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The codegree Tur\\'an density $\\pi_{\\mathrm{co}}(F)$ is the supremum over all $\\gamma \\in [0,1]$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $\\gamma n$ edges. In this paper, we prove that there is a linear $k$-graph $F$ with $0<\\pi_{co}(F) < \\varepsilon$ for any $\\varepsilon>0$. 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