Title:Three New Infinite Families of Optimal Locally Repairable Codes from Matrix-Product Codes

Abstract:Locally repairable codes have become a key instrument in large-scale distributed storage systems. This paper focuses on the construction of locally repairable codes with $(r,\delta)$-locality that achieve the equality in the Singleton-type bound. We use matrix-product codes to propose two infinite families of $q$-ary optimal $(r,\delta)$ locally repairable codes of lengths up to $q^2+q$. The ingredients in the matrix-product codes are either linear maximum distance separable codes or optimal locally repairable codes of small lengths. Further analysis and refinement yield a construction of another infinite family of optimal $(r,\delta)$ locally repairable codes. The codes in this third family have unbounded lengths not divisible by $(r+\delta-1)$.
The three families of optimal $(r,\delta)$ locally repairable codes constructed here are new. Previously constructed codes in the literature have not covered the same sets of parameters. Our construction proposals are flexible since one can easily vary $r$ and $\delta$ to come up with particular parameters that can suit numerous scenarios.