Edge Transitive Ramanujan Graphs and Highly Symmetric LDPC Good Codes
  We present a symmetric LDPC code with constant rate and constant distance (i.e. good LDPC code) that its constraint space is generated by the orbit of one constant weight constraint under a group action. Our construction provides the first symmetric LDPC good codes. This solves the main open problem raised by Kaufman and Wigderson in [4]. 
Tali Kaufman, Alexander Lubotzky
cs.IT math.CO math.GR math.IT