The Polynomial of Correctable Patterns
We present here some new combinatoric tools to evaluate the performances of a decoding algorithm correcting either errors alone, either errors and erasures simultaneously. We show how the newly defined polynomials of correctable patterns and polynomials of miscorrected patterns can be connected with the correction an miscorrection probabilities in a given symmetric memoryless channel with or without erasures.
We propose a definition of a decoding algorithm that fits to symmetric channels with errors and erasures. In particular we define the closed algorithms: for instance the Berleykamp-Massey algorithm for BCH codes. For this class of algorithm, we are able to compute the polynomials of correctable and miscorrected patterns. Thus we have a short and easily computed formula for the probabilities of correction and miscorrection of any code using a closed decoding algorithm.
KeywordsTransmission Channel Decode Algorithm Weight Enumerator Error Weight Correctable Pattern
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