Showing papers by "Swastik Kopparty published in 2023"

Improved List Decoding of Folded Reed-Solomon and Multiplicity Codes

Abstract: We show new and improved list decoding properties of folded Reed-Solomon (RS) codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the sources of recent advances in coding theory: Folded RS codes were the first known explicit construction of capacity-achieving list decodable codes (Guruswami and Rudra, IEEE Trans. Information Theory , 2010), and multiplicity codes were the first construction of high-rate locally decodable codes (Kopparty, Saraf, and Yekhanin, J. ACM , 2014). In this work, we show that folded RS codes and multiplicity codes are in fact better than was previously known in the context of list decoding and local list decoding. Our first main result shows that folded RS codes achieve list decoding capacity with constant list sizes, independent of the block length. Prior work with constant list sizes first obtained list sizes that are polynomial in the block length, and relied on pre-encoding with subspace evasive sets to reduce the list sizes to a constant (Guruswami and Wang, IEEE Trans. Information Theory , 2012; Dvir and Lovett, STOC , 2012). The list size we obtain is (1 /ε ) O (1 /ε ) where ε is the gap to capacity, which matches the list size obtained by pre-encoding with subspace evasive sets. For our second main result, we observe that univariate multiplicity codes exhibit similar behavior, and use this, together with additional ideas, to show that multivariate multiplicity codes are locally list decodable up to their minimum distance . By known reduc-tions, this gives in turn capacity-achieving locally list decodable codes with query complexity exp( ˜ O ((log N ) 5 / 6 )). This improves on the tensor-based construction of (Hemenway, Ron-Zewi, and Wootters, SICOMP , 2019), which gave capacity-achieving locally list decodable codes of query complexity

Extracting Mergers and Projections of Partitions

Abstract: We study the problem of extracting randomness from somewhere-random sources, and related combinatorial phenomena: partition analogues of Shearer's lemma on projections. A somewhere-random source is a tuple $(X_1, \ldots, X_t)$ of (possibly correlated) $\{0,1\}^n$-valued random variables $X_i$ where for some unknown $i \in [t]$, $X_i$ is guaranteed to be uniformly distributed. An $extracting$ $merger$ is a seeded device that takes a somewhere-random source as input and outputs nearly uniform random bits. We study the seed-length needed for extracting mergers with constant $t$ and constant error. We show: $\cdot$ Just like in the case of standard extractors, seedless extracting mergers with even just one output bit do not exist. $\cdot$ Unlike the case of standard extractors, it $is$ possible to have extracting mergers that output a constant number of bits using only constant seed. Furthermore, a random choice of merger does not work for this purpose! $\cdot$ Nevertheless, just like in the case of standard extractors, an extracting merger which gets most of the entropy out (namely, having $\Omega$ $(n)$ output bits) must have $\Omega$ $(\log n)$ seed. This is the main technical result of our work, and is proved by a second-moment strengthening of the graph-theoretic approach of Radhakrishnan and Ta-Shma to extractors. In contrast, seed-length/output-length tradeoffs for condensing mergers (where the output is only required to have high min-entropy), can be fully explained by using standard condensers. Inspired by such considerations, we also formulate a new and basic class of problems in combinatorics: partition analogues of Shearer's lemma. We show basic results in this direction; in particular, we prove that in any partition of the $3$-dimensional cube $[0,1]^3$ into two parts, one of the parts has an axis parallel $2$-dimensional projection of area at least $3/4$.