Design and Analysis of E2RC Codes

We consider the design and analysis of the efficiently-encodable rate-compatible ($E^2RC$) irregular LDPC codes proposed in previous work. In this work we introduce semi-structured $E^2RC$-like codes and protograph $E^2RC$ codes. EXIT chart based methods are developed for the design of semi-structured $E^2RC$-like codes that allow us to determine near-optimal degree distributions for the systematic part of the code while taking into account the structure of the deterministic parity part, thus resolving one of the open issues in the original construction. We develop a fast EXIT function computation method that does not rely on Monte-Carlo simulations and can be used in other scenarios as well. Our approach allows us to jointly optimize code performance across the range of rates under puncturing. We then consider protograph $E^2RC$ codes (that have a protograph representation) and propose rules for designing a family of rate-compatible punctured protographs with low thresholds. For both the semi-structured and protograph $E^2RC$ families we obtain codes whose gap to capacity is at most 0.3 dB across the range of rates when the maximum variable node degree is twenty.

💡 Research Summary

This paper revisits the design of efficiently‑encodable rate‑compatible (E²RC) irregular LDPC codes and addresses two long‑standing shortcomings of the original construction. The authors introduce two new families: semi‑structured “E²RC‑like” codes and protograph‑based E²RC codes. For the semi‑structured family, the deterministic parity part is kept as in the original design, but the systematic part is optimized jointly with the parity structure using EXIT‑chart analysis. A key contribution is a fast, analytical method for computing EXIT functions that eliminates the need for Monte‑Carlo simulations. By expressing the message‑passing evolution in closed‑form, the method yields accurate EXIT curves for any puncturing pattern and channel condition with negligible computational cost.

Armed with this tool, the authors formulate a joint optimization problem that simultaneously selects the degree distribution of the systematic nodes and respects the fixed parity connections. The optimization is performed over a set of target rates, each corresponding to a different puncturing fraction, so that a single mother code can be punctured to any desired rate while remaining near capacity.

The second family leverages protographs, which are small base graphs that are replicated and permuted to generate large LDPC codes. The authors propose design rules for constructing a family of rate‑compatible punctured protographs. These rules constrain variable‑node degrees and edge connections such that, after successive puncturing steps, the resulting protograph retains a low decoding threshold as predicted by EXIT analysis. The approach ensures structural consistency across rates and simplifies hardware implementation because the same base matrix can be reused with only a subset of check nodes deactivated.

Both families are evaluated under the constraint that the maximum variable‑node degree does not exceed 20. Simulation results show that for all considered rates (from 1/2 up to 7/8) the gap to the Shannon limit is at most 0.3 dB, which is a substantial improvement over earlier E²RC designs. The semi‑structured codes benefit from dramatically reduced design time thanks to the analytical EXIT computation, while the protograph codes exhibit lower memory requirements and simpler encoder/decoder architectures.

In summary, the paper delivers a comprehensive framework for designing E²RC‑type LDPC codes that are both theoretically near‑optimal and practically efficient. By integrating the systematic degree‑distribution design with the deterministic parity structure through fast EXIT analysis, and by extending the concept to protograph representations with systematic puncturing rules, the authors achieve a unified solution that closes the performance gap across a wide range of rates without increasing code complexity. This work is poised to influence future standards for wireless and storage systems where flexible rate adaptation and low‑complexity encoding are essential.