View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Computers and Mathematics with Applications 60 (2010) 2738–2752 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa A Fibonacci-polynomial based coding method with error detection and correction Mostafa Esmaeili a, Morteza Esmaeili b,c,∗ a Department of Electrical and Computer Engineering, Isfahan University of Technology, 84156-83111, Isfahan, Iran b Department of Mathematical Sciences, Isfahan University of Technology, 84156-83111, Isfahan, Iran c Department of Electrical and Computer Engineering, University of Victoria, Victoria, B.C., Canada V8W 3P6 article info a b s t r a c t Article history: A Fibonacci coding method using Fibonacci polynomials is introduced. For integers m ≥ 2, Received 20 January 2010 ≥ ≥ × n x 1 and n 1, an m m matrix Qm.x/, the nth power of Qm.x/, is considered as the Received in revised form 7 August 2010 encoding matrix, where Qm is an m × m matrix whose elements are Fibonacci polynomials. Accepted 30 August 2010 −n The decoding matrix Qm .x/ is also introduced. A simple error-detecting criterion and a simple error-correcting method for this class of codes are given. It is shown that the Keywords: probability of decoding error is almost zero for m large enough. Illustrative examples are Fibonacci polynomial provided. Fibonacci-polynomial matrices Error detecting and error correcting codes ' 2010 Elsevier Ltd. All rights reserved. 1. Introduction The Fibonacci sequence and the golden ratio have appeared in many fields of science including high energy physics, cryptography and coding [1–9]. The Fibonacci sequence is defined by the recurrence relation fn D fn−1 C fn−2; n ≥ 3, with initial values f1 D f2 D 1. This sequence has been extended in many ways. Two such extensions that will be used in this paper are the p-Fibonacci sequences [10] and the Fibonacci polynomials [11]. The p-Fibonacci sequence, Fp.n/ is defined by the following recurrence relation. Fp.n/ D Fp.n − 1/ C Fp.n − p − 1/; n > p C 1; p ≥ 0; Fp.1/ D Fp.2/ D Fp.3/ D···D Fp.p C 1/ D 1: Parts of the p-Fibonacci sequences for p D 1; 2; 3; 4 are listed in the table below. p D 1 1, 1, 2, 3, 5, 8, 11, ::: p D 2 1, 1, 1, 2, 3, 4, 6, 9, ::: p D 3 1, 1, 1, 1, 2, 3, 4, 5, 7, ::: p D 4 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, ::: The p-Fibonacci sequences have been used to construct coding and decoding matrices [12–14]. The Qp matrix of order p C 1 is introduced in [12]. For 0 ≤ p ≤ 3, the matrix Qp is given below. 01 1 0 01 1 1 0! 1 1 0 0 1 0 Q D .0/; Q D ; Q D 0 0 1 ; Q D B C : 0 1 1 0 2 3 @0 0 0 1A 1 0 0 1 0 0 0 ∗ Corresponding author at: Department of Mathematical Sciences, Isfahan University of Technology, 84156-83111, Isfahan, Iran. E-mail address: [email protected] (M. Esmaeili). 0898-1221/$ – see front matter ' 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2010.08.091 M. Esmaeili, M. Esmaeili / Computers and Mathematics with Applications 60 (2010) 2738–2752 2739 ≥ ≥ n It is shown in [12] that for n 1 and p 0, the nth power of Qp, denoted Qp , is 0 Fp.n C 1/ Fp.n/ ··· Fp.n − p C 2/ Fp.n − p C 1/ 1 − C − ··· − C − C BFp.n p 1/ Fp.n p/ Fp.n 2p 2/ Fp.n 2p 1/C n B : : : : C Q D B : : : : C p B : : : : C @ Fp.n − 1/ Fp.n − 2/ ··· Fp.n − p/ Fp.n − p − 1/ A F n F n − 1 ··· F n − p C 1 F n − p p. / p. / p. / p. / .pC1/×.pC1/ wherein Fp.n/ is the nth p-Fibonacci number. These matrices have been used in [12] to give a coding and decoding technique referred to as Fibonacci coding theory. Another extension of the Fibonacci sequence is the Fibonacci polynomial which is defined by the recurrence relation (1; n D 1I Fn.x/ D x; n D 2I (1) xFn−1.x/ C Fn−2.x/; n ≥ 3: A non-recursive expression for Fn.x/, given below, is introduced in [11]. b n c 2 − X n i n−2i FnC1.x/ D x ; n ≥ 0: i iD0 This expression will be used frequently throughout this paper. The first five Fibonacci polynomials are shown below. 81 n D 1I > >x n D 2I < 2 Fn.x/ D x C 1 n D 3I > 3 C D I >x 2x n 4 :x4 C 3x2 C 1 n D 5: There is no restriction on the Fibonacci polynomials for n ≤ 0. In this paper we set F0.x/ D 0 and Fn.x/ D 1 for n ≤ −1. An important property of Fibonacci polynomials, proved in [11], is p 2 FnC1.x/ x C x C 4 σ VD lim D : !1 n Fn.x/ 2 It is worth noting that by substituting x D 1 in the Fibonacci polynomials and p D 1 in the p-Fibonacci sequence the classical Fibonacci sequence is obtained. In the rest of the paper, for simplicity, we denote Fn.x/ by Fn. n In this paper square matrices Qm.x/ consisting of Fibonacci-polynomial entries are introduced. It is shown that these matrices are practical from the coding theory perspective. The approach presented is considered a Fibonacci-polynomial n based coding method. Obviously, this is an extension of the results in [12] based on the Qp matrices. By substituting a positive integer for x, an infinite number of matrices of order m ≥ 2 are obtained. In Sections 2 and 3 the general properties of the Fibonacci-polynomial matrices and the inverse Fibonacci-polynomial matrices, that is the encoding and decoding matrices, are given. In Section 4 the Fibonacci-polynomial based coding and decoding method is given. An interesting relation among the elements of a code-message matrix is derived in Section 5. The error-detection and error-correction methods are discussed in Section 6. The computational complexity of the decoding process is addressed in Section 7. Some conclusions are given in Section 8. 2. Fibonacci-polynomial matrices of order m Consider the 2 × 2 matrix defined below. x 1 Q .x/ D : 2 1 0 For any x, we have det.Q2.x// D −1. Setting F0 D 0 and applying induction on n ≥ 1, it is easily verified that n D FnC1 Fn Q2 .x/ : Fn Fn−1 n D − n × By using the determinant theorem, we see that det.Q2 .x// . 1/ . The following defines the m m matrix Qm.x/. Thus Qm.x/ has a recursive expression. 0x 1 0 0 ··· 01 ··· B0 x 1 0 0C 0x 1 0 01 B ··· C B0 0 x 1 0C 0 x 1 0 Qm.x/ VD B: : : : : :C Q4.x/ D B C : B: : :: :: :: :C @0 0 x 1A B: : :C @0 0 ··· 0 x 1A 0 0 1 0 0 0 ··· 0 1 0 m×m 2740 M. Esmaeili, M. Esmaeili / Computers and Mathematics with Applications 60 (2010) 2738–2752 m−2 One can easily see that det.Qm.x// D −x . The nth, n ≥ 2, power of Qm.x/ is given by the following theorem. Theorem 1. For n ≥ 2 and m ≥ 2, we have n Qm.x/ j k j k 0 n−mC2 n−mC1 1 2 2 n n − n − C X n − i − C − X n − 1 − i − C − B xn xn 1 ··· xn m 3 xn m 2 2i xn m 1 2i C B 0 1 m − 3 i C m − 2 i C m − 2 C B iD0 iD0 C B j n−mC3 k j n−mC2 k C B 2 2 C B n n n − i n − 1 − i C B n n−mC4 X n−mC3−2i X n−mC2−2i C B 0 x ··· x x x C B 0 m − 4 i C m − 3 i C m − 3 C B iD0 iD0 C B C B : :: :: : : : C B : : : : : : C B C B C D j nC1 k B b n c C : (2) B 2 − 2 − − C B n n X n i nC1−2i X n 1 i n−2i C B 0 0 ··· x x x C B 0 i C 1 i C 1 C B iD0 iD0 C B j − k C B b n c n 1 C 2 2 B X n − i − X n − 1 − i − − C B 0 0 ··· 0 xn 2i xn 1 2i C B i i C B iD0 iD0 C B C B j n−1 k j n−2 k C B 2 2 C @ X n − 1 − i − − X n − 2 − i − − A 0 0 ··· 0 xn 1 2i xn 2 2i i i iD0 iD0 Proof. For simplicity the theorem is proved for m D 4. The same argument applies to the case m 6D 4. The proof is by induction on n. The following equality shows that (2) holds for n D 1. 0 −1 −1 1 1 1 X 1 − i − − X −i − − x x 1 2i x 2 2i 0 1 i C 3 i C 2 B iD0 iD0 C B C 0 1 B 0 − −1 − C x 1 0 0 B 1 X 1 i 2i X i −1−2i C 1 D B 0 x x x C D B0 x 1 0C Q4 .x/ B 0 i C 1 i C 1 C : B iD0 iD0 C @0 0 x 1A B C B C 0 0 1 0 B 0 0 F2.x/ F1.x/ C @ A 0 0 F1.x/ F0.x/ Suppose the statement holds for n D k.