Within the framework of six-dimensional ${cal N}=(1,0)$ conformal supergravity, we introduce new off-shell multiplets ${cal O}{}^{*}(n)$, where $n=3,4,dots,$ and use them to construct higher-rank extensions of the linear multiplet action. The ${cal O}{}^{*}(n)$ multiplets may be viewed as being dual to well-known superconformal ${cal O}(n)$ multiplets. We provide prepotential formulations for the ${cal O}(n)$ and ${cal O}{}^{*}(n)$ multiplets coupled to conformal supergravity. For every ${cal O}{}^{*}(n)$ multiplet, we construct a higher derivative invariant which is superconformal on arbitrary superconformally flat backgrounds. We also show how our results can be used to construct new higher derivative actions in supergravity.
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