# The Gopala-Hemachandra universal code determined by straight lines

Volume 19, Issue 3, pp 158--170 Publication Date: May 17, 2019       Article History
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### Authors

Joydeb Pal - Department of Mathematics, National Institute of Technology Durgapur, Burdwan, W.B., Pin-713209, India. Monojit Das - Shibpur Dinobundhoo Institution (College), Howrah, W.B., Pin-711102, India.

### Abstract

Variation on the Fibonacci universal code, known as Gopala-Hemachandra (or GH) code, is mainly used in data compression and cryptography as it is a self-synchronizing code. In 2010, Basu and Prasad showed that Gopala-Hemachandra code $GH_a(n)$ exists for $-20 \leq a \leq -2$ and $1 \leq n \leq 100$ as well as there are $m$ consecutive non-existing Gopala-Hemachandra codewords in $GH_{-(4+m)}(n)$ column where $1 \leq m \leq 16$. In this paper, we have introduced GH code straight line in two-dimensional space where each integral point $(a, n)$ on the GH code straight line represents a unique GH codeword. GH code straight lines confirm the existence of GH codewords for any integer $n \geq 1$ and integer $a \leq -2$. Moreover, for a given parameter $(a, n)$, we have introduced two methods to check whether GH codeword exists or not.

### Keywords

• Fibonacci numbers
• Fibonacci coding
• Gopala-Hemachandra sequence
• Gopala-Hemachandra code
• Zeckendorf's representation

•  94B25
•  11B39
•  11T71

### References

• [1] M. Basu, B. Prasad, Long range variations on the Fibonacci universal code, J. Number Theory, 130 (2010), 1925--1931

• [2] D. E. Daykin, Representation of natural numbers as sums of generalized Fibonacci numbers, J. Lond. Math. Soc., 35 (1960), 143--160

• [3] P. Elias, Universal codeword sets and representations of the integers, IEEE Trans. Information Theory, 21 (1975), 194--203

• [4] S. Kak, A ristotle and Gautama on logic and physics, arXiv, 2005 (2005), 16 pages

• [5] S. Kak, Greek and Indian cosmology: Review of early history, arXiv, 2005 (2005), 38 pages

• [6] D. Knuth, Negafibonacci Numbers and the Hyperbolic Plane, Paper presented at the annual meeting of the Mathematical Association of America, San Jose (2008)

• [7] I. G. Pearce, Indian mathematics: Redressing the balance, University of St. Andrews, U. K. (2002)

• [8] J. H. Thomas, Variation on the Fibonacci universal code, arXiv, 2007 (2007), 4 pages

• [9] E. Zeckendorf, Representation des nombres naturels par une somme des nombres de Fibonacci ou de nombres de Lucas (French), Bull. Soc. Roy. Sci. Liege, 41 (1972), 179--182