On the minimum distance of elliptic curve codes

Computing the minimum distance of a linear code is one of the fundamental problems in algorithmic coding theory. Vardy [14] showed that it is an \np-hard problem for general linear codes. In practice, one often uses codes with additional mathematical structure, such as AG codes. For AG codes of genus $0$ (generalized Reed-Solomon codes), the minimum distance has a simple explicit formula. An interesting result of Cheng [3] says that the minimum distance problem is already \np-hard (under \rp-reduction) for general elliptic curve codes (ECAG codes, or AG codes of genus $1$). In this paper, we show that the minimum distance of ECAG codes also has a simple explicit formula if the evaluation set is suitably large (at least $2/3$ of the group order). Our method is purely combinatorial and based on a new sieving technique from the first two authors [8]. This method also proves a significantly stronger version of the MDS (maximum distance separable) conjecture for ECAG codes.Comment: 13 page

On the hardness of learning sparse parities

This work investigates the hardness of computing sparse solutions to systems of linear equations over F_2. Consider the k-EvenSet problem: given a homogeneous system of linear equations over F_2 on n variables, decide if there exists a nonzero solution of Hamming weight at most k (i.e. a k-sparse solution). While there is a simple O(n^{k/2})-time algorithm for it, establishing fixed parameter intractability for k-EvenSet has been a notorious open problem. Towards this goal, we show that unless k-Clique can be solved in n^{o(k)} time, k-EvenSet has no poly(n)2^{o(sqrt{k})} time algorithm and no polynomial time algorithm when k = (log n)^{2+eta} for any eta > 0. Our work also shows that the non-homogeneous generalization of the problem -- which we call k-VectorSum -- is W[1]-hard on instances where the number of equations is O(k log n), improving on previous reductions which produced Omega(n) equations. We also show that for any constant eps > 0, given a system of O(exp(O(k))log n) linear equations, it is W[1]-hard to decide if there is a k-sparse linear form satisfying all the equations or if every function on at most k-variables (k-junta) satisfies at most (1/2 + eps)-fraction of the equations. In the setting of computational learning, this shows hardness of approximate non-proper learning of k-parities. In a similar vein, we use the hardness of k-EvenSet to show that that for any constant d, unless k-Clique can be solved in n^{o(k)} time there is no poly(m, n)2^{o(sqrt{k}) time algorithm to decide whether a given set of m points in F_2^n satisfies: (i) there exists a non-trivial k-sparse homogeneous linear form evaluating to 0 on all the points, or (ii) any non-trivial degree d polynomial P supported on at most k variables evaluates to zero on approx. Pr_{F_2^n}[P(z) = 0] fraction of the points i.e., P is fooled by the set of points

Inapproximability of Combinatorial Optimization Problems

We survey results on the hardness of approximating combinatorial optimization problems

Distributed PCP Theorems for Hardness of Approximation in P

We present a new distributed model of probabilistically checkable proofs (PCP). A satisfying assignment $x \in \{0,1\}^n$ to a CNF formula $\varphi$ is shared between two parties, where Alice knows $x_1, \dots, x_{n/2}$, Bob knows $x_{n/2+1},\dots,x_n$, and both parties know $\varphi$. The goal is to have Alice and Bob jointly write a PCP that $x$ satisfies $\varphi$, while exchanging little or no information. Unfortunately, this model as-is does not allow for nontrivial query complexity. Instead, we focus on a non-deterministic variant, where the players are helped by Merlin, a third party who knows all of $x$. Using our framework, we obtain, for the first time, PCP-like reductions from the Strong Exponential Time Hypothesis (SETH) to approximation problems in P. In particular, under SETH we show that there are no truly-subquadratic approximation algorithms for Bichromatic Maximum Inner Product over {0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first two problems we obtain nearly-polynomial factors of $2^{(\log n)^{1-o(1)}}$; only $(1+o(1))$-factor lower bounds (under SETH) were known before

Maximum-likelihood decoding of Reed-Solomon Codes is NP-hard

Maximum-likelihood decoding is one of the central algorithmic problems in coding theory. It has been known for over 25 years that maximum-likelihood decoding of general linear codes is NP-hard. Nevertheless, it was so far unknown whether maximum- likelihood decoding remains hard for any specific family of codes with nontrivial algebraic structure. In this paper, we prove that maximum-likelihood decoding is NP-hard for the family of Reed-Solomon codes. We moreover show that maximum-likelihood decoding of Reed-Solomon codes remains hard even with unlimited preprocessing, thereby strengthening a result of Bruck and Naor.Comment: 16 pages, no figure