In Theorem 3.37, prove that if p_i= p_j, 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 Z^k₂,PGx 0. In terms of block multiplication,

[B 1] x = 0 for all x in Z A