# Adaptive Packet Coding for Reliable Underwater Acoustic Communications
## 1 Introduction
In order to reduce the energy consumption for unnecessary transmission, a window control mechanism is further proposed to estimate the expected number of packets actually needed. With the information, the sender just transmits a pre-estimated number of encoder packets (i.e., the window size), and then slows down the transmission to wait for an ACK. In [11], an underwater hybrid ARQ (UW-HARQ) is proposed which uses an NACK to feed back the receive state. Random linear packet coding (RLPC) is used as a rateless code in [14; 15; 16]. In [14], the optimal number of coded packets is investigated for a half-duplex link to minimize the time (or energy) required for the transmission of a group of packets. In [15], joint power and rate control for an acoustic link employing random linear packet coding is considered to achieve a prespecified outage/reliability criterion. To this end, ref. [16] extends the work in [15] by grouped packet coding and increases its throughput efficiency based on the S&W ARQ scheme. The sender transmits a super-group packets and waits for an ACK. This could improve efficiency by packing multiple packets together. In this paper, we propose two adaptive packet-coding schemes to achieve reliable UWA communications. We consider the time variations of UWA channels, which are ignored in most works. Since we focus on the link level performance, we model the channel state by a finite-state Markov chain (FSMC) [21]. Some works use FSMC to model the UWA channels [22; 23; 24; 25; 26; 27; 28]. In [22; 23], [PERSON] used a two-state FSMC channel model with known transition probabilities to evaluate energy-efficient schedulers for underwater acoustic point-to-point links based on experimental data. As compared to existing works, our main contributors are summarized as follows:  
1. We first propose an S&W ARQ scheme with RLPC (RLPC-ARQ). In the RLPC-ARQ scheme, the number of successfully transmitted packets in each transmission and the channel state information (CSI) are fed back to the sender in the ACK messages. The sender dynamically adjusts the coding rate based on the ACK messages to fit the time variations of UWA channels by choosing the optimal number of packetsin each transmission. Meanwhile, considering the long propagation delay and the error-prone characteristic of UWA communications, we set a maximum number of retransmissions to avoid infinite retransmission. The problem is formulated as a finite horizon optimization problem. A dynamic programming (DP) algorithm is proposed to obtain the optimal number of packets in each transmission. 2. We also propose a modified juggling-like ARQ scheme for the RLPC system (RLPC-J-ARQ) to deal with the long propagation of UWA channels. In the RLPC-J-ARQ scheme, two data blocks are alternately transmitted at the sender; there is no need to stop and wait for the ACK. Different from the J-ARQ scheme in [5] for which the duration of each transmission is fixed, the RLPC-J-ARQ scheme adopts adjustable transmission duration by exploiting the characteristics of the rateless code. The rateless code is only used when there is not enough data to transmit to avoid idleness. In addition, we also consider the effect of channel variation. In this case, the standard DP algorithm does not work. Thus, we also propose a two-step DP algorithm to find out the optimal number of packets in each transmission. The rest of this paper is organized as follows. Section 2 presents the basic system model. We formulate the optimization problem in Section 3. We propose the optimization solution based on the principle of DP in Section 4. Section 5 proposes the RLPC-J-ARQ scheme. Section 6 proposes a two-step DP approach to find out the optimal solution of the RLPC-J-ARQ scheme. Section 7 provides numerical results for the proposed solutions and Section 8 concludes the paper.