It'll probably work out: improved list-decoding through random operations

In this work, we introduce a framework to study the effect of random operations on the combinatorial list-decodability of a code. The operations we consider correspond to row and column operations on the matrix obtained from the code by stacking the codewords together as columns. This captures many natural transformations on codes, such as puncturing, folding, and taking subcodes; we show that many such operations can improve the list-decoding properties of a code. There are two main points to this. First, our goal is to advance our (combinatorial) understanding of list-decodability, by understanding what structure (or lack thereof) is necessary to obtain it. Second, we use our more general results to obtain a few interesting corollaries for list decoding: (1) We show the existence of binary codes that are combinatorially list-decodable from $1/2-\epsilon$ fraction of errors with optimal rate $\Omega(\epsilon^2)$ that can be encoded in linear time. (2) We show that any code with $\Omega(1)$ relative distance, when randomly folded, is combinatorially list-decodable $1-\epsilon$ fraction of errors with high probability. This formalizes the intuition for why the folding operation has been successful in obtaining codes with optimal list decoding parameters; previously, all arguments used algebraic methods and worked only with specific codes. (3) We show that any code which is list-decodable with suboptimal list sizes has many subcodes which have near-optimal list sizes, while retaining the error correcting capabilities of the original code. This generalizes recent results where subspace evasive sets have been used to reduce list sizes of codes that achieve list decoding capacity

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

Unconstraining Graph-Constrained Group Testing

In network tomography, one goal is to identify a small set of failed links in a network using as little information as possible. One way of setting up this problem is called graph-constrained group testing. Graph-constrained group testing is a variant of the classical combinatorial group testing problem, where the tests that one is allowed are additionally constrained by a graph. In this case, the graph is given by the underlying network topology. The main contribution of this work is to show that for most graphs, the constraints imposed by the graph are no constraint at all. That is, the number of tests required to identify the failed links in graph-constrained group testing is near-optimal even for the corresponding group testing problem with no graph constraints. Our approach is based on a simple randomized construction of tests. To analyze our construction, we prove new results about the size of giant components in randomly sparsified graphs. Finally, we provide empirical results which suggest that our connected-subgraph tests perform better not just in theory but also in practice, and in particular perform better on a real-world network topology