We continue the program of constructing cubic interactions between matter and higher spin supermultiplets. In this work we consider a complex linear superfield and we find that it can have cubic interactions only with supermultiplets with propagating spins $j=s+1$, $j=s+1/2$ for any non-negative integer $s$ (half-integer superspin supermultiplets). We construct the higher spin supercurrent and supertrace, these compose the canonical supercurrent multiplet which generates the cubic interactions. We also prove that for every $s$ there exist an alternative minimal supercurrent multiplet, with vanishing supertrace. Furthermore, we perform a duality transformation in order to make contact with the corresponding chiral theory. An interesting result is that the dual chiral theory has the same coupling constant with the complex linear theory only for odd values of $s$, whereas for even values of $s$ the coupling constants for the two theories have opposite signs. Additionally we explore the component structure of the supercurrent multiplet and derive the higher spin currents. We find two bosonic currents for spins $j=s$ and $j=s+1$ and one fermionic current for spin $j=s+1/2$.
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