Venkata Gandikota

On the NP-hardness of bounded distance decoding of Reed-Solomon codes

Guruswami and Vardy (IEEE Trans. Inf. Theory, 2005) show that given a Reed-Solomon code over a finite field F, of length n and dimension k, and given a target vector v ε Fn, it is NP-hard to decide if there is a codeword that disagrees with v on at most n - k - 1 coordinates. Understanding the complexity of this Bounded Distance Decoding problem as the amount of error in the target decreases is an important open problem in the study of Reed-Solomon codes. In this work, we extend the result of Guruswami and Vardy by proving that it is NP-hard to decide the existence of a codeword that disagrees with v on n - k - 2, and on n - k - 3 coordinates. No other NP-hardness results were known before for an amount of error <; n - k - 1. The core of our proofs is showing the NP-hardness of a parameterized generalization of the Subset-Sum problem to higher degrees (called Moments Subset-Sum) that may be of independent interest.

Nearly optimal sparse group testing

Group testing is the process of pooling arbitrary subsets from a set of n items so as to identify, with a minimal number of disjunctive tests, a “small” subset of d defective items. In “classical” non-adaptive group testing, it is known that when d = o(n^1−δ) for any δ > 0, θ(d log(n)) tests are both information-theoretically necessary, and sufficient to guarantee recovery with high probability. Group testing schemes in the literature meeting this bound require most items to be tested Ω(log(n)) times, and most tests to incorporate Ω(n/d) items. Motivated by physical considerations, we study group testing models in which the testing procedure is constrained to be “sparse”. Specifically, we consider (separately) scenarios in which (a) items are finitely divisible and hence may participate in at most γ tests; and (b) tests are size-constrained to pool no more than ρ items per test. For both scenarios we provide information-theoretic lower bounds on the number of tests required to guarantee high probability recovery. In particular, one of our main results shows that γ-finite divisibility of items forces any group testing algorithm with probability of recovery error at most ϵ to perform at least Ω(γ(n/d)^{(1−2ϵ)/((1+2ϵ)γ)}) tests. Analogously, for ρ-sized constrained tests, we show an information-theoretic lower bound of Ω(n log(n/d)/(ρ log(n/ρd))). In both scenarios we provide both randomized constructions (under both ϵ-error and zero-error reconstruction guarantees) and explicit constructions of computationally efficient group-testing algorithms (under ϵ-error reconstruction guarantees) that require a number of tests that are optimal up to constant factors in some regimes of n, d, γ and ρ. We also investigate the effect of unreliability/noise in test outcomes.

Local testing of lattices

Testing membership in lattices is of practical relevance, with applications to integer programming, error detection in lattice-based communication, and cryptography. In this work, we initiate a sys...

Deciding Orthogonality in Construction-A Lattices

Lattices are discrete mathematical objects with widespread applications to integer programs as well as modern cryptography. A fundamental problem in both domains is the Closest Vector Problem (popularly known as CVP). It is well-known that CVP can be easily solved in lattices that have an orthogonal basis \emph{if} the orthogonal basis is specified. This motivates the orthogonality decision problem: verify whether a given lattice has an orthogonal basis. Surprisingly, the orthogonality decision problem is not known to be either NP-complete or in P. In this paper, we focus on the orthogonality decision problem for a well-known family of lattices, namely Construction-A lattices. These are lattices of the form

Local Testing for Membership in Lattices

C+q\mathbb{Z}^n

NP-Hardness of Reed-Solomon Decoding and the Prouhet-Tarry-Escott Problem

, where

Deciding orthogonality in construction - A lattices

C

Brief Announcement: Relaxed Locally Correctable Codes in Computationally Bounded Channels.

is an error-correcting

Lattice-based Locality Sensitive Hashing is Optimal

q

Relaxed Locally Correctable Codes in Computationally Bounded Channels.

-ary code, and are studied in communication settings. We provide a complete characterization of lattices obtained from binary and ternary codes using Construction-A that have an orthogonal basis. We use this characterization to give an efficient algorithm to solve the orthogonality decision problem. Our algorithm also finds an orthogonal basis if one exists for this family of lattices. We believe that these results could provide a better understanding of the complexity of the orthogonality decision problem for general lattices.