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Using a finite field approach, a novel algebraic construction of low-density parity-check (LDPC) convolutional codes with fast encoding property is proposed. According to the matrices of quasi-cyclic (QC) codes constructed based on the multiplicative groups of finite fields and the algebraic property that a binary circulant matrix is isomorphic to a finite ring, we first generate a polynomial-form parity-check matrix of an LDPC convolutional code under a given rate over a given finite field. Then some related modifications are made upon the original polynomial-form matrix to obtain the new one with fast encoding property. Simulation results show that the proposed LDPC convolutional codes have good performance with the iterative belief propagation decoding algorithm.