Regular ultra-sparse graphs and related non-binary LDPC codes
This page collects some good ultra-sparse graphs that were obtained with a randomized Progressive Edge Growth (PEG) algorithm . By ultra-sparse graph, we mean a bipartite graph with two constant connection degrees (dv,dc), and with dv=2. When the obtained graph has minimum size (smallest number of nodes), we indicate that the size is minimum*.
These particular graphs can have multiple possible applications, including building good non-binary LDPC codes in high order fields [2,3]. The graphs are stored in the MacKay A-list format (http://www.inference.phy.cam.ac.uk/mackay/CodesFiles.html).

 A. Venkiah, D. Declercq and C. Poulliat, "Design of Cages with a Randomized Progressive Edge-Growth Algorithm", in IEEE Commun. Letters, vol. 12(4), pp. 1-3, April 2008.
 X.-Y. Hu, E. Eleftheriou, D.-M. Arnold, "Regular and Irregular Progressive Edge-Growth Tanner Graphs", IEEE Trans. Info. Theo., 51(1), pp. 386-398, January 2005.
 C. Poulliat, M. Fossorier and D. Declercq, "Design of regular (2,dc)-LDPC codes over GF(q) using their binary images", accepted in IEEE Trans. Commun., 2007.

Graphs with (dv=2,dc=3)
 [girth=6,N=6] graph with (dv=2,dc=3,girth=6) and minimum* size N=6 [girth=8,N=9] graph with (dv=2,dc=3,girth=8) and minimum* size N=9 [girth=10,N=15] graph with (dv=2,dc=3,girth=10) and minimum* size N=15 [girth=12,N=21] graph with (dv=2,dc=3,girth=12) and minimum* size N=21 [girth=14,N=36] graph with (dv=2,dc=3,girth=14) and minimum* size N=36 [girth=16,N=45] graph with (dv=2,dc=3,girth=16) and minimum* size N=45 [girth=18,N=114] graph with (dv=2,dc=3,girth=18) and size N=114 [girth=20,N=201] graph with (dv=2,dc=3,girth=20) and size N=201 [girth=22,N=447] graph with (dv=2,dc=3,girth=22) and size N=447 [girth=24,N=726] graph with (dv=2,dc=3,girth=24) and size N=726 [girth=26,N=1602] graph with (dv=2,dc=3,girth=26) and size N=1602

Graphs with (dv=2,dc=4)
 [girth=6,N=10] graph with (dv=2,dc=4,girth=6) and minimum* size N=10 [girth=8,N=16] graph with (dv=2,dc=4,girth=8) and minimum* size N=16 [girth=10,N=38] graph with (dv=2,dc=4,girth=10) and minimum* size N=38 [girth=12,N=52] graph with (dv=2,dc=4,girth=12) and minimum* size N=52 [girth=14,N=260] graph with (dv=2,dc=4,girth=14) and size N=260 [girth=16,N=160] graph with (dv=2,dc=4,girth=16) and minimum* size N=160

Graphs with (dv=2,dc=5)
 [girth=6,N=15] graph with (dv=2,dc=5,girth=6) and minimum* size N=15 [girth=8,N=25] graph with (dv=2,dc=5,girth=8) and minimum* size N=25 [girth=10,N=90] graph with (dv=2,dc=5,girth=10) and size N=90 [girth=12,N=105] graph with (dv=2,dc=5,girth=12) and minimum* size N=105

Graphs with (dv=2,dc=6)
 [girth=6,N=21] graph with (dv=2,dc=6,girth=6) and minimum* size N=21 [girth=8,N=36] graph with (dv=2,dc=6,girth=8) and minimum* size N=36 [girth=10,N=189] graph with (dv=2,dc=6,girth=10) and size N=189 [girth=12,N=186] graph with (dv=2,dc=6,girth=12) and minimum* size N=186

Graphs with (dv=2,dc=7)
 [girth=6,N=28] graph with (dv=2,dc=7,girth=6) and minimum* size N=28 [girth=8,N=49] graph with (dv=2,dc=7,girth=8) and minimum* size N=49 [girth=10,N=385] graph with (dv=2,dc=7,girth=10) and size N=385 [girth=12,N=840] graph with (dv=2,dc=7,girth=12) and size N=840

Graphs with (dv=2,dc=8)
 [girth=6,N=36] graph with (dv=2,dc=8,girth=6) and minimum* size N=36 [girth=8,N=64] graph with (dv=2,dc=8,girth=8) and minimum* size N=64 [girth=10,N=680] graph with (dv=2,dc=8,girth=10) and size N=680

Graphs with (dv=2,dc=9)
 [girth=6,N=45] graph with (dv=2,dc=9,girth=6) and minimum* size N=45 [girth=8,N=81] graph with (dv=2,dc=9,girth=8) and minimum* size N=81

Graphs with (dv=2,dc=10)
 [girth=6,N=55] graph with (dv=2,dc=10,girth=6) and minimum* size N=55 [girth=8,N=100] graph with (dv=2,dc=10,girth=8) and minimum* size N=100

Graphs with (dv=2,dc=20)
 [girth=6,N=210] graph with (dv=2,dc=20,girth=6) and minimum* size N=210 [girth=8,N=400] graph with (dv=2,dc=20,girth=8) and minimum* size N=400

Graphs with (dv=2,dc=30)
 [girth=6,N=465] graph with (dv=2,dc=30,girth=6) and minimum* size N=465 [girth=8,N=900] graph with (dv=2,dc=30,girth=8) and minimum* size N=900

Graphs with (dv=2,dc=40)
 [girth=6,N=820] graph with (dv=2,dc=40,girth=6) and minimum* size N=820 [girth=8,N=1600] graph with (dv=2,dc=40,girth=8) and minimum* size N=1600

Graphs with (dv=2,dc=50)
 [girth=6,N=1275] graph with (dv=2,dc=50,girth=6) and minimum* size N=1275 [girth=8,N=2500] graph with (dv=2,dc=50,girth=8) and minimum* size N=2500