For any desired rate $R \in (0,1)$, and gap to list-decoding capacity $\varepsilon>0$, we give an explicit $n^{O_{\varepsilon}(1)}$ time construction of Reed-Solomon codes over extension fields of size polynomial in the block length $n$ that can be list-decoded from a fraction $1-R-\varepsilon$ of errors with the optim...
Omar Alrabiah, Rohan Goyal, V. Guruswami· 1 citation· ⚡1
Covers in hypergraphs are frequently studied to capture various forms of dependence between hyperedges. For example, even covers--which check if each vertex appears in an even number of hyperedges--have found much success recently in the study of locally decodable codes. Inspired by a recently-emerging line of work on...
Joshua Brakensiek, V. Guruswami, Aaron L. Putterman· 1 citation
A family of constant-rate quantum error-correcting codes for which a set of bare physical qubits can be injected, i.e. fault-tolerantly encoded, into a code block and a code state can be ejected, i.e. fault-tolerantly decoded.
We show that $\bigl(\frac{\log n}{\log\log n}\bigr)$-approximate parameterized $k$-SetCover is W[1]-hard, and has no $n^{o(k/\log k)}$-time algorithms under ETH. This improves upon the previous best factors $\bigl(\frac{\log n}{\log\log n}\bigr)^{1/k}$ in (Lin, 2019) and $(\log n)^{1/\operatorname{poly}(k)}$ in (Karthi...
We derive the four principal asymptotic rate-distance tradeoffs for binary codes---Plotkin, Elias--Bassalygo, and the two McEliece--Rodemich--Rumsey--Welch (MRRW) bounds---from one theorem, the ``pretty good criterion.''If the bit error rate under the pretty good measurement (PGM)---the quantum analog of posterior samp...
We prove that for every prime power $q$ and every $p \in (0, 1-1/q)$, a random $\mathbb{F}_q$-linear code of rate $1 - h_q(p) - \epsilon$ is $(p, C_{p,q}/\epsilon)$-average-radius list-decodable with probability at least $1 - q^{-\Omega(n)}$, i.e., for every center $y \in \mathbb{F}_q^n$, the $C_{p,q}/\epsilon$ codewor...
V. Guruswami, Shi-Lun Li, Mihir Singhal· 2 citations· ⚡1
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