  In error-correcting codes, locality refers to several different ways of
quantifying how easily a small amount of information can be recovered from
encoded data. In this work, we study a notion of locality called the
s-Disjoint-Repair-Group Property (s-DRGP). This notion can interpolate between
two very different settings in coding theory: that of Locally Correctable Codes
(LCCs) when s is large---a very strong guarantee---and Locally Recoverable
Codes (LRCs) when s is small---a relatively weaker guarantee. This motivates
the study of the s-DRGP for intermediate s, which is the focus of our paper. We
construct codes in this parameter regime which have a higher rate than
previously known codes. Our construction is based on a novel variant of the
lifted codes of Guo, Kopparty and Sudan. Beyond the results on the s-DRGP, we
hope that our construction is of independent interest, and will find uses
elsewhere.
