A generalization of the ensemble of non-terminated systematic LDPC convolutional codes developed in our previous work is proposed that allows us to design codes with lower rates than the original structure. We show that over the BEC, the modified codes have improved asymptotic and finite-length behavior and we determine the operational anytime exponent. Having shown the advantages of lowering the rate of the code, we propose a feedback protocol that permits encoder and decoder to operate at a variable rate. The rate is set on-the-fly and depends on the decoding success of the decoder. We describe the construction of the variable rate code structure and demonstrate by simulations the superiority of the variable rate scheme as compared to a scheme using a fixed rate.

Here we propose an ensemble of non-terminated systematic LDPC convolutional codes with increasing memory, and show that, over the binary erasure channel (BEC), these codes achieve anytime reliability asymptotically when decoded with an expanding-window message-passing decoder. The corresponding anytime exponents are determined through protograph-based extrinsic information transfer charts. Fundamental complications arising when transmitting with finite block lengths are identified and a combinatorial performance analysis, when transmitting over a static BEC with a fixed number of erasures per codeword block, is developed. Based on the performance analysis, we explore the use of feedback for achieving anytime behavior with constraints on block length. To meet complexity constraints, with or without feedback, the code memory can be limited at the cost of an error floor emerging with a delay proportional to the memory constraint. Although the analysis is developed for a static BEC we show numerically that we can design efficient low-complexity finite-length codes with anytime properties even for the conventional BEC.