# Rate Optimal Binary Linear Locally Repairable Codes with Small Availability

## TL;DR

This paper derives tight bounds and structural insights for rate-optimal binary linear locally repairable codes with small availability, and introduces geometric constructions based on polyhedra.

## Contribution

It establishes new upper bounds, proves uniqueness of certain codes, and introduces geometric code constructions related to Platonic solids.

## Key findings

- Tight upper bounds on code rate for (r,2) and (2,3) availability.
- Uniqueness results showing codes are direct sums of shorter codes.
- Construction of codes from convex polyhedra, including Platonic solids.

## Abstract

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 $(r,2)$ 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.

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Source: https://tomesphere.com/paper/1701.02456