Lifted Multiplicity Codes and the Disjoint Repair Group Property

Abstract

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, for t ≤ √ N , lifted multiplicity codes with length N and redundancy O(t 0.585 √ 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 < √ N. We also give an alternative analysis of lifted Reed Solomon codes using dual codes, which may be of independent interest.

Lifted Reed-Solomon Codes and Lifted Multiplicity Codes

“…In this work we continue the study of lifted RS codes and lifted multiplicity codes by generalizing the results on the bivariate case of [11], [16] to an arbitrary number of variables. Since lifted RS codes represent a specific class of lifted multiplicity codes, when derivatives are not taken into account, we focus on the description of lifted multiplicity codes in the following.…”

“…Since lifted RS codes represent a specific class of lifted multiplicity codes, when derivatives are not taken into account, we focus on the description of lifted multiplicity codes in the following. Essentially, we investigate the same class of codes as defined in [11], [19]. Informally, the [m, s, d, q] lifted multiplicity code consists of the evaluation (together with the derivatives up to the sth order) of polynomials from F q [X 1 , .…”

“…These properties are of interest in a variety of applications, such as load balancing in distributed data storage, cryptography, and low-complexity error correction/detection. Several different notions related to these parameters have been considered in the literature, including, but not limited to, locally recoverable codes (LRCs) [3], [4], locally decodable/correctable codes (LDCs/LCCs) [5], [6], relaxed LCCs [7] and LDCs [8], batch codes [9], PIR codes [10], and codes with the disjoint repair group property (DRGP) [11].…”

“…As both lifted RS codes and multiplicity codes are based on generalizations of RM codes, it is a natural question whether these techniques can be combined to further improve the parameters of the respective codes. Some progress in the study of these lifted multiplicity codes has recently been made in [11], [19]. In [19], the authors show asymptotic results for any number of variables.…”

“…In [19], the authors show asymptotic results for any number of variables. Paper [11] is devoted to improving the existing bounds on the required redundancy in the bi-variate case.…”