50 research outputs found
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 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 <=sqrt{N}, 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
On taking advantage of multiple requests in error correcting codes
In most notions of locality in error correcting codes -- notably locally
recoverable codes (LRCs) and locally decodable codes (LDCs) -- a decoder seeks
to learn a single symbol of a message while looking at only a few symbols of
the corresponding codeword. However, suppose that one wants to recover r > 1
symbols of the message. The two extremes are repeating the single-query
algorithm r times (this is the intuition behind LRCs with availability,
primitive multiset batch codes, and PIR codes) or simply running a global
decoding algorithm to recover the whole thing. In this paper, we investigate
what can happen in between these two extremes: at what value of r does
repetition stop being a good idea? In order to begin to study this question we
introduce robust batch codes, which seek to find r symbols of the message using
m queries to the codeword, in the presence of erasures. We focus on the case
where r = m, which can be seen as a generalization of the MDS property.
Surprisingly, we show that for this notion of locality, repetition is optimal
even up to very large values of
Locality via Partially Lifted Codes
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
Batch Codes from Hamming and Reed-Muller Codes
Batch codes, introduced by Ishai \textit{et al.}, encode a string into an -tuple of strings, called buckets. In this paper we consider multiset batch codes wherein a set of -users wish to access one bit of information each from the original string. We introduce a concept of optimal batch codes. We first show that binary Hamming codes are optimal batch codes. The main body of this work provides batch properties of Reed-Muller codes. We look at locality and availability properties of first order Reed-Muller codes over any finite field. We then show that binary first order Reed-Muller codes are optimal batch codes when the number of users is 4 and generalize our study to the family of binary Reed-Muller codes which have order less than half their length
Study of Optimal Linear Batch Codes
Lineaarseid partiikoode saab kasutada koormuse ühtlustamiseks hajusandmetalletussüsteemides. Selleks, et tagada efektiivne sooritusvõime, on tarvis optimeeritud parameetritega koode. Selliste koodide leidmine on aga keerukas matemaatiline probleem. Selles töös esitatakse algoritme ja tarkvara, mille abil on võimalik uurida lineaarseid partiikoode. Tuletatakse kaks uut ülemtõket lineaarsetele partiikoodidele. Lõpuks võrreldakse tarkvara abil leitud lühimaid süstemaatiliste lineaarsete partiikoodide pikkuseid seni teadaolevate tõketega.Linear batch codes can be used for load balancing in distributed storage systems. In order to obtain efficient performance, it is important to have codes with optimized parameters, which is a complicated mathematical problem.Specifically, in this thesis, algorithms and software for searching for linear batch codes are presented. Two upper bounds for systematic linear batch codes are derived. The shortest lengths of systematic linear batch codes, which have been found with the help of the software, are compared to known upper and lower bounds