Constructions Of Optimal Cyclic Locally Repairable Codes

In order to enable the distributed storage system to tolerate multiple node failures,Prakash et al.introduced the(r,?)-locally repairable code(LRC)in[20],which is a generalization of the concept of r-locally repairable code that Gopalan et al.introduced in[6].In[28],Tamo et al.made the complete defining set be the form of an union of its two parts,one part controls the minimum distance,the other explains the repair locality,and constructed a family of optimal cyclic r-locally repairable code over (?)_qwith code length n satisfying(r+1)|n and n|(q-1).Chen et al.generalized this way to the case of(r,?)-LRC and constructed several classes of optimal cyclic(r,?)-LRCs over F_qof lengths n satisfying(r+?-1)|n and n|(q-1)or n|(q+1)[3].In this thesis we first derive a generic description of the(r,?)-locality of a cyclic code via decomposing its complete defining set.Specifically,if the complete defining set of a cyclic code contains the product of two sets of n-th roots of unity,then the cyclic code has the(r,?)-locality,where r and?are fully determined by the two sets.After that we proceed to give a different way of constructing optimal cyclic(r,?)-LRCs over F_qby setting complete defining set as the product of two sets of n-th roots of unity,where n satisfies n|(q-1)or n|(q+1).Our constructs include all optimal cyclic(r,?)-LRCs proposed in[3,4]and in a more intuitive way,especially when r k.Moreover,our construction style provides a way of finding optimal cyclic(r,?)-LRCs such that(r+?-1)(?)n and enable us to construct optimal cyclic(r,?)-LRCs with Betti-Sala bound.