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This paper is concerned with threshold performance analysis of implement-oriented low-density parity check (LDPC) ensemble on binary erasure channel and design of its H-matrix structure and encoder hardware architecture. We construct a class of the special LDPC ensemble which has capacity-achieving degree distribution pair for infinity length code over the binary erasure channel (BEC). Its special characteristic different from the other LDPC ensembles is that the coefficient lambda1 of its degree distribution pair equal 1 instead of 0. Its H-matrix is constructed by combining zigzag matrix and partitioned permutation matrices. This paper completed a design from capacity-achieving sequence to algebraic structural H-matrix, and the H-matrix can directly be used to design low complexity recursive encoding circuit.