Therefore,we also refer to the Tanner graph of a code as the Tanner graph of its parity-check matrix H. [Cycle and Girth:A cycle of length I in a Tanner graph is a path comprised of edges which closes back on itself.The minimum cycle length of the graph is called the girth of the graph. Clearly,the shortest possible cycle in a bipartite graph is a length-4 cycle,which manifest thems elves in the H matrix as four I's that lie an th corners of a submatrix of H. Figure 4.8.2 As stated in Section 4.7,the cycles,particularly short cycles will degrade the perfor rmance of the iterative decoding algo e for LDPC cod The cycles that affec the decoding performance the most are the cycles of length 4.Therefore.in constructing LDPC codes.cycles of length 4 must be avoided. If the parity-check matrix H of a binary linear block code satisfies the RC-constraint,then wocoddcan e checked simultanously by two parity-check sums.As a result the girth of at least 6.Since we Fan LDPC cod regularor,satisfies the RC-constraint the Tanner graph of an LDPC code has a girth of at least 6. Example 4.8.2:The Tanner graph of the (7,3.3)LDPC code given in Example 1 is shown in Fig.4.8.3. 4 Therefore, we also refer to the Tanner graph of a code as the Tanner graph of its parity-check matrix H. Definition 4.8.4 [Cycle and Girth]: A cycle of length l in a Tanner graph is a path comprised of l edges which closes back on itself. The minimum cycle length of the graph is called the girth of the graph. Clearly, the shortest possible cycle in a bipartite graph is a length-4 cycle, which manifest themselves in the H matrix as four 1’s that lie an the corners of a submatrix of H. . 1 1 . 1 1 . ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ H Figure 4.8.2 As stated in Section 4.7, the cycles, particularly short cycles will degrade the performance of the iterative decoding algorithm used for LDPC codes. The cycles that affect the decoding performance the most are the cycles of length 4. Therefore, in constructing LDPC codes, cycles of length 4 must be avoided. If the parity-check matrix H of a binary linear block code satisfies the RC-constraint, then no two coded bits can be checked simultaneously by two parity-check sums. As a result, the Tanner graph of the code is free of cycles of length 4 and has a girth of at least 6. Since we require that the matrix H of an LDPC code, regular or irregular, satisfies the RC-constraint, the Tanner graph of an LDPC code has a girth of at least 6. „ Example 4.8.2: The Tanner graph of the (7, 3, 3) LDPC code given in Example 1 is shown in Fig. 4.8.3