Bounds on the Reliability of RaptorQ Codes in the Finite-Length Regime

In this paper we analyze the bounds on the reliability of RaptorQ codes under maximum likelihood decoding, especially in the finite-length regime. RaptorQ code ensembles by a high-order low density generator-matrix (LDGM) code as pre-code and an inner Luby transform (LT) code. By investigating the rank of the product of two random coefficient matrices of the high-order LDGM code and the inner LT code, we derive the expressions for the upper and lower bounds of decoding failure probability (DFP) on the RaptorQ code. Then, the accuracy of our derived theoretical bounds are verified through the Monte Carlo simulations with different degree distributions and short packets. The high accuracy bounds are then exploited to design near-optimum finite-length RaptorQ codes, enabling a tight control on the tradeoff between decoding complexity and DFP.