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 ReedSolomon 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 tradeoff between redundancy and a notion of locality called the tdisjointrepairgroup property than previously known constructions. More precisely, we show that lifted multiplicity codes with length N and redundancy O(t0.585 sqrtN) 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 tradeoff for this problem for any superconstant t u003c sqrtN. We also give an alternative analysis of lifted Reed Solomon codes using dual codes, which may be of independent interest.

New stabilizer codes from the construction of dualcontaining matrixproduct codes Abstract   It is known from the CSS code construction that an          m  ,  2  k  xe2x88x92  m  ,  xe2x89xa5  d        q      stabilizer code can be obtained from a (Euclidean) dualcontaining        m  ,  k  ,  d      q      code. In  5 , Blackmore and Norton introduced an interesting code called matrixproduct code, which is very useful in constructing new quantum codes of large lengths. Recently, Galindo et al.  16  constructed several classes of stabilizer codes from the dualcontaining matrixproduct codes of (generalized) ReedMuller, hyperbolic and affine variety ones. In this paper, we first provide a more general approach to construct dualcontaining matrixproduct codes and then further study it in two cases. The first case generalizes the result by Galindo et al. and constructs dualcontaining matrixproduct codes more explicitly since the matrices involved are not restricted to be orthogonal. The second case presents a different way to construct dualcontaining matrixproduct codes in which some of the constituent codes are not required to be dualcontaining. Through the construction of dualcontaining matrixproduct codes of ReedMuller and affine variety ones, the CSS code construction and Steaneu0027s enlargement, we supply several classes of new stabilizer codes over the fields      F    5     ,      F    7      and      F    9      either having minimum distances larger than the ones achieved from the first case or the technique in  16 , or having lengths that are not studied in  16 .