BATS Coding Scheme for Multi-hop Data Transport

3.1. Common Parts

The T octets of a source packets are treated as a column vector of T elements in GF(256). Linear algebra and matrix operations over finite fields are assumed in this section.¶

Suppose that a pseudorandom number generator Rand() which generates an unsigned integer of 32 bits is shared by both encoding and decoding. The pseudorandom generator can be initialized by Rand_Init(S) with seed S. When S is not provided, the pseudorandom generator is initialized arbitrarily. One example of such a pseudorandom generator is defined in RFC 8682 [RFC8682].¶

A function called BatchSampler is used in both encoding and decoding. The function takes two integers j and d as input, and generates an array idx of d integers and a d x M matrix G. The function first initializes the pseudorandom generator with j, sample d distinct integers from 0 to K-1 as idx, and sample d*M integers from 0 to 255 as G. See the pseudocode in Figure 6.¶

function BatchSampler(j,d)
    // initialize the pseudorandom generator by seed j.
    Rand_Init(j)
    // sample d distinct integers between 0 and K-1.
    for k = 0, ..., d-1 do
        r = Rand() % K
        while r already exists in idx do
            r = Rand() % K
        idx[k] = r

    // sample d x M matrix
    for r = 0, ..., d-1 do
        for c = 0,...,M-1 do
            G[r,c] = Rand() % 256

    return idx, G

3.2. Outer Code Encoder

Define a function called DegreeSampler that return an integer d using the degree distribution DD. We expect that the empirical distribution of the returning d converges to DD(d) when d < K. One design of DegreeSampler is illustrated in Figure 7.¶

function DegreeSampler(j, DD)
    Let CDF be an array
    CDF[0] = 0
    for i = 1, ..., MAX_DEG do
        CDF[i] = CDF[i-1] + DD[i]
    Rand_Init()
    r = Rand() % CDF[MAX_DEG]
    for d = 1, ..., MAX_DEG do
        if r >= CDF[d] do
            return min(d,K)
    return min(MAX_DEG,K)

Let b[0], b[1], ..., b[K-1] be the K source packets. A batch with BID j is generated using the following steps.¶

• Obtain a degree d by calling DegreeSampler with input j.¶
• Obtain idx and G[j] by calling BatchSampler with input j and d.¶
• Let B[j] = (b[idx[0]], b[idx[1]], ..., b[idx[d-1]]). Form the batch X[j] = B[j]*G[j], whose dimension is T x M.¶
• Form the TO x M matrix Xr[j], where the first O rows of Xr[j] form the M x M identity matrix I with entries in GF(q), and the last T rows of Xr[j] is X[j].¶

See the pseudocode of the batch generating process in Figure 8.¶

function GenBatch(j)
    d = DegreeSampler(j)
    (idx, G) = BatchSampler(j,d)
    B = (b[idx[0]], b[idx[i]], ..., b[idx[d-1]])
    X = B * G
    Xr = [I_M; X]
    return Xr

3.3. Inner Code Encoder (Recoder)

The inner code comprises (random) linear network coding applied on the coded packets belonging to the same batch. At a particular network node, recoded packets are generated by (random) linear combinations of the received coded packets of a batch. The recoded packets have the same BID, degree and coded packet length.¶

The number Mr of recoded packets for a batch is decided first by the recoder. Mr can be set as M. When the link statistics is known, the recoder can try to obtain the link packet loss rate e for the link to transmit the recoded batch, and set Mr to be (1+e)M.¶

Suppose that coded packets xr[i], i = 0, 1, ..., r-1, which have the same BID j, have been received at an intermediate node. Using the recommended packet format, it can be verified whether the corresponding packet headers of these coded packets are the same. Then a recoded packet can be generated by one of the following two approaches:¶

• forwarding: when receiving xr[i], directly use xr[i] as a recoded packet.¶
• linear combination recoding: (randomly) choose a sequence of coefficients c[i], i = 0, 1, ..., r-1 from GF(q). Generate c[0]xr[0]+c[1]xr[1]+...+c[r-1]xr[r-1] as a recoded packet.¶

A recoder can combine these two approaches to generate recoded packets. For example, the recoder will output xr[i], i = 0, 1, ..., r-1 as r systematic recoded packets and generate Mr-r recoded packets using linear combinations of randomly chosen coefficients.¶

3.4. Belief Propagation Decoder

The decoder receives a sequence of batches Yr[j], j = 0, 1, ..., n-1, each of which is a TO-row matrix over GF(256). The degree d[j] of batch j is also known. Let Y[j] be the submatrix of the last T rows of Yr[j]. When q = 256, let H[j] be the first M rows of Yr[j]; when q = 2, let H[j] be the matrix over GF(256) formed by embedding each bit in the first M/8 rows of Yr[j] into GF(256).¶

By calling BatchSampler with input j and d[j], we obtain idx[j] and G[j]. According to the encoding and recoding processes described in Section 3.2 and Section 3.3, we have the system of linear equations Y[j] = B[j]G[j]H[j] for each received batch with ID j, where B[j] = (b[idx[j,0]], b[idx[j,1]], ..., b[idx[j,d-1]]) is unknown.¶

We describe a belief propagation (BP) decoder that can efficiently solve the source packets when a sufficient number of batches have been received. A batch j is said to be decodable if rank(G[j]H[j]) = d[j] (i.e., the system of linear equations Y[j] = B[j]G[j]H[j] with B[j] as the variable matrix has a unique solution). The BP decoding algorithm has multiple iterations. Each iteration is formed by the following steps:¶

• Decoding step: Find a batches j that is decodable. Solve the corresponding system of linear equations Y[j] = B[j]G[j]H[j] and decode B[j].¶
• Substitution step: Substitute the decoded source packets into undecodable batches. Suppose that a decoded source packet b[k] is used in generating a undecodable Y[j]. The substitution involves 1) removing the entry in idx[j] corresponding to k, 2) removing the row in G[j] corresponding to b[k], and 3) reducing d[j] by 1.¶

The BP decoder repeats the above steps until no batches are decodable during the decoding step.¶

4.1. Coding Design Issues

The BATS code specification in Section 3 has nearly optimal throughput when the number of source packets K is sufficiently large. But when K is small, the degree sampler function in Figure 7 and the BatchSampler function in Figure 6 based on a pseudorandom generator may not sample all the source packets evenly, so that some of the source packets are not well protected. One approach to solve this issue is to generate a deterministic degree sequence when the number of batches is relatively small, and design a special pseudorandom generator that has a good sampling performance when K is small.¶

The belief propagation decoder in Section 3.4 guarantees the recovery of a given fraction of the source packets. To recover all the source packets, a precode can be applied to the source packets to generate a fraction of redundant packets before applying the outer code encoding. Moreover, when the belief propagation decoder stops, it is possible to continue with inactivation decoding, where certain source packets are treated inactive so that a similar belief propagation process can be resumed. The reader is referred to RFC 6330 [RFC6330] for the design of a precode with a good inactivation decoding performance.¶

There are research issues related to recoding discussed in Section 3.3. One question is how many recoded packets to generate for each batch. Though it is asymptotically optimal when using the same number of recoded packets for all batches, it has been shown that transmitting a different number of recoded packets for different batches can improve the recoding efficiency. The intuition is that for a batch with a lower rank, a smaller number of recoded packets need to be transmitted. This kind of recoding scheme is called adaptive recoding [Yin19].¶

Packet loss in network communication is usually bursty, which may harm the recoding performance. One way to resolve this issue is to transmit the packets of different batches in a mixed order, which is also called batch interleaving [Yin20]. How to efficiently interleave batches without increasing too much end-to-end latency is a research issue.¶

Though we only focus on the BATS coding scheme with one source node and one destination node, a BATS coding scheme can be used for multiple source and destination nodes. To benefit from multiple source nodes, we would need different source nodes to generate statistically independent batches. For communicating the same data to multiple destination nodes, which is also call multicast, it is well-known that linear network coding [Li03] achieves the mulicast capacity. BATS codes can benefit from network coding due to its inner code, but how to efficiently implement multicast needs further research.¶

4.2. Protocol Design Issues

The baseline scheme in this document focuses on reliable communication. There are other issues to be considered towards designing a fully functionally DDP based on a BATS coding scheme. Here we discuss some network management issues that are closely related to a BATS coding scheme: routing, congestion control and media access control.¶

The outer code of a BATS code can be regarded as a channel code for the channel induced by the inner code, and hence the network management algorithms should try to maximize the capacity of the channel induced by the inner code. A network utility maximization problem [Dong20] for BATS coding can be applied to study routing, congestion control and media access control jointly. Compared with the network utility maximization for Internet, there are two major differences. First, the network flow rate is not measured by the rate of the raw packets. Instead, a rank based measurement induced by the inner code is applied for BATS coding schemes. Second, due to recoding, the raw packet rate of a flow may not be the same for different links, i.e., no flow conservation for BATS coding schemes. These differences affect both the objective and the constraints of the utility maximization problem.¶

Practical congestion control, routing and media access control algorithms for BATS coding schemes deserve more research efforts. Due to the recoding operation, congestion control cannot be only performed end-to-end. The rate of transmitting batches can be controlled end-to-end, but the number of recoded packets generated for a batch must be controlled at the intermediate nodes, which introduces new research issues for congestion control. For routing, the BATS coding scheme is flexible for implementing multi-path data transmission, and different batches can be transmitted on a different path between a source node and a destination node. Under the scenario of BATS coding schemes, media access control can have some different considerations: Retransmission is not necessary, and a reasonably high packet loss rate can be tolerated.¶

4.3. Application Related Issues

There are more research issues pertaining to different applications. The reliable communication technique provided by BATS codes can be used for a broad range of network communication scenarios. In general, a BATS coding scheme is suitable for data delivery in networks with multiple hops and unreliable links.¶

One class of typical application scenario is wireless mesh and ad hoc networks [Toh02], including vehicular networks, wireless sensor networks, smart-lamppost networks, etc. These networks are characterized by a large number of network devices connected wirelessly with each other without a centralized network infrastructure. A BATS coding scheme is suitable for high data load delivery in such networks without the requirement that the point-to-point/one-hop communication is highly reliable. Therefore, employing a BATS coding scheme can provide more freedom for media access control, including power control so that the overall network throughput can be improved.¶

Another typical application scenario of BATS coding schemes is underwater acoustic networks [Sprea19], where the propagation delay of acoustic waves in underwater can be as long as several seconds. Due to the long delay, feedback based mechanisms become inefficient. Moreover, point-to-point/one-hop underwater acoustic communication (for both the forward and reverse directions) is highly unreliable. Due to these reasons, traditional networking techinques developed for radio and wireline networks cannot be directly applied to underwater networks. As a BATS coding scheme does not rely on the feedback for reliability communication and can tolerate highly unreliable links, it makes a good candidate for developing data delivery protocols for underwater acoustic networks.¶

Last but not least, due to its capability of performing multi-source multi-destination communications, a BATS coding scheme can be applied in various content distribution scenarios. For example, a BATS coding scheme can be a candidate for the erasure code used in the liquid data networking framework [Byers20] of CCN (content centric networking), and provides the extra benefit of network coding [Zhang16].¶