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This paper presents algebraic methods for constructing efficiently encodable and high performance nonbinary quasi-cyclic LDPC codes based on hyperplanes of Euclidean geometries and masking. Codes constructed from these methods perform very well over the AWGN channel. With iterative decoding using a Fast Fourier Transform based sum-product algorithm, they achieve significantly large coding gains over Reed-Solomon codes of the same lengths and rates decoded with either the algebraic hard-decision Berlekamp-Massey algorithm or the algebraic soft-decision KÃ¶tter-Vardy algorithm. Due to their quasi-cyclic structure, these nonbinary LDPC codes on Euclidean geometries can be encoded with simple shift-registers with linear complexity. Structured nonbinary LDPC codes have a great potential to replace Reed-Solomon codes for some applications in either communication systems or storage systems for combating mixed types of noise and interferences.