  Gopala-Hemachandra codes are a variation of the Fibonacci universal code and
have applications in cryptography and data compression. We show that
$GH_{a}(n)$ codes always exist for $a=-2,-3$ and $-4$ for any integer $n \geq
1$ and hence are universal codes. We develop two new algorithms to determine
whether a GH code exists for a given set of parameters $a$ and $n$. In 2010,
Basu and Prasad showed experimentally that in the range $1 \leq n \leq 100$ and
$1 \leq k \leq 16$, there are at most $k$ consecutive integers for which
$GH_{-(4+k)}(n)$ does not exist. We turn their numerical result into a
mathematical theorem and show that it is valid well beyond the limited range
considered by them.
