irreducible and of the expected dimension $\beta_g(d, k)$, see [@T1]. The first case not governed by classical Brill-Noether theory is $k=4$, and we note that $$\beta_g(d, 4)=4d-4g-11.$$ It is natural to ask whether in this case too, the Brill-Noether number, determines the non-emptiness of $\mathcal{BN}_C(d, 4)$. Teixidor [@T2] has provided almost optimal answers to this question, and we summarize her results for a general curve $[C]\in \cM_g$: $$\mathcal{BN}_C(d, 4)\neq \emptyset, \ \ \mbox{ provided that  } d\geq \begin{cases}
2a+3, \text{ if } g=2a \Leftrightarrow \beta_g(d, 4)\geq 1;\\
2a+5, \text{ if } g=2a+1 \Leftrightarrow \beta_g(d, 4)\geq 5.\\
 \end{cases}$$ This leaves the case $g=2a+1$ and $d=2a+4$, as the only remaining possibility when $\beta_g(d, 4)\geq 0$. We prove the following result:

\[existence\] For a general curve $[C]\in \cM_{2a+1}$, the Brill-Noether locus $\mathcal{BN}_C(2a+4, 4)$ is non-empty and has at least one component of dimension $2$.

Note that since $\beta_g(d, 4)=1$, unlike in the case $k\leq 3$, the Brill-Noether number no longer predicts the dimension of $\mathcal{BN}_C(d, 4)$. This is a phenomenon which propagates beyond control as $k$ grows, and appears for the first time when $k=4$. This result combined with [@T2], settles the existence problem for bundles of rank $2$ with $4$ sections:

For 