This paper proposes a method to derive equivalent coding structures of elements to construct low density parity check (LDPC) codes. We propose stairs LDPC (SLDPC) codes to demonstrate the effectiveness of the proposed method, which is expected to be beneficial for short block-length transmissions, but providing high coding rate. The equivalent coding structures are both for transmitter and receiver to: (i) reduce the encoding and decoding computational complexity, and (ii) search possibility of finding new coding scheme and observe their performances. We evaluate the validity of the method by confirming the equality in performances of the SLDPC codes in terms of bit-error-rate (BER) followed by investigation on their performance gaps to the Shannon limit via a series of computer simulations. The results show that the SLDPC codes have the same BER performance with that of the low density generator matrix (LDGM) codes confirming the validity of the proposed equivalent matrix derivation. This result indicates that different graphs can provide the same performances, because their equivalent matrices are the same. This result is expected to open new insight for the designing simple channel coding for short block-length LDPC codes having high coding rate for future less power consumption applications.

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