Three New Constructions 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(r,\delta)$ </tex-math></inline-formula> -locality that achieve equality in the Singleton-type bound. We use matrix-product codes to propose two constructions of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -ary optimal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(r,\delta)$ </tex-math></inline-formula> locally repairable codes of lengths up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q^{2}+q$ </tex-math></inline-formula> . The ingredients in the matrix-product codes are linear maximum distance separable codes. We give another construction of optimal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(r,\delta)$ </tex-math></inline-formula> locally repairable codes by using optimal locally repairable codes as ingredients in the matrix-product approach. The codes in this third construction have unbounded lengths not divisible by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(r+\delta -1)$ </tex-math></inline-formula> . The three constructions of optimal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(r,\delta)$ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula> to come up with particular parameters that can suit numerous scenarios.