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 Extended Cauchy matrix that gives the first row of 1 for "free". This results in N+M <= 256+1. So, DISKS = 257 - PARITY -> 251 = 257 - 6 A brief introduction of Cauchy and Extended Cauchy matrix can be found in: Vinocha, "On Generator Cauchy Matrices of GDRS/GTRS Codes", 2012 http://www.m-hikari.com/ijcms/ijcms-2012/45-48-2012/brarIJCMS45-48-2012.pdf (just check the Introduction, the rest is not related) More details can be found in: Roth, "Introduction to Coding Theory", 2006 http://carlossicoli.free.fr/R/Roth_R.-Introduction_to_coding_theory-Cambridge_University_Press%282006%29.pdf (search for "Extended Cauchy") Ciao, Andrea On Tue, Nov 19, 2013 at 12:25 AM, H. Peter Anvin <hpa@xxxxxxxxx> 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. >> >> Besides the mathematical proof, I've also inverted all the >> 377,342,351,231 possible submatrices for up to 6 parities and 251 data >> disks, and got an experimental confirmation of this. >> > > Nice. > >> >> The only limit is coming from the GF(2^8). You have a maximum number >> of disk = 2^8 + 1 - number_of_parities. For example, with 6 parities, >> you can have no more of 251 data disks. Over this limit it's not >> possible to build a Cauchy matrix. >> > > 251? Not 255? > >> Note that instead with a Vandermonde matrix you don't have the >> guarantee to always have all the submatrices not singular. This is the >> reason because using power coefficients, before or late, it happens to >> have unsolvable equations. >> >> You can find the code that generate the Cauchy matrix with some >> explanation in the comments at (see the set_cauchy() function) : >> >> http://sourceforge.net/p/snapraid/code/ci/master/tree/mktables.c > > OK, need to read up on the theoretical aspects of this, but it sounds > promising. > > -hpa > > -- To unsubscribe from this list: send the line "unsubscribe linux-raid" in the body of a message to majordomo@xxxxxxxxxxxxxxx More majordomo info at http://vger.kernel.org/majordomo-info.html
