep. At present to multiply for 2^-1 I'm using in C: static inline uint64_t d2_64(uint64_t v) { uint64_t mask = v 0x0101010101010101U; mask = (mask 8) - mask; (mask 7) I assume... -- To unsubscribe from this list:

### Re: Triple parity and beyond

Hi Jim, I build the matrix in a way that results in coefficients matching Linux RAID for the first two rows, and at the same time gives the guarantee that all the square submatrices are not singular, resulting in a MDS code. I start forming a Cauchy matrix setting each element to 1/(xi+yj) where

### Re: Triple parity and beyond

On 11/20/2013 11:05 AM, Andrea Mazzoleni wrote: For the first row with j=0, I use xi = 2^-i and y0 = 0, that results in: How can xi = 2^-i if x is supposed to be constant? That doesn't mean that your approach isn't valid, of course, but it might not be a Cauchy matrix and thus needs

### Re: Triple parity and beyond

Peter, I think I understand it differently. Concrete example in GF(256) for k=6, m=4: First, create a 3 by 6 cauchy matrix, using x_i = 2^-i, and y_i = 0 for i=0, and y_i = 2^i for other i. In this case: x = { 1, 142, 71, 173, 216, 108 } y = { 0, 2, 4). The cauchy matrix is: 1 2 4

### Re: Triple parity and beyond

Hi Peter, static inline uint64_t d2_64(uint64_t v) { uint64_t mask = v 0x0101010101010101U; mask = (mask 8) - mask; (mask 7) I assume... No. It's (mask 8) - mask. We want to expand the bit at position 0 (in each byte) to the full byte, resulting in 0xFF if the bit is at

### Re: Triple parity and beyond

On 11/20/2013 01:04 PM, Andrea Mazzoleni wrote: Hi Peter, static inline uint64_t d2_64(uint64_t v) { uint64_t mask = v 0x0101010101010101U; mask = (mask 8) - mask; (mask 7) I assume... No. It's (mask 8) - mask. We want to expand the bit at position 0 (in each byte)

### Re: Triple parity and beyond

Hi Peter, Now, that doesn't sound like something that can get neatly meshed into the Cauchy matrix scheme, I assume. You are correct. Multiplication by 2^-1 cannot be used for the Cauchy method. I used it to implement an alternate triple parity not requiring PSHUFB that I used as reference for

### Re: Triple parity and beyond

Hi, First, create a 3 by 6 cauchy matrix, using x_i = 2^-i, and y_i = 0 for i=0, and y_i = 2^i for other i. In this case: x = { 1, 142, 71, 173, 216, 108 } y = { 0, 2, 4). The cauchy matrix is: 1 2 4 8 16 32 244 83 78 183 118 47 167 39 213 59 153 82 Divide row 2 by

### Re: Triple parity and beyond

On 11/20/2013 12:30 PM, James Plank wrote: Peter, I think I understand it differently. Concrete example in GF(256) for k=6, m=4: First, create a 3 by 6 cauchy matrix, using x_i = 2^-i, and y_i = 0 for i=0, and y_i = 2^i for other i. In this case: x = { 1, 142, 71, 173, 216, 108 } y

### Re: Triple parity and beyond

Hi Piergiorgio, In RAID-6 (as per raid6check) there is an easy way to verify where an HDD has incorrect data. I suspect, for each 2 parity block it should be possible to find 1 error (and if this is true, then quad parity is more attractive than triple one). Yes. The theory say that with

### Re: Triple parity and beyond

On Wed, Nov 20, 2013 at 11:44:39AM +0100, David Brown wrote: [...] In RAID-6 (as per raid6check) there is an easy way to verify where an HDD has incorrect data. I think the way to do that is just to generate the parity blocks from the data blocks, and compare them to the existing parity

### Re: Triple parity and beyond

On Mon, Nov 18, 2013 at 11:08:59PM +0100, Andrea Mazzoleni wrote: [...] I've a side question, a bit OT, but maybe you could help with the answer. How about par2? How does this work? They claim Vendermonde matrix and they seem to be quite flexible in amount of parities. The could be in GF(2^16),

### Re: Triple parity and beyond

For myself or any machines I managed for work that do not need high IOPS, I would definitely choose triple- or quad-parity over RAID 51 or similar schemes with arrays of 16 - 32 drives. No need to go into detail here on a subject Adam Leventhal has already covered in detail in an article

### Re: Triple parity and beyond

On 11/20/2013 12:44 PM, Andrea Mazzoleni wrote: Yes. There are still AMD CPUs sold without SSSE3. Most notably Athlon. Instead, Intel is providing SSSE3 from the Core 2 Duo. I hate branding discontinuity, due to the resulting confusion... Athlon, Athlon64, Athlon64 X2, Athlon X2 (K10), Athlon

### Re: Triple parity and beyond

On Wed, Nov 20, 2013 at 10:52 PM, Stan Hoeppner s...@hardwarefreak.com wrote: On 11/20/2013 8:46 PM, John Williams wrote: For myself or any machines I managed for work that do not need high IOPS, I would definitely choose triple- or quad-parity over RAID 51 or similar schemes with arrays of 16

### Re: Triple parity and beyond

On 19/11/13 00:25, H. Peter Anvin wrote: On 11/18/2013 02:35 PM, Andrea Mazzoleni wrote: Hi Peter, The Cauchy matrix has the mathematical property to always have itself and all submatrices not singular. So, we are sure that we can always solve the equations to recover the data disks.

### Re: Triple parity and beyond

Hi Peter, Yes, 251 disks for 6 parity. To build a NxM Cauchy matrix you need to pick N+M distinct values in the GF(2^8) and we have only 2^8 == 256 available. This means that for every row we add for an extra parity level, we have to remove one of the disk columns. Note that in true, I use an

### Re: Triple parity and beyond

Hi David, Just to say that I know your good past work, and it helped me a lot. Thanks for that! Unfortunately the Cauchy matrix is not compatible with a triple parity implementation using power coefficients. They are different and incompatible roads. I partially agree on your considerations,

### Re: Triple parity and beyond

On Mon, Nov 18, 2013 at 11:08:59PM +0100, Andrea Mazzoleni wrote: Hi, I want to report that I recently implemented a support for arbitrary number of parities that could be useful also for Linux RAID and Btrfs, both currently limited to double parity. In short, to generate the parity I use

### Re: Triple parity and beyond

On 11/19/2013 12:28 PM, Andrea Mazzoleni wrote: Hi Peter, Yes, 251 disks for 6 parity. To build a NxM Cauchy matrix you need to pick N+M distinct values in the GF(2^8) and we have only 2^8 == 256 available. This means that for every row we add for an extra parity level, we have to remove one

### Re: Triple parity and beyond

I'm not going to claim any expert status on this discussion (the theory makes my head spin) but I will say I agree with Andrea as far as prefering his implementation for triple parity and beyond. PSHUFB has been around the intel platform since the Core2 introduced it as part of SSSE3 back in Q1

### Re: Triple parity and beyond

On Nov 19, 2013, at 3:51 PM, Drew drew@gmail.com wrote: I'm not going to claim any expert status on this discussion (the theory makes my head spin) but I will say I agree with Andrea as far as prefering his implementation for triple parity and beyond. PSHUFB has been around the intel

### Re: Triple parity and beyond

On Tue, Nov 19, 2013 at 4:54 PM, Chris Murphy li...@colorremedies.com wrote: If anything, I'd like to see two implementations of RAID 6 dual parity. The existing implementation in the md driver and btrfs could remain the default, but users could opt into Cauchy matrix based dual parity which

### Re: Triple parity and beyond

On 11/18/2013 02:08 PM, Andrea Mazzoleni wrote: Hi, I want to report that I recently implemented a support for arbitrary number of parities that could be useful also for Linux RAID and Btrfs, both currently limited to double parity. In short, to generate the parity I use a Cauchy matrix

### Re: Triple parity and beyond

Hi Peter, The Cauchy matrix has the mathematical property to always have itself and all submatrices not singular. So, we are sure that we can always solve the equations to recover the data disks. Besides the mathematical proof, I've also inverted all the 377,342,351,231 possible submatrices for

### Re: Triple parity and beyond

On 11/18/2013 02:35 PM, Andrea Mazzoleni wrote: Hi Peter, The Cauchy matrix has the mathematical property to always have itself and all submatrices not singular. So, we are sure that we can always solve the equations to recover the data disks. Besides the mathematical proof, I've also