Design of Nonbinary LDPC Cycle Codes with Large Girth from Circulants and Finite Fields

Table 1

Some optimal circulants with row/column weight and location set .

		Location set			Location set			Location set			Location set
4	4	{0, 1, 2, 3}	5	5	{0, 1, 2, 3, 4}	6	6	{0, 1, 2, 3, 4, 5}	7	7	{0, 1, 2, 3, 4, 5, 6}
5	4	{0, 1, 2, 3}	6	5	{0, 1, 2, 3, 4}	7	6	{0, 1, 2, 3, 4, 5}	8	7	{0, 1, 2, 3, 4, 5, 6}
6	4	{0, 1, 2, 3}	7	5	{0, 1, 2, 3, 4}	8	6	{0, 1, 2, 3, 4, 5}	9	7	{0, 1, 2, 3, 4, 5, 6}
7	4	{0, 1, 2, 4}	8	5	{0, 1, 2, 3, 5}	9	6	{0, 1, 2, 3, 4, 6}	10	7	{0, 1, 2, 3, 4, 5, 7}
8	4	{0, 1, 2, 4}	9	5	{0, 1, 2, 3, 5}	10	6	{0, 1, 2, 3, 5, 6}	11	7	{0, 1, 2, 3, 4, 6, 7}
9	4	{0, 1, 2, 4}	10	5	{0, 1, 2, 3, 6}	11	6	{0, 1, 2, 4, 5, 7}	12	7	{0, 1, 2, 3, 4, 7, 9}
10	4	{0, 1, 2, 5}	11	5	{0, 1, 2, 4, 7}	12	6	{0, 1, 2, 3, 5, 8}	13	7	{0, 1, 2, 3, 4, 6, 9}
11	4	{0, 1, 2, 5}	12	5	{0, 1, 2, 4, 7}	13	6	{0, 1, 2, 3, 5, 9}	14	7	{0, 1, 2, 3, 5, 6, 9}
12	4	{0, 1, 3, 7}	13	5	{0, 1, 2, 4, 7}	14	6	{0, 1, 2, 3, 5, 9}	15	7	{0, 1, 2, 4, 5, 8, 10}
13	4	{0, 1, 3, 9}	14	5	{0, 1, 2, 4, 7}	15	6	{0, 1, 2, 3, 6, 10}	16	7	{0, 1, 2, 3, 5, 8, 12}
14	4	{0, 1, 4, 6}	15	5	{0, 1, 2, 4, 7}	16	6	{0, 1, 2, 3, 6, 10}	17	7	{0, 1, 2, 3, 5, 8, 12}
15	4	{0, 1, 3, 7}	16	5	{0, 1, 2, 5, 8}	17	6	{0, 1, 2, 3, 6, 10}	18	7	{0, 1, 2, 3, 5, 8, 12}
16	4	{0, 1, 3, 7}	17	5	{0, 1, 2, 4, 12}	18	6	{0, 1, 2, 4, 8, 13}	19	7	{0, 1, 2, 3, 5, 9, 14}
17	4	{0, 1, 3, 7}	18	5	{0, 1, 2, 5, 11}	19	6	{0, 1, 2, 4, 7, 11}	20	7	{0, 1, 2, 3, 6, 10, 15}
18	4	{0, 1, 3, 7}	19	5	{0, 1, 2, 6, 9}	20	6	{0, 1, 2, 4, 7, 12}	21	7	{0, 1, 2, 4, 8, 11, 16}
19	4	{0, 1, 3, 8}	20	5	{0, 1, 2, 5, 14}	21	6	{0, 1, 2, 4, 7, 12}	22	7	{0, 1, 2, 4, 6, 14, 17}
20	4	{0, 1, 3, 14}	21	5	{0, 1, 4, 14, 16}	22	6	{0, 1, 2, 4, 8, 13}	23	7	{0, 1, 2, 3, 8, 13, 17}
21	4	{0, 1, 3, 9}	22	5	{0, 1, 3, 7, 12}	23	6	{0, 1, 2, 4, 7, 15}	24	7	{0, 1, 2, 4, 7, 15, 19}
22	4	{0, 1, 3, 9}	23	5	{0, 1, 3, 8, 14}	24	6	{0, 1, 2, 4, 12, 19}	25	7	{0, 1, 2, 4, 7, 12, 16}
23	4	{0, 1, 3, 10}	24	5	{0, 1, 3, 11, 20}	25	6	{0, 1, 2, 4, 9, 15}	26	7	{0, 1, 2, 4, 7, 13, 18}
24	4	{0, 1, 3, 10}	25	5	{0, 1, 3, 15, 21}	26	6	{0, 1, 2, 5, 9, 15}	27	7	{0, 1, 2, 4, 7, 12, 21}
25	4	{0, 1, 3, 10}	26	5	{0, 1, 3, 7, 12}	27	6	{0, 1, 2, 5, 13, 22}	28	7	{0, 1, 2, 4, 7, 17, 21}
26	4	{0, 1, 3, 11}	27	5	{0, 1, 3, 7, 18}	28	6	{0, 1, 4, 15, 20, 22}	29	7	{0, 1, 2, 4, 7, 12, 16}
27	4	{0, 1, 4, 10}	28	5	{0, 1, 3, 13, 24}	29	6	{0, 1, 2, 4, 18, 23}	30	7	{0, 1, 2, 4, 7, 15, 25}
28	4	{0, 1, 3, 12}	29	5	{0, 1, 3, 7, 19}	30	6	{0, 1, 2, 5, 14, 24}	30	5	{0, 1, 3, 12, 25}