Gopala-Hemachandra codes revisited

Abstract

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 ≥ 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 ≤ n ≤ 100 and 1 ≤ k ≤ 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.