In Theorem 3.37, prove that if p i = p j , then we cannot determine whether

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In Theorem 3.37, prove that if pi= pj, then we cannot determine whether an error occurs in the ith or the jth component of the received vector.

(Throughout this proof we denote by ai the ith column of a matrix A.) With P and G as in the statement of the theorem, assume first that they are standard parity check and generator matrices for the same binary code. Therefore, for every x in Zk2,PGx 0. In terms of block multiplication,

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