Long range variations on the Fibonacci universal code

Abstract

Fibonacci coding is based on Fibonacci numbers and was defined by Apostolico and Fraenkel (1987) [1]. Fibonacci numbers are generated by the recurrence relation F i = F i − 1 + F i − 2 ∀ i ⩾ 2 with initial terms F 0 = 1 , F 1 = 1 . Variations on the Fibonacci coding are used in source coding as well as in cryptography. In this paper, we have extended the table given by Thomas [8]. We have found that there is no Gopala–Hemachandra code for a particular positive integer n and for a particular value of a ∈ Z . We conclude that for n = 1 , 2 , 3 , 4 , Gopala–Hemachandra code exists for a = − 2 , − 3 , … , − 20 . Also, for 1 ⩽ n ⩽ 100 , there is at most m consecutive not available (N/A) Gopala–Hemachandra code in GH − ( 4 + m ) column where 1 ⩽ m ⩽ 16 . And, for 1 ⩽ n ⩽ 100 , as m increases the availability of Gopala–Hemachandra code decreases in GH − ( 4 + m ) column where 1 ⩽ m ⩽ 16 .