  Let $U_\theta$ be a unital defined in a shift plane of odd order $q^2$, which
are constructed recently by the authors. In particular, when the shift plane is
desarguesian, $U_\theta$ is a special Buekenhout-Metz unital formed by a union
of ovals. We investigate the dimensions of the binary codes derived from
$U_\theta$.
  By using Kloosterman sums, we obtain a new lower bound on the aforementioned
dimensions which improves the result obtained by Leung and Xiang in 2009. In
particular, for $q=3^m$, this new lower bound equals
$\frac{2}{3}(q^3+q^2-2q)-1$ for even $m$ and $\frac{2}{3}(q^3+q^2+q)-1$ for odd
$m$.
