A Tight Estimate for Decoding Error-Probability of LT Codes Using Kovalenkos Rank Distribution

A new approach for estimating the Decoding Error-Probability (DEP) of LT codes with dense rows is derived by using the conditional Kovalenko’s rank distribution. The estimate by the proposed approach is very close to the DEP approximated by Gaussian Elimination, and is significantly less complex. As a key application, we utilize the estimates for obtaining optimal LT codes with dense rows, whose DEP is very close to the Kovalenko’s Full-Rank Limit within a desired error-bound. Experimental evidences which show the viability of the estimates are also provided.

💡 Research Summary

The paper presents a novel analytical framework for accurately estimating the decoding error probability (DEP) of Luby‑Transform (LT) codes when the parity‑check matrix contains a mixture of sparse rows and a controlled fraction of dense rows. Traditional approaches rely on Gaussian elimination (GE) to compute the probability that the random binary matrix formed by the received symbols is full‑rank. While GE yields exact results, its computational complexity grows cubically with the matrix dimension, making it impractical for the large block lengths typical of modern communication and storage systems. Moreover, existing probabilistic approximations lose accuracy in the presence of dense rows, which are often introduced deliberately to improve the rank growth of the matrix and thus reduce the error floor.

To overcome these limitations, the authors exploit Kovalenko’s rank distribution, a closed‑form expression that gives the exact probability that a random binary matrix has a particular rank under the assumption of independent rows. They extend this theory to a conditional setting: the parity‑check matrix H is partitioned into a sparse sub‑matrix (H_s) (the traditional LT part) and a dense sub‑matrix (H_d) (the added dense rows). First, the rank distribution of (H_s) is obtained using standard LT analysis (e.g., degree distribution, tree‑like neighborhood arguments). Let (r) denote the rank realized by (H_s). Then, conditioned on this rank, the authors apply Kovalenko’s formula to evaluate the probability that the addition of (m_d) dense rows raises the overall rank to the full‑rank value (n) (the number of columns). The conditional probability can be written as

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