Error control coding for wireless communication technologies Background material for Hamming codes EU-USA Atlantis Programme FIT & Budapest University of Technology and Economics

Objective Design a code which can correct every single error. Motivation: If the channel is good then it is enough to have a limited error correcting capability.

Probability distribution of the weight of the error vector BSC with n 5 6 7 P(w(e)>1) 0,00001 0,000014 0,00002

Error detection with matching gates syndrome vector s 1 1 1 Identified error vector e 1

Error detection with matching gates (operation) 1 1 e=(010) 1 1 1

Error detection with matching gates (operation) 1 1 1 e=(100) 1 1

Error detection with matching gates (operation) 1 1 e=(001) 1 1 1

The full coding scheme 1 1 1 G H 1 Tr 1

Suggested readings D. Costello: Error control codes, Wiley, 2005, Chapter 3

Expected Quiz question Given a generator matrix of a linear binary systematic code, determine the error group belonging to a given error vector Give the parity check matrix of the C(15,11) Hamming code

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