Rate Optimal Binary Linear Locally Repairable Codes with Small Availability · arXivDesk
1701.02456Jan 10, 2017Shorter version of the paper was presented at ISIT 2017. The updated second version contains a new section on rate upper bounds for binary linear codes with (2,t) and (r,3)-availability
Rate Optimal Binary Linear Locally Repairable Codes with Small Availability
A locally repairable code with availability has the property that every code symbol can be recovered from multiple, disjoint subsets of other symbols of small size. In particular, a code symbol is said to have (r,t)-availability if it can be recovered from t disjoint subsets, each of size at most r. A code with availability is said to be 'rate-optimal', if its rate is maximum among the class of codes with given locality, availability, and alphabet size. This paper focuses on rate-optimal binary, linear codes with small availability, and makes four contributions. First, it establishes tight upper bounds on the rate of binary linear codes with (r,2)
Nearby in the stack
and
(2,3)
availability. Second, it establishes a uniqueness result for binary rate-optimal codes, showing that for certain classes of binary linear codes with
(r,2)
and
(2,3)
-availability, any rate optimal code must be a direct sum of shorter rate optimal codes. Third, it presents novel upper bounds on the rates of binary linear codes with
(2,t)
and
(r,3)
-availability. In particular, the main contribution here is a new method for bounding the number of cosets of the dual of a code with availability, using its covering properties. Finally, it presents a class of locally repairable linear codes associated with convex polyhedra, focusing on the codes associated with the Platonic solids. It demonstrates that these codes are locally repairable with
t=2
, and that the codes associated with (geometric) dual polyhedra are (coding theoretic) duals of each other.