Channel Coding Reed-Solomon


Kode Reed-Solomon diperkenalkan oleh Irving Reed dan Gus Solomon pada tahun 1960 [1].

Cara Coding untuk Kode Reed-Solomon ada dua cara:

1. Dengan menggunakan polynom generator g(X) pada domain waktu
2. Dengan menggunakan DFT pada domain frekuensi

Satu kode Reed-Solomon (7,3) pada GF(8) yang memiliki kemampuan koreksi 2 posisi error disusun menggunakan polinom generator:

RS(N,K), N=7, K=3
2t=N-K=7-3=4

g(X)=(X-\alpha)(X-\alpha^2)(X-\alpha^3)(X-\alpha^4)
=(X^2-(\alpha+\alpha^2)X+\alpha^3)(X^2-(\alpha^3+\alpha^4)X+\alpha^7)
=(X^2-\alpha^4X+\alpha^3)(X^2-(\alpha+1+\alpha^2+\alpha)X+\alpha^7)
=(X^2-\alpha^4X+\alpha^3)(X^2-\alpha^6X+\alpha^0)
=X^4-(\alpha^4+\alpha^6)X^3+(\alpha^0+\alpha^3+\alpha^9)X^2-(\alpha^4+\alpha^9)X+\alpha^3\cdot\alpha^0
=\alpha^0X^4-(\alpha^2+\alpha+\alpha^2+1)X^3+(1+\alpha+1+\alpha^{10-7})X^2
-(\alpha^2+\alpha+\alpha^{9-7})X+\alpha^3
=\alpha^0X^4-(\alpha+1)X^3+(\alpha+\alpha^3)X^2-\alpha X+\alpha^3
=\alpha^0X^4-\alpha^3X^3+\alpha^0X^2-\alpha^1X+\alpha^3

Karena -\alpha^i=\alpha^i disebabkan pada biner +1 = -1, maka kita dapatkan polynom generator yaitu

=\alpha^0X^4+\alpha^3X^3+\alpha^0X^2+\alpha^1X+\alpha^3

Contoh Decoding Kode Reed-Solomon

Jika polynom yang diterima adalah

r(X)=\alpha^2X^6+\alpha^2X^4+X^3+\alpha^5X^2

Mencari sindrom dengan cara:
Jumlah sindrom adalah N-K = 7-3 = 4 buah sindrom

S_i=r(\alpha^i), i=1,2,3,...,N-K

S_1=r(\alpha^1)=\alpha^2\cdot\alpha^6+\alpha^2\cdot\alpha^4+\alpha^3+\alpha^5\alpha^2
= \alpha^8+\alpha^6+\alpha^3+\alpha^7
=\alpha^{8-7}+\alpha^6+\alpha^3+\alpha^{7-7}
=\alpha+\alpha^2+1+\alpha+1+1
=\alpha^2+1=\alpha^6

S_2=r(\alpha^2)=\alpha^2\cdot\alpha^{2\cdot 6}+\alpha^2\cdot\alpha^{2\cdot 4}+\alpha^{2\cdot 3}+\alpha^5\cdot\alpha^{2\cdot 2}

Blibiografi:

[1] I. S. Reed and G. Solomon, “Polynomial Codes over Certain Finite Fields,”J. Soc. Ind. Appl. Math., 8, pp. 300-304, June 1960.
[2] B. Sklar, Reed-Solomon Code,
[3] Sugihartono, Slide Kuliah Pengkodean Kanal, STEI, ITB, 2011

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