It focuses the systematic LRCs where each group have one local parity
blocks. Especially, the global parities have hamming weight equals to k.
Structure Theorem. For any systematic LRCS with information locality
(locality for data blocks r) and code distance d, prove the lower bound
of number of parity blocks n - k, where n >= k + ceil(k/n) + d - 2.
basic Pyramid codes matches the lower bound.
I didn’t go into the details of the proof.
Canonical codes
Data blocks are divisible by r, where k | r, and each is assigned
with a local parity block. Global parity blocks have hamming weight k.
Example: Pyramid codes
Theorem: let d < r + 3. Any systematic code with locality r is
systematic code. The physical meaning is that, when code distance is
less than r + 3, which means that the local group size is sufficiently
large, then the code is canonical and thus optimal in distance.
Provides an lower bound and upper bound for the locality of parity blocks
Non-canonical codes
Existance proof of all-symbol locality, given the sufficiently large
field size