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Patterson algorithm converts a syndrome to a vector of errors. The syndrome of a word <math>c=(c_0,\dots,c_{n-1})</math> is expected to take a form of
: <math>s(x) \equiv \sum_{
Alternative form of a parity-check matrix based on formula for <math>s(x)</math> can be used to produce such syndrome with a simple matrix multiplication.
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If the original codeword was decodable and the <math>e=(e_0,e_1,\dots,e_{n-1})</math> was the error vector, then
: <math>\sigma(x) = \prod_{
Factoring or evaluating all roots of <math>\sigma(x)</math> therefore gives enough information to recover the error vector and fix the errors.
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