Low-Rate Regular Concatenated Zigzag Codes are Capacity-Approaching over the BEC

Concatenated zigzag (CZ) codes have been widely studied for practical applications due to their loss encoding complexity and excellent performance. However, there still lacks rigorous analysis and optimization techniques for CZ codes. This paper presents an analysis technique for CZ codes under belief propagation (BP) decoding over the binary erasure channel (BEC). We derive a pair of closed-form density evolution (DE) equations to characterize the asymptotic behavior of a CZ code under BP decoding. We show that capacity-approaching CZ codes can be designed by matching the curves corresponding to the DE equations. We also prove that the gap between the BP threshold of a regular CZ code and the corresponding Shannon threshold diminishes when the rate approaches zero. This suggests that regular CZ codes are good candidates for low-rate communication systems, where good low-density parity-check codes are difficult to construct.