This paper is concerned with the systematic Bernoulli generator matrix~(BGM) codes, which have been proved to be capacity-achieving over binary-input output-symmetric~(BIOS) channels in terms of bit-error rate~(BER). We prove that the systematic BGM codes are also capacity-achieving over BIOS channels in terms of frame-error rate (FER). To this end, we present a new framework to prove the coding theorems for binary linear codes. Different from the widely-accepted approach via ensemble enlargement, the proof directly applies to the systematic binary linear codes. The new proof indicates that the pair-wise independence condition is not necessary for proving the binary linear code ensemble to achieve the capacity of the BIOS channel. The Bernoulli parity-check~(BPC) codes, which fall within the framework of the systematic BGM codes with parity-check bits known at the decoder can also be proved to achieve the capacity. The presented framework also reveals a new mechanism pertained to the systematic linear codes that the systematic bits and the corresponding parity-check bits play different roles. Precisely, the noisy systematic bits are used to limit the list size of candidate codewords, while the noisy parity-check bits are used to select from the list the maximum likelihood codeword. For systematic BGM codes with finite length, we derive the lower bounds on the BER and FER, which can be used to predict the error floors. Numerical results show that the systematic BGM codes match well with the derived error floors. The performance in water-fall region can be improved with approaches in statistical physics and the error floors can be significantly improved by implementing the concatenated codes with the systematic BGM codes as the inner codes.