Gopala-Hemachandra codes revisited

04/02/2020

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by   L. Childers, et al.

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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.