Research Article
Design of Nonbinary LDPC Cycle Codes with Large Girth from Circulants and Finite Fields
Table 2
The nonzero field elements in the parity-check matrix of the proposed 64-ary
LDPC cycle code in Example
1.
| Row index | Nonzero field elements | Row index | Nonzero field elements | Row index | Nonzero field elements |
| 1 | 37 | 9 | 22 | 0 | 2 | 7 | 0 | 18 | 44 | 3 | 0 | 37 | 9 | 19 | 4 | 0 | 37 | 19 | 9 | 5 | 7 | 18 | 44 | 0 | 6 | 7 | 18 | 0 | 44 | 7 | 7 | 18 | 44 | 0 | 8 | 0 | 18 | 44 | 7 | 9 | 19 | 9 | 0 | 37 | 10 | 0 | 18 | 44 | 7 | 11 | 0 | 9 | 37 | 22 | 12 | 18 | 44 | 7 | 0 | 13 | 9 | 19 | 37 | 0 | 14 | 18 | 0 | 7 | 44 | 15 | 22 | 37 | 0 | 9 | 16 | 0 | 44 | 7 | 18 | 17 | 9 | 19 | 0 | 37 | 18 | 37 | 22 | 0 | 9 | 19 | 44 | 18 | 7 | 0 | 20 | 9 | 37 | 22 | 0 | 21 | 19 | 37 | 9 | 0 | 22 | 0 | 22 | 37 | 9 | 23 | 44 | 7 | 0 | 18 | 24 | 0 | 7 | 44 | 18 | 25 | 9 | 37 | 0 | 22 | 26 | 19 | 9 | 37 | 0 | 27 | 22 | 37 | 0 | 9 | 28 | 9 | 22 | 37 | 0 | 29 | 9 | 37 | 22 | 0 | 30 | 37 | 9 | 22 | 0 | 31 | 37 | 9 | 22 | 0 | 32 | 22 | 9 | 37 | 0 | 33 | 7 | 18 | 44 | 0 | 34 | 37 | 0 | 9 | 22 | 35 | 0 | 37 | 22 | 9 | 36 | 7 | 44 | 0 | 18 | 37 | 0 | 44 | 7 | 18 | 38 | 9 | 0 | 37 | 19 | 39 | 37 | 9 | 22 | 0 | 40 | 0 | 9 | 19 | 37 | 41 | 44 | 7 | 18 | 0 | 42 | 37 | 19 | 9 | 0 | 43 | 9 | 37 | 0 | 19 | 44 | 37 | 9 | 22 | 0 | 45 | 0 | 9 | 19 | 37 | 46 | 19 | 9 | 37 | 0 | 47 | 0 | 19 | 37 | 9 | 48 | 18 | 0 | 44 | 7 | 49 | 0 | 19 | 9 | 37 | 50 | 22 | 37 | 9 | 0 | 51 | 37 | 0 | 9 | 22 | 52 | 18 | 7 | 0 | 44 | 53 | 0 | 9 | 37 | 22 | 54 | 44 | 7 | 0 | 18 | 55 | 7 | 0 | 44 | 18 | 56 | 9 | 0 | 37 | 22 | 57 | 9 | 19 | 37 | 0 | 58 | 22 | 9 | 37 | 0 | 59 | 9 | 19 | 37 | 0 | 60 | 0 | 44 | 18 | 7 | 61 | 18 | 7 | 0 | 44 | 62 | 9 | 37 | 0 | 19 | 63 | 22 | 0 | 9 | 37 | 64 | 7 | 0 | 18 | 44 | 65 | 18 | 44 | 0 | 7 | 66 | 19 | 37 | 9 | 0 | 67 | 9 | 37 | 22 | 0 | 68 | 18 | 7 | 0 | 44 | 69 | 0 | 9 | 22 | 37 | 70 | 37 | 22 | 0 | 9 | 71 | 18 | 7 | 44 | 0 | 72 | 7 | 0 | 18 | 44 | 73 | 0 | 22 | 37 | 9 | 74 | 7 | 0 | 44 | 18 | 75 | 9 | 0 | 19 | 37 | 76 | 37 | 0 | 9 | 19 | 77 | 9 | 37 | 0 | 22 | 78 | 0 | 37 | 9 | 22 | 79 | 37 | 9 | 0 | 22 | 70 | 9 | 0 | 37 | 19 | 81 | 7 | 44 | 18 | 0 | 82 | 0 | 44 | 18 | 7 | 83 | 9 | 37 | 22 | 0 | 84 | 44 | 0 | 7 | 18 | 85 | 9 | 22 | 37 | 0 | 86 | 9 | 19 | 37 | 0 | 87 | 22 | 9 | 0 | 37 | 88 | 0 | 18 | 7 | 44 | 89 | 9 | 37 | 22 | 0 | 90 | 18 | 0 | 7 | 44 | 91 | 37 | 22 | 9 | 0 | 92 | 0 | 22 | 37 | 9 | 93 | 37 | 0 | 9 | 22 | 94 | 9 | 22 | 37 | 0 | 95 | 44 | 0 | 7 | 18 | 96 | 0 | 9 | 19 | 37 | 97 | 7 | 44 | 18 | 0 | 98 | 9 | 19 | 37 | 0 | 99 | 7 | 44 | 18 | 0 | 100 | 37 | 22 | 0 | 9 | 101 | 0 | 22 | 9 | 37 | 102 | 22 | 0 | 37 | 9 | 103 | 44 | 0 | 7 | 18 | 104 | 22 | 9 | 0 | 37 | 105 | 22 | 0 | 37 | 9 | 106 | 44 | 7 | 18 | 0 | 107 | 37 | 19 | 0 | 9 | 108 | 0 | 9 | 37 | 22 | 109 | 9 | 37 | 19 | 0 | 110 | 44 | 18 | 0 | 7 | 111 | 37 | 22 | 9 | 0 | 112 | 7 | 0 | 18 | 44 | 113 | 0 | 18 | 44 | 7 | 114 | 37 | 22 | 0 | 9 | 115 | 0 | 9 | 37 | 19 | 116 | 7 | 18 | 0 | 44 | 117 | 9 | 0 | 22 | 37 | 118 | 0 | 37 | 19 | 9 | 119 | 0 | 9 | 22 | 37 | 120 | 19 | 37 | 9 | 0 | 121 | 37 | 0 | 22 | 9 | 122 | 37 | 0 | 9 | 22 | 123 | 0 | 7 | 44 | 18 | 124 | 19 | 37 | 0 | 9 | 125 | 7 | 44 | 18 | 0 | 126 | 7 | 44 | 0 | 18 | 127 | 9 | 37 | 22 | 0 | 128 | 44 | 18 | 0 | 7 | 129 | 7 | 18 | 0 | 44 | 130 | 0 | 37 | 22 | 9 | 131 | 7 | 0 | 44 | 18 | 132 | 0 | 37 | 9 | 22 | 133 | 0 | 44 | 18 | 7 | 134 | 44 | 18 | 7 | 0 | 135 | 9 | 37 | 0 | 22 | 136 | 9 | 37 | 19 | 0 | 137 | 0 | 9 | 37 | 22 | 138 | 18 | 44 | 7 | 0 | 142 | 0 | 18 | 7 | 44 | 143 | 44 | 18 | 0 | 7 | 144 | 9 | 37 | 0 | 19 | 145 | 0 | 18 | 7 | 44 | 146 | 18 | 0 | 7 | 44 | 147 | 19 | 9 | 37 | 0 | 148 | 44 | 18 | 0 | 7 | 149 | 37 | 0 | 19 | 9 | 150 | 19 | 0 | 9 | 37 | 151 | 0 | 37 | 22 | 9 | 152 | 7 | 18 | 44 | 0 | 153 | 44 | 0 | 7 | 18 | 154 | 0 | 37 | 9 | 22 | 155 | 18 | 0 | 7 | 44 | 156 | 44 | 7 | 0 | 18 | 157 | 0 | 37 | 9 | 19 | 158 | 22 | 37 | 9 | 0 | 159 | 7 | 18 | 44 | 0 | 160 | 0 | 37 | 9 | 19 | 161 | 0 | 18 | 7 | 44 | 162 | 0 | 37 | 9 | 22 | 163 | 7 | 18 | 44 | 0 | 164 | 37 | 22 | 9 | 0 | 165 | 0 | 9 | 37 | 22 | 166 | 0 | 37 | 19 | 9 | 167 | 18 | 44 | 7 | 0 | 168 | 9 | 0 | 22 | 37 | 169 | 0 | 37 | 9 | 19 | 170 | 44 | 0 | 18 | 7 | 171 | 9 | 0 | 19 | 37 | 172 | 9 | 0 | 37 | 19 | 173 | 9 | 37 | 19 | 0 | 174 | 44 | 7 | 18 | 0 | 175 | 19 | 0 | 37 | 9 | 176 | 0 | 22 | 9 | 37 | 177 | 9 | 0 | 37 | 19 | 178 | 0 | 18 | 7 | 44 | 179 | 0 | 18 | 44 | 7 | 180 | 37 | 0 | 9 | 22 | 181 | 9 | 37 | 0 | 22 | 182 | 0 | 37 | 22 | 9 | 183 | 37 | 0 | 22 | 9 | 184 | 0 | 9 | 22 | 37 | 185 | 9 | 19 | 37 | 0 | 186 | 44 | 7 | 18 | 0 | 187 | 19 | 9 | 0 | 37 | 188 | 37 | 0 | 9 | 22 | 189 | 18 | 44 | 7 | 0 | 190 | 22 | 37 | 9 | 0 | 191 | 37 | 19 | 0 | 9 | 192 | 37 | 9 | 19 | 0 | 193 | 0 | 37 | 19 | 9 | 194 | 22 | 0 | 37 | 9 | 195 | 7 | 0 | 18 | 44 | 196 | 37 | 19 | 0 | 9 | 197 | 9 | 37 | 22 | 0 | 198 | 22 | 9 | 0 | 37 | 199 | 44 | 18 | 0 | 7 | 200 | 22 | 37 | 9 | 0 | 201 | 7 | 18 | 0 | 44 | 202 | 0 | 19 | 9 | 37 | 203 | 0 | 44 | 18 | 7 | 204 | 9 | 0 | 19 | 37 | 205 | 7 | 18 | 0 | 44 | 206 | 37 | 19 | 0 | 9 | 207 | 7 | 44 | 18 | 0 | 208 | 44 | 0 | 18 | 7 | 209 | 19 | 37 | 0 | 9 | 210 | 44 | 18 | 0 | 7 | 211 | 19 | 37 | 0 | 9 | 212 | 18 | 0 | 7 | 44 | 213 | 37 | 9 | 22 | 0 | 214 | 0 | 9 | 37 | 22 | 215 | 22 | 37 | 0 | 9 | 216 | 19 | 37 | 9 | 0 | 217 | 7 | 44 | 18 | 0 | 218 | 44 | 0 | 7 | 18 | 219 | 44 | 0 | 7 | 18 | 220 | 37 | 0 | 19 | 9 | 221 | 22 | 9 | 0 | 37 | 222 | 44 | 18 | 7 | 0 | 223 | 19 | 9 | 37 | 0 | 224 | 18 | 0 | 44 | 7 | 225 | 37 | 22 | 9 | 0 | 226 | 22 | 9 | 37 | 0 | 227 | 19 | 37 | 9 | 0 | 228 | 18 | 44 | 7 | 0 | 229 | 9 | 19 | 37 | 0 | 230 | 18 | 44 | 7 | 0 | 231 | 18 | 7 | 0 | 44 | 232 | 44 | 7 | 18 | 0 | 233 | 44 | 18 | 0 | 7 | 234 | 37 | 22 | 9 | 0 | 235 | 22 | 9 | 37 | 0 | 236 | 22 | 9 | 0 | 37 | 237 | 44 | 0 | 18 | 7 | 238 | 0 | 37 | 22 | 9 | 239 | 37 | 22 | 0 | 9 | 240 | 37 | 19 | 0 | 9 | 241 | 0 | 18 | 7 | 44 | 242 | 19 | 37 | 0 | 9 | 243 | 22 | 9 | 0 | 37 | 244 | 44 | 7 | 18 | 0 | 245 | 9 | 37 | 19 | 0 | 246 | 37 | 22 | 0 | 9 | 247 | 44 | 7 | 18 | 0 | 248 | 19 | 9 | 0 | 37 | | | | | |
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