The Parameterized Inapproximability Hypothesis (PIH), which is an analog of the PCP theorem in parameterized complexity, asserts that, there is a constant $\varepsilon> 0$ such that for any computable function $f:\mathbb{N}\to\mathbb{N}$, no $f(k)\cdot n^{O(1)}$-time algorithm can, on input a $k$-variable CSP instance with domain size $n$, find an assignment satisfying ... more >>>

We prove that a binary linear code of block length $n$ that is locally correctable with $3$ queries against a fraction $\delta > 0$ of adversarial errors must have dimension at most $O_{\delta}(\log^2 n \cdot \log \log n)$. This is almost tight in view of quadratic Reed-Muller codes being a ... more >>>

The Parameterized Inapproximability Hypothesis (PIH) asserts that no fixed parameter tractable (FPT) algorithm can distinguish a satisfiable CSP instance, parameterized by the number of variables, from one where every assignment fails to satisfy an $\varepsilon$ fraction of constraints for some absolute constant $\varepsilon > 0$. PIH plays the role of ... more >>>

Classical locally recoverable codes, which permit highly efficient recovery from localized errors as well as global recovery from larger errors, provide some of the most useful codes for distributed data storage in practice. In this paper, we initiate the study of quantum locally recoverable codes (qLRCs). In the long ... more >>>

The Parameterized Inapproximability Hypothesis (PIH) is the analog of the PCP theorem in the world of parameterized complexity. It asserts that no FPT algorithm can distinguish a satisfiable 2CSP instance from one which is only $(1-\varepsilon)$-satisfiable (where the parameter is the number of variables) for some constant $0<\varepsilon<1$.

We ... more >>>

We prove that the Minimum Distance Problem (MDP) on linear codes over any fixed finite field and parameterized by the input distance bound is W[1]-hard to approximate within any constant factor. We also prove analogous results for the parameterized Shortest Vector Problem (SVP) on integer lattices. Specifically, we prove that ... more >>>

In the range avoidance problem, the input is a multi-output Boolean circuit with more outputs than inputs, and the goal is to find a string outside its range (which is guaranteed to exist). We show that well-known explicit construction questions such as finding binary linear codes achieving the Gilbert-Varshamov bound ... more >>>

A code $C \colon \{0,1\}^k \to \{0,1\}^n$ is a $q$-locally decodable code ($q$-LDC) if one can recover any chosen bit $b_i$ of the message $b \in \{0,1\}^k$ with good confidence by randomly querying the encoding $x = C(b)$ on at most $q$ coordinates. Existing constructions of $2$-LDCs achieve $n = ... more >>>