## 3 Our Contribution
### Systematic-RLNC Based Data Transmission System
#### 3.2.1 Systematic Random Linear Network Coding (SRLNC)  
Our proposed Systematic-RLNC (SRLNC) algorithm applies different optimization measures to strength multipath transmission for UWSNs. The goal of the proposed transmission model is focused on ensuring delay sensitive multipath transmission over UWSNs to enable reliable and mission critical communication. SRLNC introduces Coefficient Vector Overhead (CVO) optimization, where it performs linear combination of the data packets using a set of vectors called Coefficient Vectors (CV). Generally, connected sinks have the CV information that helps them to decode the data. A matrix named the coefficient matrix (CM) contains all the CVs. Despite transmitting CVs themselves, the information signifying row (i.e., index location) of the coefficient matrix is applied to generate a particular linear combination to be transmitted along with the coded data packets. The combined CV and CM are stated to be the coefficient information (CI). Once receiving CI, the intermediate node in the developed UWSN network model decompresses it and, after appending it, forwards the (appended) data to the next hop node towards sink. Since the initial CM used at the source node remains unshared to the intermediate nodes, thus, the probability of data decoding at any intermediate node is avoided. This makes SRLNC based communication seamless across multipath transmission based UWSN. Retrieving the packets, the sink node decompresses CI and the substitute coefficient index with the original CV in the correct manner. The retrieved CI is decoded to obtain the original transmitted data. Obtaining the original data, the sink node sends acknowledgement (ACK) to the transmitter and, thus, the transmitter stops transmitting the data packets. With the multipath transmission over UWSN, the SRLNC function can be visualized in three phases:  
1. Process at the source node,
2. Process at the intermediate node, and
3. Process at the sink node. A brief discussion of the RLNC processes at the source, intermediate and the sink node is presented as follows:  
Process at the Source Node  
In SRLNC transmission, at first, the RLNC coefficients are obtained at the source node and, accordingly, n data packets \((\mathrm{s}_{1},\mathrm{s}_{2},\ldots\ldots\mathrm{s}_{n})\) are generated, where each packet signifies a \(1\times\mathrm{s}\) vector from the Galois field (GF)2F. Implementing m \(\times\) n CM, a total of n packets are combined to generate m linear combinations. The generated linear combinations are (61):  
\[ \begin{bmatrix}\mathrm{x}_{1}\\ \mathrm{x}_{2}\\ \vdots\\ \mathrm{x}_{m}\end{bmatrix} _{\mathrm{m}\times 1}= \begin{bmatrix}\mathrm{q}_{1,1}& \mathrm{q}_{1,2}&\ldots&\mathrm{q}_{1,n}\\ \mathrm{q}_{2,1}&\mathrm{q}_{2,2}&\ldots&\mathrm{q}_{2,n}\\ \vdots&\vdots&&\vdots\\ \mathrm{q}_{m,1}&\mathrm{q}_{m,2}&\ldots&\mathrm{q}_{m,n}\end{bmatrix} _{ \mathrm{m}\times n} \begin{bmatrix}\mathrm{s}_{1}\\ \mathrm{s}_{2}\\ \vdots\\ \mathrm{s}_{n}\end{bmatrix} _{\mathrm{n}\times 1} \tag{61}\]where \((x_{1},x_{2},\ldots x_{m})\) represents the generated linear combinations. To improve multipath transmission, \(m-n\) packet combinations are transmitted in each generation. Further, to enhance the computational efficiency, we used relatively lower size \(m\times n\) CM than the original one. Here, CM is known to all connected nodes in UWSN. Being a non-zero element, CM taken from the GF is generated in such a way that the total rank of the \(n\times n\) part is always \(n\). It enables SRLNC to perform linear combination of the individual data packets over generations and enhances the likelihood of data packet combinations to reach the sink node without suffering packet loss. In most of the NC schemes, the coefficients applied for generating the linear combinations are also needed to be transmitted along with the data packet, and thus the output data packet \(x_{p}\) from the transmitter becomes \(\left[q_{p,1}\ q_{p,2}\ \cdots\ q_{p,n}\ \middle|\ x_{p}\right]\), where \(1\ \leq\ p\ \leq\ m\).