LetC be a smooth irreducible projective algebraic curve defined over the complex numbers. The notion of the Clifford index of C was extended a few years ago to semistable bundles of any rank. Recentwork has been focused mainly on the rank-2 Clifford index, although interesting results have also been obtained for the case of rank 3. In this paper we extend this work, obtaining improved lower bounds for the rank-3 Clifford index. This allows the first computations of the rank-3 index in nontrivial cases and examples for which the rank-3 index is greater than the rank-2 index.
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