In recent years the explosion in the volumes of data being stored online has resulted in distributed storage systems transitioning to erasure coding based schemes. Local Reconstruction Codes (LRCs) have emerged as the codes of choice for these applications. An $(n,r,h,a,q)$-LRC is a $q$-ary code, where encoding is as a ... more >>>

A square matrix $V$ is called rigid if every matrix $V^\prime$ obtained by altering a small number of entries of $V$ has sufficiently high rank. While random matrices are rigid with high probability, no explicit constructions of rigid matrices are known to date. Obtaining such explicit matrices would have major ... more >>>

Consider a systematic linear code where some (local) parity symbols depend on few prescribed symbols, while other (heavy) parity symbols may depend on all data symbols. Local parities allow to quickly recover any single symbol when it is erased, while heavy parities provide tolerance to a large number of simultaneous ... more >>>

Consider a linear $[n,k,d]_q$ code $\mc{C}.$ We say that that $i$-th coordinate of $\mc{C}$ has locality $r,$ if the value at this coordinate can be recovered from accessing some other $r$ coordinates of $\mc{C}.$ Data storage applications require codes with small
redundancy, low locality for information coordinates, large distance, and ... more >>>

Let $f\in F_q[x]$ be a polynomial of degree $d\leq q/2.$ It is well-known that $f$ can be uniquely recovered from its values at some $2d$ points even after some small fraction of the values are corrupted. In this paper we establish a similar result for sparse polynomials. We show that ... more >>>

Locally decodable codes are error-correcting codes that admit efficient decoding algorithms; any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword. The tradeoff between the rate of a code and the locality/efficiency of its decoding algorithms has been ... more >>>

An $(r,\delta,\epsilon)$-locally decodable code encodes a $k$-bit message $x$ to an $N$-bit codeword $C(x),$ such that for every $i\in [k],$ the $i$-th message bit can be recovered with probability $1-\epsilon,$ by a randomized decoding procedure that reads only $r$ bits, even if the codeword $C(x)$ is corrupted in up to ... more >>>

The Nearest Codeword Problem (NCP) is a basic algorithmic question in the theory of error-correcting codes. Given a point v in an n-dimensional space over F_2 and a linear subspace L in F_2^n of dimension k NCP asks to find a point l in L that minimizes the (Hamming) distance ... more >>>

A k-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that one can probabilistically recover any bit x_i of the message by querying only k bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. The major ... more >>>

A q-query Locally Decodable Code (LDC) encodes an n-bit message
x as an N-bit codeword C(x), such that one can
probabilistically recover any bit x_i of the message
by querying only q bits of the codeword C(x), even after
some constant fraction of codeword bits has been corrupted.

We give ... more >>>

A two server private information retrieval (PIR) scheme
allows a user U to retrieve the i-th bit of an
n-bit string x replicated between two servers while each
server individually learns no information about i. The main
parameter of interest in a PIR scheme is its communication
complexity, namely the ... more >>>

A t-private private information retrieval (PIR) scheme allows a user to retrieve the i-th bit of an n-bit string x replicated among k servers, while any coalition of up to t servers learns no information about i. We present a new geometric approach to PIR, and obtain (1) A t-private ... more >>>