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Based on the construction of linear codes proposed by Niederriter and Xing (see Applicable Algebra in Engineering, Communication and Computing, vol.10, p.425-32, 2000), a corresponding generator matrix is derived. The duality operator and the propagation operator of these extended linear codes are studied. A sufficient and necessary condition under which these two operators are commutative is obtained. This commutative property is found useful for constructing the corresponding parity-check matrix and, thus, the error detection/recovery codes.