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Golay codes

# Golay codes - Extended(24 12 Binary Golay Code Encoding and...

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Extended (24, 12) Binary Golay Code: Encoding and Decoding Procedures © Czeslaw Koscielny 2006 Academy of Management in Legnica, Legnica, Poland, Faculty of Computer Science, Wroclaw University of Applied Informatics, Wroclaw, Poland e mail: [email protected] Abstract It has been shown in the worksheet how to implement encoding and decoding of triple error correcting (24, 12) binary Golay code. The worksheet proves that Maple is an excellent (but underestimated) tool for teaching error-correcting codes. 1. Introduction The extended (24, 12) binary Golay code [1] considered in this submission can correct three or fewer errors. Due to the 11 x 11 matrix Bc , having a cyclic structure and being a component of both the generator and the parity check matrices of this code, its decoding procedure is very simple. Therefore, the discussed (24, 12) code was used about 25 years ago in the spacecraft Voyager. As it is known, this spacecraft delivered to the Earth many perfect photographs of Jupiter and Saturn. 2. Generator and Parity Check Matrices of (24, 12) Golay Code Let

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be the 11 x 11 matrix over GF (2), where ,
. The (24, 12) Golay code has the following generator and parity check matrices, correspondingly: , , where I - identity matrix 12 x 12, and . Therefore

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It can be seen that , , 3. Encoding and Decoding of (24, 12) Golay Code As in the case of any linear code, to generate a code vector it suffices to multiply the vector i , containing 12 information symbols i = [ by the G matrix: wherefrom

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The decoding algorithm of the extended Golay code, shown below, consists in determining the error pattern u = v + w , where w denotes the vector received and v the nearest to w code vector. In the content of the algorithm wt(x) denotes the weight of the vector x , (i.e. the number of "ones" contained in x ), i -th row of the matrix B , the word of length 12 with 1 in the i -th position and 0 elsewhere. After determining u we assume that the corrected received vector will be v = w + u . Here are the steps of the algorithm [1]: Step 1. Compute the syndrome Step 2. If u = [ s , 000000000000].
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Golay codes - Extended(24 12 Binary Golay Code Encoding and...

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