## 1 Introduction  
Communication systems are naturally prone to varying channel conditions. Before the recent information explosion, it was common for systems to use conservative configurations which allowed them to operate in a wide range of conditions, but this came at the expense of performance. In order to accommodate the ever-growing traffic requirements of next generation communication devices, researchers are now using adaptive schemes to maximize bandwidth efficiency and squeeze as much throughput as possible in every situation. A significant amount of work has been devoted to designing algorithms for adapting physical layer parameters such as the transmit power, modulation and coding rate based on the channel state information [1; 2; 3; 4; 5], but there is not as much literature on adaptive retransmissions when failures occur despite it has been shown that they can provide significant gains in terms of both throughput [6; 7; 8] and outage probability [9; 10]. Traditional automatic repeat request (ARQ) forces the receiver to send an acknowledgment (ACK) back to the transmitter for every packet it successfully decodes, and a negative-acknowledgment (NACK) otherwise. If the transmitter does not receive an ACK before the timeout expires, the entire packet will be resent, assuming that it is still within the latency limit. Retransmitting the whole packet is justified when the previous one has been completely lost, but in many cases, the received packet can be partially recovered, and it still contains useful information for the decoder, even if it cannot be entirely decoded. In those cases, it is more efficient if the receiver can recover the whole packet with the help of a few additional bits sent from the transmitter, referred to as incremental redundancy (IR). This is commonly known as Type-II hybrid automatic repeat request (Type-II HARQ) [11], and it is the focus of this paper. The achievable data rate (throughput) with Type-II HARQ has been upper-bounded under the assumption of unlimited single bit IR and perfect feedback [6; 12] and several methods have been proposed to construct IR bits [13; 14] or optimize their block lengths under a finite number of retransmissions [15; 16]. However, most of these works have focused on extending idealized error correcting codes (ECC) in known channels with either infinite or single-bit feedback. Some works have proposed more realistic models accounting for system-level constraints [17; 18; 19], bundling multiple packets in one resource block [8; 20] and imperfect channel information [21]. The first part of this paper takes one step further in this direction by introducing a new type of IR bits and proposing frameworks to optimize the number and type of IR bits to be sent in scenarios with imperfect ECC, limited feedback, packet bundling, and overhead costs for each round of incremental redundancy. It models the problem as a Markov decision process (MDP) which minimizes the average cost per information bit delivered, relying on a code-specific Gaussian model for the probability of decoding failure as a function of signal-to-noise ratio (SNR) and code rate. By adjusting the relative costs associated to decoding and retransmissions, this method can be used to model practical constraints such as latency and hardware. HARQ has been included and widely deployed in recent cellular networks such as Long-Term Evolution (LTE) [22; 23] and 5G NR [24], and there are studies that evaluate its performance [25]. However, most standards proposed the use of fixed-length IRs due to practical constraints. LTE generally assigns one bit feedback per transport block, equivalent to one bit per codeword. The 5G NR standard includes multiple types of operations and is a little more flexible, but does not rise to the level that we propose in this paper. It still uses ACK or NACK and pre-fixed IR lengths. This paper shows that our proposed HARQ strategies can potentially achieve higher throughput by allowing even more flexibility than 5G NR in the types and lengths of IR retransmissions.