The Gopala-Hemachandra universal code determined by straight lines

Volume 19, Issue 3, pp 158--170 http://dx.doi.org/10.22436/jmcs.019.03.03

Publication Date: May 17, 2019 Submission Date: November 23, 2018 Revision Date: February 22, 2019 Accteptance Date: April 19, 2019

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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.

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Keywords

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

MSC

•  94B25
•  11B39
•  11T71

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