  Lifted Reed Solomon Codes (Guo, Kopparty, Sudan 2013) were introduced in the
context of locally correctable and testable codes. They are multivariate
polynomials whose restriction to any line is a codeword of a Reed-Solomon code.
We consider a generalization of their construction, which we call lifted
multiplicity codes. These are multivariate polynomial codes whose restriction
to any line is a codeword of a multiplicity code (Kopparty, Saraf, Yekhanin
2014). We show that lifted multiplicity codes have a better trade-off between
redundancy and a notion of locality called the $t$-disjoint-repair-group
property than previously known constructions. More precisely, we show that
lifted multiplicity codes with length $N$ and redundancy $O(t^{0.585}
\sqrt{N})$ have the property that any symbol of a codeword can be reconstructed
in $t$ different ways, each using a disjoint subset of the other coordinates.
This gives the best known trade-off for this problem for any super-constant $t
< \sqrt{N}$. We also give an alternative analysis of lifted Reed Solomon codes
using dual codes, which may be of independent interest.
