Internet Research Task Force (IRTF)                              S. Yang
Request for Comments: 9426                 CUHK(SZ) & n-hop technologies
Category: Informational                                         X. Huang
ISSN: 2070-1721                                                     CUHK
                                                                R. Yeung
                                               CUHK & n-hop technologies
                                                                  J. Zao
                                                               July 2023

    BATched Sparse (BATS) Coding Scheme for Multi-hop Data Transport


   In general, linear network coding can improve the network
   communication performance in terms of throughput, latency, and
   reliability.  BATched Sparse (BATS) code is a class of efficient
   linear network coding scheme with a matrix generalization of fountain
   codes as the outer code and batch-based linear network coding as the
   inner code.  This document describes a baseline BATS coding scheme
   for communication through multi-hop networks and discusses the
   related research issues towards a more sophisticated BATS coding
   scheme.  This document is a product of the Coding for Efficient
   Network Communications Research Group (NWCRG).

Status of This Memo

   This document is not an Internet Standards Track specification; it is
   published for informational purposes.

   This document is a product of the Internet Research Task Force
   (IRTF).  The IRTF publishes the results of Internet-related research
   and development activities.  These results might not be suitable for
   deployment.  This RFC represents the consensus of the Coding for
   Efficient Network Communications Research Group of the Internet
   Research Task Force (IRTF).  Documents approved for publication by
   the IRSG are not candidates for any level of Internet Standard; see
   Section 2 of RFC 7841.

   Information about the current status of this document, any errata,
   and how to provide feedback on it may be obtained at

Copyright Notice

   Copyright (c) 2023 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (https://trustee.ietf.org/license-info) in effect on the date of
   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.

Table of Contents

   1.  Introduction
     1.1.  Requirements Language
   2.  A Use Case of the BATS Coding Scheme
     2.1.  Introduction
     2.2.  DDP Procedures
       2.2.1.  Source Node Data Partitioning and Padding
       2.2.2.  Source Node Outer Code Encoding Procedure
       2.2.3.  Recoding Procedures
       2.2.4.  Destination Node Procedures
     2.3.  Recommendation for the Parameters
     2.4.  Coding Parameters in DDP Packets
       2.4.1.  Coding Parameter Format
       2.4.2.  Coded Packet Format
   3.  BATS Code Specification
     3.1.  Common Parts
     3.2.  Outer Code Encoder
     3.3.  Inner Code Encoder (Recoder)
     3.4.  Outer Decoder
   4.  Research Issues
     4.1.  Coding Design Issues
     4.2.  Protocol Design Issues
     4.3.  Usage Scenario Considerations
   5.  IANA Considerations
   6.  Security Considerations
     6.1.  Preventing Eavesdropping
     6.2.  Privacy Preservation against Traffic Analysis
     6.3.  Countermeasures against Pollution Attacks
   7.  References
     7.1.  Normative References
     7.2.  Informative References
   Authors' Addresses

1.  Introduction

   This document specifies a baseline BATched Sparse (BATS) code
   [Yang14] scheme for data delivery in multi-hop networks and discusses
   the related research issues towards a more sophisticated scheme.  The
   BATS code described here includes an outer code and an inner code.
   The outer code is a matrix generalization of fountain codes (see also
   the RaptorQ code described in [RFC6330]), which inherits the
   advantages of reliability and efficiency and possesses the extra
   desirable property of being compatible with network coding.  The
   inner code, also called recoding, is formed by linear network coding
   for combating packet loss, improving the multicast efficiency, etc.
   A detailed design and analysis of BATS codes are provided in the BATS
   monograph [Yang17].

   A BATS coding scheme can be applied in multi-hop networks formed by
   wireless communication links, which are inherently unreliable due to
   interference, fading, multipath, etc.  Existing transport protocols
   like TCP use end-to-end retransmission, while network protocols such
   as IP might enable store-and-forward at the relays so that packet
   loss would accumulate along the way.

   A BATS coding scheme can be used for various data delivery
   applications, like file transmission, video streaming over wireless
   multi-hop networks, etc.  Different from the forward error correction
   (FEC) schemes that are applied either hop-by-hop or end-to-end, the
   BATS coding scheme combines the end-to-end coding (the outer code)
   with certain hop-by-hop coding (the inner code) and hence can
   potentially achieve better performance.

   The baseline coding scheme described here considers a network with
   multiple communication flows.  For each flow, the source node encodes
   the data for transmission separately.  However, inside the network,
   it is possible to mix the packets from different flows for recoding.
   In this document, we describe a simple case where recoding is
   performed within each flow.  Note that the same encoding/decoding
   scheme described here can be used with different recoding schemes as
   long as they follow the principle we illustrate in this document.

   In this document, we focus on the case that each flow has only one
   destination node.  Communicating the same data to multiple
   destination nodes is called multicast.  Refer to Section 4 for the
   further discussion of multicast.

   The purpose of the baseline BATS coding scheme is twofold.  First, it
   provides researchers and engineers a starting point for developing
   network communication applications/protocols based on BATS codes.
   Second, it helps to make the research issues clearer towards a
   sophisticated network protocol based on BATS codes.  Important
   research directions include the security issues, congestion control,
   routing algorithms for BATS codes, etc.

   This document is a product of and represents the collaborative work
   and consensus of the Coding for Efficient Network Communications
   Research Group (NWCRG).  It is not an IETF product and is not an IETF

1.1.  Requirements Language

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "OPTIONAL" in this document are to be interpreted as described in
   BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
   capitals, as shown here.

2.  A Use Case of the BATS Coding Scheme

   The BATS coding scheme described in this document can be used, for
   example, by a Data Delivery Protocol (DDP).  Though this document is
   not about a DDP, in this section, we briefly illustrate how a BATS
   coding scheme is employed by a DDP to make the role of the coding
   scheme clear.  Some terms that will be used in this section are
   summarized here:

   DDP:  Data Delivery Protocol

   DDP packet:  the packet formed by a DDP employing a BATS coding

   source packet:  the packet formed by the data for delivery

   outer encoder:  the outer code encoder of a BATS code

   recoder:  the inner code encoder of a BATS code

   outer decoder:  the outer code decoder of a BATS code

   coded packet:  the packet generated by the outer code encoder or a

   batch:  a set of coded packets generated by a BATS coding scheme from
      the same subset of the source packets

   recoded packet:  a coded packet generated by a recoder

   degree:  the number of source packets used to generate a batch by the
      outer encoder.  (The degree can be different for a different

   Other common terms can be found in [RFC8406].

2.1.  Introduction

   We describe a DDP that involves one source node, one destination
   node, and multiple intermediate nodes in between.  A BATS coding
   scheme includes an outer code encoder (also called outer encoder), an
   inner code encoder (also called recoder), and an outer decoder that
   decodes the outer code and the inner code jointly, as illustrated in
   Figure 1.  The functions of the outer encoder, recoder, and outer
   decoder are described below:

                            |  {set of source packets}
                    | outer encoder |
                    |       v       | source node
                    |    recoder    |
                            |  {set of DDP packets}
                    |               |
                    |    recoder    | intermediate node
                    |               |
                            |  {set of DDP packets}
                            |  {set of DDP packets}
                    |               |
                    | outer decoder | destination node
                    |               |
                            |  {set of source packets}

                Figure 1: A Network Model for Data Delivery

   At the source node, the DDP first processes the data to be delivered
   into a number of source packets, each of the same number of bits (see
   details in Section 2.2.1), and then provides all the source packets
   to the outer encoder.  The outer encoder will further generate a
   sequence of batches, each consisting of a fixed number of coded
   packets (see the description in Section 2.2.2).

   Each batch generated at the source node is further processed by the
   recoder separately.  The recoder may generate a number of new coded
   packets using the existing coded packets of the batch (see the
   description in Section 2.2.3).  After it is processed by the recoder,
   The DDP forms and transmits the DDP packets using the coded packets,
   together with the corresponding batch information.

   We assume that a DDP packet is either correctly received or
   completely erased during the communication.  The DDP extracts the
   coded packets and the corresponding batch information from its
   received DDP packets.  A recoder is employed at an intermediate node
   that does not need the data and generates recoded packets for each
   batch (see the description in Section 2.2.3).  The DDP forms and
   transmits DDP packets using the recoded packets and the corresponding
   batch information.

   The outer decoder is employed at the destination node that needs the
   data.  The DDP extracts the coded packets and the corresponding batch
   information from its received DDP packets.  The outer decoder tries
   to recover the transmitted data using the received batches (see the
   description in Section 2.2.4).  The DDP sends the decoded data to the
   application that needs the data.

2.2.  DDP Procedures

   Suppose that the DDP has F octets of data for transmission.  We
   describe the procedures of one BATS session for transmitting the F
   octets.  There is a limit on F of a single BATS session.  If the
   total data has more than the limit, the data needs to be transmitted
   using multiple BATS sessions.  The limit on F of a single BATS
   session depends on the coding parameters that are discussed in this
   section and the calculations described at the end of Section 2.4.2.

2.2.1.  Source Node Data Partitioning and Padding

   The DDP first determines the following parameters:

   *  batch size (M): the number of coded packets in a batch generated
      by the outer encoder

   *  recoding field size (q): the number of elements in the finite
      field for recoding, i.e., q is 2 or 2^8

   *  BATS payload size (TO): the number of payload octets in a BATS
      packet, including the coded data and the coefficient vector

   Based on the above parameters, the parameters T, CO, and K are
   calculated as follows:

   *  CO: the number of octets of a coefficient vector, calculated as CO
      = ceil(M * log2(q) / 8), which is also called the coefficient
      vector overhead

   *  T: the number of data octets of a coded packet, calculated as T =
      TO - CO

   *  K: number of source packets, calculated as K = floor(F / T) + 1

   The data MUST be padded to have T*K octets, which will be partitioned
   into K source packets b[0], ..., b[K - 1], each of T octets.  In our
   padding scheme, b[0], ..., b[K - 2] are filled with data octets, and
   b[K - 1] is filled with the remaining data octets and padding octets.
   Let P = K * T - F denote the number of padding octets.  We use b[K -
   1, 0], ..., b[K - 1, T - P - 1] to denote the T - P source octets and
   b[K - 1, T - P], ..., b[K - 1, T - 1] to denote the P padding octets
   in b[K - 1], respectively.  The padding insertion process is shown in
   Figure 2.

       Z = T - P
       j = 1
       v = 1
       Let bl be the last source packet b[K - 1]
       for i = Z, Z + 1, ..., T - 1 do
         bl[i] = j
         if i + 1 >= v + Z do
           j += 1
           v += j

                  Figure 2: Data Padding Insertion Process

2.2.2.  Source Node Outer Code Encoding Procedure

   The DDP provides the BATS encoder with the following information:

   *  batch size (M): the number of coded packets in a batch

   *  recoding field size (q): the number of elements in the finite
      field for recoding

   *  maximum degree (MAX_DEG): a positive integer that specifies the
      largest degree can be used

   *  degree distribution (DD): an unsigned integer array of size
      MAX_DEG+1.  The i-th entry DD[i] is the probability that i is
      chosen as the degree, where i is between 0 and MAX_DEG.

   *  a sequence of batch IDs (BIDs) (j, j = 0, 1, ...)

   *  number of source packets (K)

   *  packet size (T): the number of octets in a source packet

   *  source packets (b[i], i = 0, 1, ..., K - 1)

   Using this information, the outer encoder generates M coded packets
   for each BID using the following steps that are described in detail
   in Section 3.2:

   *  Obtain a degree d by sampling DD.  Roughly, the value d is chosen
      with probability DD[d].

   *  Choose d source packets uniformly at random from all the K source
      packets.  It is allowed that a source packet is used by multiple

   *  Generate M coded packets using the d source packets.

   From the outer encoder, the DDP receives a sequence of batches, where
   the batch with ID j has M coded packets (x[j,i], i =0, 1, ..., M -
   1), each containing TO octets.

   The DDP will use the batches to form DDP packets to be transmitted to
   other network nodes towards the destination nodes.  The DDP MUST
   deliver each coded packet with its batch ID, which will be further
   used by both the recoder and decoder.

   The DDP MUST deliver some of the information used by the encoder to
   each of the recoders and the decoder, either embedded in the DDP
   packets or transmitted using separated protocol messages.  For
   recoders, the DDP MUST deliver the following information to each

   *  M: batch size

   *  q: recoding field size

   For the decoder, the DDP MUST deliver the following information to
   the decoder:

   *  M: batch size

   *  q: recoding field size

   *  K: the number of source packets

   *  T: the number of octets in a source packet

   *  DD: the degree distribution

   The BID is used by both recoders and decoders.  In Section 2.4, we
   illustrate how to embed BID, M, q, and K into DDP packets.  The
   degree distribution DD does not need to be changed frequently.  See
   Section 6 of [Yang17] about how to design a good degree distribution.
   Once designed, the degree distribution can be shared between the
   source node and the destination node by the DDP, which is not further
   discussed here.

2.2.3.  Recoding Procedures

   Both the source node and the intermediate nodes perform recoding on
   the batches before transmission.  At the source node, the recoder
   receives the batches from the outer code encoding procedure.  At an
   intermediate node, the DDP receives the DDP packets from the other
   network nodes.  If the DDP chooses not to recode, it just forwards
   the DDP packets it received.  Otherwise, the DDP should be able to
   extract coded packets and the corresponding batch information from
   these packets.

   For a received batch, the DDP determines a positive integer (Mr) and
   the number of recoded packets to be transmitted for the batch, and
   DDP provides the recoder with the following information:

   *  M: batch size

   *  q: recoding field size

   *  a number of received coded packets of the same batch, each
      containing TO octets

   *  Mr: the number of recoded packets to be generated

   The recoder uses the information provided by the DDP to generate Mr
   recoded packets, each containing TO octets, which are described in
   Section 3.3.  The DDP uses the Mr recoded packets to form the DDP
   packets for transmitting.

2.2.4.  Destination Node Procedures

   At the destination node, the DDP receives DDP packets from an
   intermediate network node and should be able to extract coded packets
   and the corresponding batch information from these packets.  The DDP
   then employs an outer decoder to recover the data transmitted by the
   source node.

   The DDP provides the outer decoder (to be described in Section 3.4)
   with the following information:

   *  M: batch size

   *  q: recoding field size

   *  K: the number of source packets

   *  T: the number of octets of a source packet

   *  a sequence of batches, each of which is formed by a number of
      coded packets belonging to the same batch, with their
      corresponding BIDs

   The decoder uses this information to decode the outer code and the
   inner code jointly and recover the K source packets (see details in
   Section 3.4).  If successful, the decoder returns the recovered K
   source packets to the DDP, which will use the K source packets to
   form the F octets data.  The recommended padding deletion process is
   shown as follows:

       // this procedure returns the number P of padding octets
       // at the end of b[K - 1]
       Let bl be the last decoded source packet b[K - 1]
       PL = bl[T - 1]
       if PL == 1 do
           return P = 1
       WI = T - 1
       while bl[WI] == PL do
           WI = WI - 1
       return P = (1 + bl[WI]) * bl[WI] + T - WI - 1

                  Figure 3: Data Padding Deletion Process

2.3.  Recommendation for the Parameters

   The recommendation for the parameters M and q is shown as follows:

   *  when q = 2, M = 16, 32, 64, 128

   *  when q = 256, M = 4, 8, 16, 32

   It is RECOMMENDED that K is at least 128.  The encoder/decoder SHALL
   support an arbitrary positive integer value less than 2^16.  However,
   the BATS coding scheme to be described is not optimized for small K.

2.4.  Coding Parameters in DDP Packets

   Here, we provide an example of embedding the aforementioned BATS
   coding parameters into the DDP packets that will be used for recoding
   and decoding.  A DDP can form a DDP packet using a coded packet by
   adding necessary information that can help to deliver the DDP packet
   to the next node (e.g., the version of the DDP, addresses, and
   session identifiers).  We will not go into the details of formatting
   these fields in a DDP packet, but we focus on how to format the
   coding parameters and the coded packet in a DDP packet.

2.4.1.  Coding Parameter Format

   Here, we provide an example of using 32 bits (4 octets) to embed the
   parameters K, M, q, and BID.  The 32 bits are separated into three
   subfields, denoted as K, Mq, and BID, respectively, as illustrated in
   Figure 4.

      0                   1                   2                   3
      0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
     |                K              | Mq  |          BID            |

                  Figure 4: Coding Parameter Field Format

   K:  16-bit unsigned integer, specifying the number of source packets
      of the BATS session

   Mq:  3-bit unsigned integer, specifying the value of M and q, as
      shown in Table 1

   BID:  13-bit unsigned integer, specifying the batch ID of the batch
      the packet belongs to

                            | Mq  | M   | q   |
                            | 000 | 16  | 2   |
                            | 010 | 32  | 2   |
                            | 100 | 64  | 2   |
                            | 110 | 128 | 2   |
                            | 001 | 4   | 256 |
                            | 011 | 8   | 256 |
                            | 101 | 16  | 256 |
                            | 111 | 32  | 256 |

                           Table 1: Values of the
                                  Mq Field

   The choice of the coding parameters depends on the computation cost,
   the network conditions, and the expected end-to-end coding
   performance.  Usually, a larger batch size M will have a better
   coding performance but higher computation cost for encoding,
   recoding, and decoding.  The field size q affects the coefficient
   vector overhead and also the computation cost for recoding.  Within a
   BATS session, the BID field should be different for all batches, and
   hence, the maximum number of batches that can be generated for the
   outer encoder is 2^13.  For different BATS sessions, batches can use
   the same BID.

2.4.2.  Coded Packet Format

                     CO                          T
         |   coefficient vector  |          coded data           |

                Figure 5: Code Packet Format in a DDP Packet

   A coded packet has TO=T+CO octets, where the first CO octets contain
   the coefficient vector and the remaining T octets contain the coded

   coefficient vector:  CO = M * log2(q) / 8 octets.  For the values of
      M and q in Table 1, CO is at most 32 octets when q is 256 and 6
      octets when q is 2.

   coded data:  T octets.  T should be chosen so that the whole DDP
      packet is at most Path MTU (PMTU).

   Using the above formation, we can calculate the largest file size (F)
   for different parameters.  For example, when q = 2 and M = 128, we
   have CO = 6 octets.  Counting the 4 octets for embedding the coding
   parameters, we can choose T = PMTU - H - 10, where H is the header
   length of a DDP packet.  As K can be at most 2^16 - 1, F can be at
   most (PMTU - H - 10)(2^16 - 1) octets.  In this case, K / M is about
   2^9 and the BID field allows transmitting 2^4 * K / M batches.

3.  BATS Code Specification

3.1.  Common Parts

   The T octets of a source packet are treated as a column vector of T
   elements in GF(256).  The CO octets of a coefficient vector are
   treated as a column vector of CO elements in GF(q), where q = 2 or
   q = 256.  Linear algebra and matrix operations over finite fields are
   assumed in this section.

   For the two elements of GF(2), their multiplication corresponds to a
   logical AND operation, and their addition is a logical XOR operation.
   An element of the field GF(256) can be represented by a polynomial
   with binary coefficients and degree lower or equal to 7.  The
   addition between two elements of GF(256) is defined as the addition
   of the two binary polynomials.  The multiplication between two
   elements of GF(256) is the multiplication of the two binary
   polynomials modulo a certain irreducible polynomial of degree 8,
   called a primitive polynomial.  One example of such a primitive
   polynomial for GF(256) is:

      x^8 + x^4 + x^3 + x^2 + 1

   A common primitive polynomial MUST be specified for all the finite
   field multiplications over GF(256).  Note that a binary polynomial of
   degree less than 8 can be represented by a binary sequence of 8 bits,
   i.e., an octet.

   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 [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.
       // 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

                      Figure 6: Batch Sampler Function

3.2.  Outer Code Encoder

   Define a function called DegreeSampler that returns 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.  Note that when K
   < MAX_DEG, the degree value returned by DegreeSampler does not
   exactly follow the distribution DD, which however would not affect
   the practical decoding performance for the outer decoder to be
   described in Section 3.4.

   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]
       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)

                     Figure 7: Degree Sampler Function

   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 CO 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; X]
       return Xr

                    Figure 8: Batch Generation Function

3.3.  Inner Code Encoder (Recoder)

   In general, the inner code of a BATS code comprises (random) linear
   network coding applied on the coded packets belonging to the same
   batch.  The recoded packets have the same BID.  Suppose that coded
   packets xr[i], i = 0, 1, ..., r - 1, which have the same BID j, have
   been received at an intermediate node and Mr recoded packets are
   supposed to be generated.  Following random linear network coding, a
   recoded packet can be generated by a random linear combination:
   (randomly) choose a sequence of coefficients c[i], i = 0, 1, ..., r -
   1 from GF(q) and generate c[0]xr[0] + c[1]xr[1] + ... + c[r - 1]xr[r
   - 1] as a recoded packet.  This recoding approach, called random
   linear recoding, achieves good network coding performance for
   multicast when the batch size is sufficiently large.

   For unicast communications in a single path, as illustrated in
   Figure 1, it is not necessary to generate all the Mr recoded packets
   using a random linear combination.  Instead, xr[i], i = 0, 1, ..., r
   - 1 are directly used as recoded packets, and max(Mr - r, 0) recoded
   packets are generated using linear combinations.  Compared with
   random linear recoding, this recoding approach, called systematic
   recoding, can reduce both the computation cost and the recoding
   latency that accumulates linearly with the number of nodes.  Note
   that the use of systematic recoding may not always achieve the
   optimal network coding performance as random linear recoding in more
   complicated communication scenarios that include multiple paths and
   multiple destination nodes.

3.4.  Outer 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).  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).  For successful decoding, we require that the total
   rank of all the batches is at least K.

   The same degree distribution DD used for the outer encoder is
   supposed to be known by the outer decoder.  By calling DegreeSampler
   and BatchSampler with input j, we obtain d[j], idx[j], and G[j].
   According to the encoding and recoding processes described in
   Sections 3.2 and 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 first 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 batch 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 an 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.

   When the degree distribution DD in the outer code encoder (see
   Section 3.2) is properly designed, the BP decoder guarantees a high
   probability for the recovery of a given fraction of the source
   packets when K is large.  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 BP decoder stops, which may happen with a high
   probability when K is relatively small, it is possible to continue
   with inactivation decoding, where certain source packets are treated
   as inactive so that a similar belief propagation process can be
   resumed.  The reader is referred to [RFC6330] for the design of a
   precode with a good inactivation decoding performance.

4.  Research Issues

   The baseline BATS coding scheme described in Sections 2 and 3 needs
   various refinements and complements towards becoming a more
   sophisticated network communication application.  Various related
   research issues are discussed in this section, but the security-
   related issues are left to Section 6.

4.1.  Coding Design Issues

   When the number of batches is sufficiently large, the BATS code
   specification in Section 3 has nearly optimal performance in the
   sense that the decoding can be successful with a high probability
   when the total rank of all the batches used for decoding is just
   slightly larger than the number of source packet K.  But when K is
   small, the DegreeSampler 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.

   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.  For a batch with a
   lower rank, the intuition is that 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 end-to-end latency too much 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.  It is well-known that linear
   network coding [Li03] achieves the multicast capacity.  BATS codes
   can benefit from network coding due to its inner code.  For
   multicast, each destination node independently performs the same
   operations as described in this document, but the inner code should
   be improved to take multiple destination nodes into consideration.
   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 functional 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 the 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 may not be the same for
   different links of a flow, 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.
   The rate of transmitting batches can be controlled end-to-end.  Due
   to the recoding operation, however, congestion control cannot be only
   performed end-to-end.  The number of recoded packets generated for a
   batch must be controlled at the intermediate nodes, which introduces
   new research issues for congestion control.  A BATS coding scheme is
   an extension of forward erasure correction coding.  See [RFC9265] for
   more discussion of forward erasure correction coding and congestion

   For routing, the BATS coding scheme is flexible for implementing data
   transmission on multiple paths simultaneously.  For unicast, it is
   optimal to transmit all the packets of a batch on the same path
   between the source node and the destination node, and for multicast,
   network coding gain can be obtained by transmitting packets of the
   same batch on different paths [Yang17].  Lastly, 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.  Usage Scenario Considerations

   There are more research issues pertaining to various usage scenarios.
   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 usage 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 or one-hop communication is highly reliable.  Therefore,
   employing a BATS coding scheme can provide more freedom for media
   access control, including power control, and physical-layer design so
   that the overall network throughput can be improved.

   Another typical usage scenario of BATS coding schemes is underwater
   acoustic networks [Sprea19], where the propagation delay of acoustic
   waves 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, the networking techniques 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 content-centric
   networking (CCN) and provides the extra benefit of network coding

5.  IANA Considerations

   This document has no IANA actions.

6.  Security Considerations

   Subsuming both random linear network codes (RLNCs) and fountain
   codes, BATS codes naturally inherit both their desirable security
   capability of preventing eavesdropping as well as their vulnerability
   towards pollution attacks.  In this section, we discuss some related
   research issues.

6.1.  Preventing Eavesdropping

   Suppose that an eavesdropper obtains a batch where the degree value d
   is strictly larger than the batch size M.  Even if the eavesdropper
   has all the related encoding information, the system of linear
   equations related to this batch does not have a unique solution, and
   the probability that the eavesdropper can guess the d source packets
   used for encoding the batch correctly is 2^(-(d-M)T)<=2^(-T), where T
   is the number of octets of a source packet (see also [Bhattad07]).
   When inactivation decoding is applied, we can design the degree
   distribution DD so that the smallest degree is M+1 and hence prevent
   the eavesdropper from decoding source packets from individual

   If we allow the eavesdropper to collect multiple batches and use
   inactivation decoding, the same security holds if the total rank of
   all the batches collected by the eavesdropper is less than the number
   of source packet.  Therefore, if the DDP can manage to restrict the
   eavesdropper from collecting a sufficient number of coded packets,
   the security of BATS code is effective when T is sufficiently large.
   Here, by "intrinsic security", we mean the security protection
   provided by the BATS coding scheme without extra enhancement.

   If the eavesdropper can collect a sufficient number of coded packets
   for correctly decoding, the intrinsic security of BATS code is
   ineffective.  One solution in this case is to encrypt the whole data
   before using the BATS code scheme.  Better schemes are desired
   towards reducing the computation cost of the whole data encryption
   solution.  This is a research issue that depends on specific BATS
   code schemes and will not be further discussed here.

   The threat exists for eavesdropping on the initial encoding process,
   which takes place at the encoding nodes.  In these nodes, the
   transported data are presented in plain text and can be read along
   their transfer paths.  Hence, information isolation between the
   encoding process and all other user processes running on the source
   node MUST be assured.

   In addition, the authenticity and trustworthiness of the encoding,
   recoding, and decoding program running on all the nodes MUST be
   attested by a trusted authority.  Such a measure is also necessary in
   countering pollution attacks.

6.2.  Privacy Preservation against Traffic Analysis

   A security issue closely related to eavesdropping is traffic
   analysis.  Even when eavesdropping is prevented, tracking the traffic
   flow patterns can help an attacker to know certain information about
   the communication.  Preventing traffic analysis is critical for
   communications that need to be anonymous.  In [Fan09], an approach
   based on homomorphic encryption is proposed for network coding to
   prevent traffic analysis.  However, homomorphic encryption could be
   too computationally expensive for practical applications and cannot
   help with the traffic analysis by monitoring the frequency and timing
   of network traffic.

   The network traffic using network coding does not necessarily satisfy
   the flow conservation property, and hence, network coding can be used
   as a tool for defeating traffic analysis.  For example, redundant
   network traffic can be generated by network coding to make it harder
   for an attacker to learn the true communication.  Moreover, traffic
   analysis countermeasures can benefit from multipath communication
   [Yang15], and network coding makes multipath communication more
   flexible and efficient.  Therefore, using network coding brings new
   research issues for both traffic analysis and its countermeasure.

6.3.  Countermeasures against Pollution Attacks

   Like all network codes, BATS codes are vulnerable to pollution
   attacks.  In these attacks, one or more compromised coding node(s)
   can pollute the coded messages by injecting forged packets into the
   network and thus prevent the receivers from recovering the
   transported data correctly.  Moreover, a small number of polluted
   packets can infect a large number of packets by recoding and decoding

   The research community has long been investigating the use of
   homomorphic signatures to identify the forged packets and stall the
   attacks (see [Zhao07], [Yu08], and [Agrawal09]).  In these schemes,
   the source node attaches a signature to each packet to transmit, and
   the signature is allowed to be processed by network coding in the
   same way as the payload.  All the intermediate nodes and the
   destination node can verify the signature attached to a received
   packet.  However, these countermeasures are regarded as being too
   computationally expensive to be employed in broadband communications.

   A system-level approach based on trusted computing [TC-Wikipedia] can
   provide an alternative to protect BATS codes against pollution
   attacks.  Trusted computing consists of software and hardware
   technologies so that a computer behaves as expected.  Suppose that
   all the network nodes employ trusted computing.  Two nodes will first
   gain trust with each other and then negotiate an authentication
   method for exchanging the coded packets of the BATS coding scheme.  A
   network node would not use any packets received from other nodes
   without trust to avoid the pollution attack.

7.  References

7.1.  Normative References

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <https://www.rfc-editor.org/info/rfc8174>.

   [RFC8406]  Adamson, B., Adjih, C., Bilbao, J., Firoiu, V., Fitzek,
              F., Ghanem, S., Lochin, E., Masucci, A., Montpetit, M.,
              Pedersen, M., Peralta, G., Roca, V., Ed., Saxena, P., and
              S. Sivakumar, "Taxonomy of Coding Techniques for Efficient
              Network Communications", RFC 8406, DOI 10.17487/RFC8406,
              June 2018, <https://www.rfc-editor.org/info/rfc8406>.

   [RFC8682]  Saito, M., Matsumoto, M., Roca, V., Ed., and E. Baccelli,
              "TinyMT32 Pseudorandom Number Generator (PRNG)", RFC 8682,
              DOI 10.17487/RFC8682, January 2020,

7.2.  Informative References

              Agrawal, S. and D. Boneh, "Homomorphic MACs: MAC-Based
              Integrity for Network Coding", International Conference on
              Applied Cryptography and Network Security,
              DOI 10.1007/978-3-642-01957-9_18, May 2009,

              Bhattad, K. and K. Narayanan, "Weakly Secure Network
              Coding", April 2005.

   [Byers20]  Byers, J. and M. Luby, "Liquid Data Networking",
              Proceedings of the 7th ACM Conference on Information-
              Centric Networking, DOI 10.1145/3405656.3418710, September
              2020, <https://doi.org/10.1145/3405656.3418710>.

   [Dong20]   Dong, Y., Jin, S., Yang, S., and H. Yin, "Network Utility
              Maximization for BATS Code Enabled Multihop Wireless
              Networks", ICC 2020 - 2020 IEEE International Conference
              on Communications (ICC),
              DOI 10.1109/ICC40277.2020.9148834, June 2020,

   [Fan09]    Yanfei, Y., Yixin, Y., Haojin, H., and X. Sherman, "An
              Efficient Privacy-Preserving Scheme against Traffic
              Analysis Attacks in Network Coding", IEEE INFOCOM 2009,
              DOI 10.1109/INFCOM.2009.5062146, April 2009,

   [Li03]     Li, S.-Y., Yeung, R., and N. Cai, "Linear network coding",
              IEEE Transactions on Information Theory,
              DOI 10.1109/TIT.2002.807285, February 2003,

   [RFC6330]  Luby, M., Shokrollahi, A., Watson, M., Stockhammer, T.,
              and L. Minder, "RaptorQ Forward Error Correction Scheme
              for Object Delivery", RFC 6330, DOI 10.17487/RFC6330,
              August 2011, <https://www.rfc-editor.org/info/rfc6330>.

   [RFC9265]  Kuhn, N., Lochin, E., Michel, F., and M. Welzl, "Forward
              Erasure Correction (FEC) Coding and Congestion Control in
              Transport", RFC 9265, DOI 10.17487/RFC9265, July 2022,

   [Sprea19]  Sprea, N., Bashir, M., Truhachev, D., Srinivas, K.,
              Schlegel, C., and C. Sacchi, "BATS Coding for Underwater
              Acoustic Communication Networks", OCEANS 2019 - Marseille,
              DOI 10.1109/OCEANSE.2019.8867299, June 2019,

              Wikipedia, "Trusted Computing", April 2023,

   [Toh02]    Toh, C., "Ad Hoc Mobile Wireless Networks", Prentice Hall
              Publishers, December 2001.

   [Yang14]   Yang, S. and R. Yeung, "Batched Sparse Codes", IEEE
              Transactions on Information Theory, Vol. 60, Issue 9, pgs.
              5322-5346, DOI 10.1109/TIT.2014.2334315, September 2014,

   [Yang15]   Yang, L. and F. Fengjun, "mTor: A multipath Tor routing
              beyond bandwidth throttling", 2015 IEEE Conference on
              Communications and Network Security (CNS),
              DOI 10.1109/CNS.2015.7346860, September 2015,

   [Yang17]   Yang, S. and R. Yeung, "BATS Codes: Theory and Practice",
              Morgan & Claypool Publishers, September 2017.

   [Yin19]    Yin, H., Tang, B., Ng, K., Yang, S., Wang, X., and Q.
              Zhou, "A Unified Adaptive Recoding Framework for Batched
              Network Coding", 2019 IEEE International Symposium on
              Information Theory (ISIT), DOI 10.1109/ISIT.2019.8849277,
              July 2019, <https://doi.org/10.1109/ISIT.2019.8849277>.

   [Yin20]    Yin, H., Yeung, R., and S. Yang, "A Protocol Design
              Paradigm for Batched Sparse Codes", Entropy,
              DOI 10.3390/e22070790, July 2020,

   [Yu08]     Yu, Z., Wei, Y., Ramkumar, B., and Y. Guan, "An Efficient
              Signature-Based Scheme for Securing Network Coding Against
              Pollution Attacks", IEEE INFOCOM 2008 - The 27th
              Conference on Computer Communications,
              DOI 10.1109/INFOCOM.2008.199, April 2008,

   [Zhang16]  Zhang, G. and Z. Xu, "Combing CCN with network coding: An
              architectural perspective", Computer Networks,
              DOI 10.1016/j.comnet.2015.11.008, January 2016,

   [Zhao07]   Zhao, F., Kalker, T., Medard, M., and K. Han, "Signatures
              for content distribution with network coding", 2007 IEEE
              International Symposium on Information Theory,
              DOI 10.1109/ISIT.2007.4557283, June 2007,


   The authors would like to thank the NWCRG chairs, Vincent Roca (our
   shepherd) and Marie-Jose Montpetit, as well as all those who provided
   comments, namely (in alphabetical order), Emmanuel Lochin, David
   Oran, and Colin Perkins.

Authors' Addresses

   Shenghao Yang
   CUHK(SZ) & n-hop technologies
   Phone: +86 755 8427 3827
   Email: shyang@cuhk.edu.cn

   Xuan Huang
   Hong Kong
   Hong Kong SAR,
   Phone: +852 3943 8375
   Email: 1155136647@link.cuhk.edu.hk

   Raymond W. Yeung
   CUHK & n-hop technologies
   Hong Kong
   Hong Kong SAR,
   Phone: +852 3943 8375
   Email: whyeung@ie.cuhk.edu.hk

   John K. Zao
   Hong Kong
   Hong Kong SAR,
   Phone: +852 3943 8346
   Email: johnzao@ie.cuhk.edu.hk