Locally repairable codes and implied parity
For instance if OSD X3 was lost, OSDs X1, X2, X4 to X10 and P1 to P4 are retrieved by the primary OSD and the erasure code plugin uses them to rebuild the content of X3.
Locally repairable codes are designed to lower the bandwidth requirements when recovering from the loss of a single OSD. A local parity block is calculated for each five blocks : S1 and S2. When the X3 OSD is lost, instead of retrieving blocks from 13 OSDs, it is enough to retrieve X1, X2, X4, X5 and S1, that is 5 OSDs.
In some cases, local parity blocks can help recover from the loss of more blocks than any individual encoding function can. In the example above, let say five blocks are lost: X1, X2, X3, X4 and X8. The block X8 can be recovered from X6, X7, X9, X10 and S2. Now that only four blocks are missing, the initial parity blocks are enough to recover. The combined effect of local parity blocks and the global parity blocks acts as if there was implied parity block.