Some Constructions And Bounds Of Optimal Locally Repairable Codes

Locally repairable codes（LRCs）are a family of erasure codes that can reduce the cost of data repair in distributed storage systems.It has attracted the attention of many researchers in recent years.In this thesis,we study the LRCs from two perspectives and give some new upper bounds and constructions.In 2020,Chen et al.studied the Singleton-optimal LRCs with d=6,r=2.Explicit constructions of optimal LRCs with code length up to 3（2q-4）were obtained,and some upper bounds on the code length were given.Inspired by them,we prove the existence of optimal LRCs with d=7,r=2 and length at least 3₃^√2q,and obtain some new upper bounds on the length of optimal LRCs.In 2014,Tamo and Barg constructed the well-known RS-like LRCs by polynomial evaluation with the help of good polynomials.We extend their construction for some good polynomials with special forms,such that the code length can be slightly increased.Our main innovation is to use two evaluation polynomials,so that the multiple roots of a good polynomial can also be the evaluation points.And we find that when one of the evaluation polynomials is always a constant polynomial,it no longer needs any evaluation point,which makes it possible that the code length exceeds the field size.Combining it with some known good polynomials,we can obtain Singleton-optimal（r,δ）-LRCs with code length q+δ-1.