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 $G{H}_a(n)$ 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 $1\le n\le 100$ and $1\le k\le 16$, there are at most $k$ consecutive integers for which $G{H}_{-(4+k)}(n)$ 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.