Gopala-Hemachandra codes revisited

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.