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a₁ alphabet assume average probability B₁ binary matrix binary sequences binary symmetric channel channel capacity check digits code word code-word length column converges corrector corresponding coset cyclic code decision scheme decoding set define discrete memoryless channel e-tuple eigenvalue equality equations error pattern example finite follows function Gaussian channel given Hamming bound hence information source input distribution input sequence instantaneous code L₂ Lemma linear linearly independent log p(y Markov chain Markov source memoryless channel minimal polynomial modulo n-sequences nonzero P{X₁ P₁ parity check code parity check matrix positive integer probability of error Problem prove r₁ random variables received sequence result S₁ S₂ sequence of codes sequences of length stationary distribution steady state probabilities symbols time-discrete transmission rate transmitted uncertainty uniquely decipherable vector w₁ W₂ words of length X₁ X₂ Y₁ zero