Locally Recoverable Codes correcting many erasures over small fields

E Ballico

doi:10.1007/s10623-021-00905-4

Abstract

We define linear codes which are s-Locally Recoverable Codes (or s-LRC), i.e. codes which are LRC in s ways, the case s=1 roughly corresponding to the classical case of LRC codes. We use them to describe codes which correct many erasures, although they have small minimum distance. Any letter of a received word may be corrected using s different local codes. We use the Segre embedding of s local codes and then a linear projection.

Highlights

Cs , and if at least one local code, say Ci , has only a few erasures or errors we may correct them using only Ci (Remark 2.4)
Our codes have not good minimum distance, but they are excellent for erasures, because they have the following additional structure
We stress that the minimum distance of these codes is low, but we find that this type of long codes structured in layers may correct a huge number of erasures

Summary

Introduction

Cs , and if at least one local code, say Ci , has only a few erasures or errors we may correct them using only Ci (Remark 2.4). Theorem 1.2 Fix a prime power q, an integer s ≥ 2 and s linear evaluation codes C1, .

Results

Conclusion