Kovalenkos Full-Rank Limit and Overhead as Lower Bounds for Error-Performances of LDPC and LT Codes over Binary Erasure Channels

We present Kovalenko’s full-rank limit as a tight lower bound for decoding error probability of LDPC codes and LT codes over BEC. From the limit, we derive a full-rank overhead as a lower bound for stable overheads for successful maximum-likelihood decoding of the codes.

💡 Research Summary

The paper investigates the fundamental limits of decoding performance for Low‑Density Parity‑Check (LDPC) codes and Luby‑Transform (LT) codes when transmitted over a Binary Erasure Channel (BEC). The authors adopt Kovalenko’s full‑rank limit—a precise expression for the probability that a random binary matrix is of full rank—as the cornerstone of their analysis. By modeling the parity‑check matrix of an LDPC code or the generator matrix of an LT code as a random binary matrix, they show that successful maximum‑likelihood (ML) decoding on a BEC is equivalent to the associated linear system being full rank. Consequently, the probability of ML decoding failure is bounded below by the complement of Kovalenko’s full‑rank probability.

Kovalenko’s formula for an (n \times m) binary matrix with independent, equally likely entries (p = ½) is
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