Lifted Multiplicity Codes and the Disjoint Repair Group Property

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 <italic>lifted multiplicity codes</italic>. 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 <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-disjoint-repair-group property than previously known constructions. As a corollary, they also give better tradeoffs for PIR codes in the same parameter regimes. More precisely, we show that, for <inline-formula> <tex-math notation="LaTeX">$t\le \sqrt {N}$ </tex-math></inline-formula>, lifted multiplicity codes with length <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> and redundancy <inline-formula> <tex-math notation="LaTeX">$O(t^{0.585} \sqrt {N})$ </tex-math></inline-formula> have the property that any symbol of a codeword can be reconstructed in <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$t < \sqrt {N}$ </tex-math></inline-formula>. We also give an alternative analysis of lifted Reed-Solomon codes using dual codes, which may be of independent interest.