Notice
1.6. Decoding (A Difficult Problem)
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Descriptif
The process of correctingerrors and obtaining back the message is calleddecoding. In this sequence, we will focus on this process, the decoding. We would like that thedecoder of the received vector, which is the encoding ofthe original message plus a certain vector, is again theoriginal message, for every message and everyreasonable noisy pattern. The basis of decoding is thefollowing principle, called Minimum Distance Decoding.Given a received vector, we look for a codeword thatminimizes the Hamming distance with the received vector One of the most obviousdecoding algorithms is Brute Force. What do we have to do? First of all, enumerate allcodewords until we find one at distance t or lessfrom the received vector. And we return such codeword. The complexity of thisBrute Force is n*q^k since we have to compare thereceived vectors, and all the codewords; recall thatwe have q^k codewords. Thus, largeparameters make any Brute Force method impractical. Let us explain a decodingmethod using parity check matrices. A vector is in the code ifan only if this product is 0.
Intervention
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