RMT Working Group M. Luby
INTERNET DRAFT L. Vicisano
Expires 8 September 2000 L. Rizzo
J. Gemmell
J. Crowcroft
B. Lueckenhoff
8 March 2000
Reliable Multicast Transport Building Block:
Forward Error Correction Codes
<draft-ietf-rmt-bb-fec-00.txt>
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Copyright Notice
Copyright (C) The Internet Society (2000). All Rights Reserved.
Abstract
This memo describes the use of Forward Error Correction (FEC) codes
within the context of reliable IP multicast transport and provides an
introduction to some commonly-used FEC codes.
1. Rationale and Overview
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There are many ways to provide reliability for transmission proto-
cols. A common method is to use ARQ, automatic request for
retransmission. With ARQ, receivers use a back channel to the sender
to send requests for retransmission of lost packets. ARQ works well
for one-to-one reliable protocols, as evidenced by the pervasive suc-
cess of TCP/IP. ARQ has also been an effective reliability tool for
one-to-many reliability protocols, and in particular for some reli-
able IP multicast protocols. However, for one-to-very many reliabil-
ity protocols, ARQ has limitations, including the feedback implosion
problem because many receivers are transmitting back to the sender,
and the need for a back channel to send these requests from the
receiver. Another limitation is that receivers may experience dif-
ferent loss patterns of packets, and thus receivers may be delayed by
retransmission of packets that other receivers have lost that but
they have already received. This may also cause wasteful use of
bandwidth used to retransmit packets that have already been received
by many of the receivers.
In environments where ARQ is either costly or impossible because
there is either a very limited capacity back channel or no back chan-
nel at all, such as satellite transmission, a Data Carousel approach
to reliability is sometimes used [AFZ95]. With a Data Carousel, the
sender partitions the object into equal length source symbols, places
them into packets, and then continually cycles through and sends
these packets. Receivers continually receive packets until they have
received a copy of each packet. Data Carousel has the advantage that
it requires no back channel because there is no data that flows from
receivers to the sender. However, Data Carousel also has limita-
tions. For example, if a receiver loses a packet in one round of
transmission it must wait an entire round before it has a chance to
receive that packet again. This may also cause wasteful use of
bandwidth, as the sender continually cycles through and transmits the
packets until no receiver is missing a packet.
FEC codes provide a reliability method that can be used to augment or
replace other reliability methods, especially for one-to-many relia-
bility protocols such as reliable IP multicast. Ideally, FEC codes
in the context of IP multicast can be used to encode an object into
packets in such a way that each received packet is fully useful to a
receiver to reassemble the object regardless of previous packet
reception patterns. Thus, if some packets are lost in transit between
the sender and the receiver, instead of asking for retransmission
using ARQ or waiting till the packets are resent using Data Carousel,
the receiver can use any other subsequent equal number of packets
that arrive to reassemble the object. This implies that the same
packet is fully useful to all receivers to reassemble the object,
even though the receivers may have previously experienced different
packet loss patterns. This property can reduce or even eliminate the
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problems mentioned above associated with ARQ and Data Carousel and
thereby dramatically increase the scalability of the protocol to ord-
ers of magnitude more receivers.
For some reliable IP multicast protocols, FEC codes are used in con-
junction with ARQ to provide reliability. For example, in a first
round all of the source symbols could be transmitted, and then
receivers could report back to the sender the number of symbols they
are missing from each block. Then, the sender could compute the max-
imum number of missing symbols from each block among all receivers,
and then transmit that number of redundant symbols for each block.
In this case, even if different receivers are missing different sym-
bols in different blocks, transmitted redundant symbols for a given
block are useful to all receivers missing symbols from that block.
For others, FEC codes are used in a Data Carousel fashion to provide
reliability, by cycling through and transmitting the encoding symbols
instead of the source symbols. For example, suppose an FEC code is
applied to the entire object consisting of k source symbols to gen-
erate n encoding symbols with the property that the entire object can
be reassembled from any k encoding symbols, and the sender cycles
through and transmits the n encoding symbols in the same order in
each round. Then, a receiver can join the transmission at any point
in time, and as long as the receiver receives at least k encoding
symbols during the transmission of n encoding symbols then the
receiver can completely recover the object. This is true even if the
receiver joins the data carousel in the middle of a round.
For yet other reliable IP multicast protocols the sole method to
obtain reliability is to use FEC codes. For example, the sender can
decide a priori how many encoding symbols it will transmit, use an
FEC code to generate that number of encoding symbols from the object,
and then transmit the encoding symbols to all receivers. This method
is for example applicable to streaming protocols, where the stream is
partitioned into objects, each object is encoded into encoding sym-
bols using an FEC code, and then the sets of encoding symbols for
each object are transmitted one after the other using IP multicast.
The large on demand codes described below have the property that the
FEC encoder can generate sequentially as many encoding symbols as are
desired on demand. Thus, reliable IP multicast protocols that use
large on demand codes generally rely solely on these codes for relia-
bility.
In the general literature, FEC refers to the ability to overcome both
erasures (losses) and bit-level corruption. However, in the case of
IP multicast, lower network layers will detect corrupted packets and
discard them. Therefore, an IP multicast protocol need not be con-
cerned with corruption; the focus is solely on erasure codes. The
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payloads are generated and processed using an FEC erasure encoder and
objects are reassembled from reception of packets using the
corresponding FEC erasure decoder.
The primary purpose of using FEC codes is to ensure that minimal
number of packets need be received in order for a receiver to
reassemble an object. Reception overhead is used to measure how
close a protocol comes to achieving this minimum. Reception overhead
is the aggregate length of packets needed to recover the object
beyond the object length, measured as a percentage of the object
length. For example, if it takes 15 MB of packets in order to
recover a 10 MB object, then the reception overhead is (15 10)/10
times 100, or 50%. The minimal reception overhead possible is 0%.
2. FEC Codes
2.1. Simple codes
There are some very simple codes that are effective for repairing
packet loss under very low loss conditions. For example, one simple
way to provide protection from a single loss is to partition the
object into fixed size source symbols and then add a redundant symbol
that is the parity (XOR) of all the source symbols. The size of a
source symbol is chosen so that it fits perfectly into the payload of
a packet, i.e. if the packet payload is 512 bytes then each source
symbol is 512 bytes. The header of each packet contains enough
information to identify the payload. In this case, this includes a
symbol ID. The symbol IDs are numbered consecutively starting from
zero independently for the source symbols and for the redundant sym-
bol. Thus, the packet header also contains an encoding flag that
indicates whether they symbol in the payload is a source symbol or a
redundant symbol, where 1 indicates source symbol and 0 indicates
redundant symbol. For example, if the object consists of four source
symbols that have values a, b, c and d, then the value of the redun-
dant symbol is e = a XOR b XOR c XOR d. Then, the packets carrying
these symbols look like (0, 1: a), (1, 1: b), (2, 1: c), (3, 1: d),
(0, 0: e). In this example, the first two fields are in the header
of the packet, where the first field is the symbol ID and the second
field is the encoding flag. The portion of the packet after the
colon is the payload. Any single symbol of the object can be
recovered as the parity of all the other symbols. For example, if
packets (0, 1: a), (1, 1: b), (3, 1: d), (0, 0: e) are received then
the symbol value for the missing source symbol with ID 2 can be
recovered by computing a XOR b XOR d XOR e = c.
When the number of source symbols in the object is large, a simple
block code variant of the above can be used. In this case, the
source symbols are grouped together into source blocks of k
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consecutive symbols each, and then for each block of source symbols a
redundant symbol is added to form encoding blocks of k+1 symbols
each. Then, a source block can be recovered from any k of the k+1
symbols from the associated encoding block.
Slightly more sophisticated ways of adding redundant symbols using
parity can also be used. For example, one can group the k source sym-
bols in the object into a p x p square matrix, where p = sqrt(k).
Then, for each row a redundant symbol is added that is the parity of
all the source symbols in the row. Similarly, for each column a
redundant symbol is added that is the parity of all the source sym-
bols in the column. Then, Any row of the matrix can be recovered
from any p of the p+1 symbols in the row, and similarly for any
column. Higher dimensional product codes using this technique can
also be used. However, one must be wary of using these constructions
without some thought towards the possible loss patterns of symbols.
Ideally, the property that one would like to obtain is that if k
source symbols are encoded into n encoding symbols (the encoding sym-
bols consist of the source symbols and the redundant symbols) then
the k source symbols can be recovered from any k of the n encoding
symbols. Using the simple constructions described above does not
yield codes that come close to obtaining this ideal behavior.
2.2. Small block codes
Reliable IP multicast protocols may use a block (n, k) FEC erasure
code [BLA84]. A popular example of this types of codes is a class of
Reed-Solomon codes. For such codes, k source symbols are encoded into
n > k encoding symbols, such that any k of the encoding symbols can
be used to reassemble the original k source symbols. Thus, these
codes have 0% reception overhead when used to encode the entire
object directly. Block codes are usually systematic, which means
that the n encoding symbols consist of the k source symbols and n-k
redundant symbols generated from these k source symbols, where the
size of a redundant symbol is the same as that for a source symbol.
For example, the first simple code (XOR) described in the previous
subsection is a (k+1, k) code. In general, the freedom to choose n
larger than k+1 is desirable, as this can provide much better protec-
tion against losses. Codes of this sort are often based on alge-
braic methods using finite fields. Some of the most popular such
codes are based on linear block codes. Implementations of (n, k)
FEC erasure codes are efficient enough to be used by personal comput-
ers [RIZ97c, NON97]. For example, [Riz97b] describes an implementa-
tion where the encoding and decoding speeds decay as C/j, where the
constant C is on the order of 10 to 80 Mbytes/second for Pentium
class machines of various vintages and j is upper bounded by min(k,
n-k).
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In practice, the values of k and n must be small for these codes as
large values make encoding and decoding prohibitively expensive. As
many objects are longer than k symbols for reasonable values of k and
the symbol length (e.g. larger than 16 KB for k = 16 using 1 KB sym-
bols), they are divided into m source blocks consisting of k source
symbols each. An erasure code is used to encode each source block
into an encoding block consisting of n encoding symbols. For a
receiver to completely recover the object, k distinct encoding sym-
bols (i.e., with different symbol IDs) must be received for each of
the encoding blocks. For some encoding blocks, more than k encoding
symbols may be received, in which case any additional encoding sym-
bols are discarded. An example encoding structure is shown in Figure
1.
| source symbols | source symbols |
| of source block 0 | of source block 1 |
| |
v v
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|0 |1 |2 |3 |4 |5 |6 |7 |0 |1 |2 |3 | 4|5 |6 |7 |
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|
encode
|
v
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|0 |1 |2 |3 | 4| 5| 6| 7| 8| 9| 0| 1| 2| 3| 4| 5| 6| 7| 8| 9|
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
^ ^
| |
| encoding symbols | encoding symbols |
| of encoding block 0 | of encoding block 1 |
Figure 1. Encoding structure for object divided into m = 2 source
blocks, k = 8 and n = 10
When using small block codes for objects that are larger than k
source symbols in length, the source symbols in the object are
assigned to blocks. Typically, each k contiguous source symbols of
the object is assigned to a block, i.e., block c consists of the
range of source symbols [ck, (c+1)k-1]. This ensures that memory
reference are local when the sender reads source symbols to encode,
and when the receiver reads encoding symbols to decode.Locality of
reference is particularly important when the object is stored on
disk, as it reduces the disk seeks required.
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The block number and the source symbol ID within that block can be
used to uniquely specify a source symbol within the object. If the
size of the object is not a multiple of k source symbols, then the
last source block will contain less than k symbols.
Encoding symbols can be uniquely identified by block number and
encoding symbol ID. The block numbers can be numbered consecutively
starting from zero. One way of identifying encoding symbols within a
block are to use symbol IDs and an encoding flag that is used to
specify whether an encoding symbol is a source symbol or a redundant
symbol, where for example 1 indicates source symbol and 0 indicate
redundant symbol. The symbol IDs can be numbered consecutively
starting from zero for each block independently for the source sym-
bols and for the redundant symbols. Thus, an encoding symbol can be
identified by its block number, the encoding flag, and the symbol ID.
For example, if the object consists 10 source symbols with values a,
b, c, d, e, f, g, h, i, and j, and k = 5 and n = 8, then there are
two source blocks consisting of 5 symbols each, and there are two
encoding blocks consisting of 8 symbols each. Let p, q and r be the
values of the redundant symbols for the first encoding block, and let
x, y and z be the values of the redundant symbols for the second
encoding block. Then the encoding symbols together with their iden-
tifiers are (0, 0, 1:a), (0, 1, 1: b), (0, 2, 1: c), (0, 3, 1: d),
(0, 4, 1: e), (0, 0, 0: p), (0, 1, 0: q), (0, 2, 0: r), (1, 0, 1: f),
(1, 1, 1: g), (1, 2, 1: h), (1, 3, 1: i), (1, 4, 1: j), (1, 0, 0: x),
(1, 1, 0: y) and (1, 2, 0: z). In this example, the first three
fields identify the encoding symbol, where the first field is the
block number, the second field is the symbol ID and the third field
is the encoding flag. The value of the encoding symbol is written
after the colon. Each block can be recovered from any 5 of the 8
encoding symbols associated with that block. For example, reception
of (0, 1, 1: b), (0, 2, 1: c), (0, 3, 1: d), (0, 0, 0: p) and (0, 1,
0: q) are sufficient to recover the first source block and reception
of (1, 0, 1: f), (1, 1, 1: g), (1, 0, 0: x), (1, 1, 0: y) and (1, 2,
0: z) are sufficient to recover the second source block.
2.3. Large block codes
Tornado codes [LUB97] are block FEC erasure codes that provide an
alternative to small block codes. A (n, k) Tornado code requires
slightly more than k out of n encoding symbols to reassemble k source
symbols. However, the advantage is that the value of k may be on the
order of tens of thousands and still run efficiently. Because of
memory considerations, in practice the value of n is restricted to be
a small multiple of k, e.g., n = 2k. For example, [BYE98] describes
an implementation of Tornado codes where the encoding and decoding
speeds are proportional to 10 Mbytes/second to 80 Mbytes/second for
Pentium class machines of various vintages when k is in the tens of
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thousands and n = 2k. The reception overhead for such values of k
and n is in the 5-10% range. Tornado codes require a large amount of
out of band information to be communicated to all senders and
receivers for each different object length, and require an amount of
memory on the encoder and decoder which is proportional to the object
length times 2n/k.
Tornado codes are designed to have low reception overhead on average
with respect to reception of a random portion of the encoding pack-
ets. Thus, to ensure that a receiver can reassemble the object with
low reception overhead, the packets are permuted into a random order
before transmission.
2.4. On demand codes
All of the FEC erasure codes described up to this point are block
codes. There is a different type of FEC erasure code that we call on
demand codes. Like block codes, an on demand encoder operates on an
object of known size that is partitioned into equal length source
symbols. Unlike block codes, there is no predetermined number of
encoding symbols that can be generated for on demand codes.
Instead, an on demand encoder can generate as few or as many encoding
symbols as required on demand. Also unlike block codes, optimal on
demand codes have the additional attractive property that encoding
symbols for the same object can be generated and transmitted from
multiple servers and concurrently received by a receiver and yet the
receiver incurs a 0% reception overhead.
LT codes are an example of a large on demand FEC code. An LT encoder
uses randomization to generate each encoding symbol randomly and
independently of all other encoding symbols. An LT encoder satisfies
the on demand property, as it can generate as few or as many encoding
symbols as required on demand. Let k be the number of source symbols
that the object is partitioned into. An LT decoder has the property
that with very high probability the receipt of any set of slightly
more than k encoding symbols is sufficient to reassemble the object.
Like Tornado codes, the value of k may be very large, i.e., on the
order of tens or hundreds of thousands, and the encoder and decoder
run efficiently in software. For example the encoding and decoding
speeds for LT codes are proportional to 3 Mbytes/second to 20
Mbytes/second for Pentium class machines of various vintages when k
is in the high tens of thousands. The reception overhead for such
values of k is in the 2-4% range.
When a new encoding symbol is to be generated, it is based on a key
that uniquely describes how the encoding symbol is to be generated.
Because encoding symbols are randomly and independently generated, LT
codes have the property that encoding symbols for the same object can
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be generated and transmitted from multiple servers and concurrently
received by a receiver with no more reception overhead than if all
the encoding symbols were generated by a single sender.
There is a tradeoff between the number of source symbols and the
reception overhead, and the larger the number of source symbols the
smaller the reception overhead. Thus, for shorter objects, it is
sometimes advantageous to include multiple symbols in each packet.
Normally, and in the discussion below, there is only one symbol per
packet.
Like small block codes, there is a point when the object is large
enough that it makes sense to partition it into blocks when using LT
codes. Generally the object is partitioned into blocks whenever the
number of source symbols times the packet payload length is less than
the size of the object. Thus, if the packet payload is 1024 bytes
and the number of source symbols is 64,000 then any object over 64 MB
will be partitioned into more than one block. One can choose the
number of source symbols to partition the object into, depending on
the desired encoding and decoding speed versus reception overhead
tradeoff desired. Encoding symbols can be uniquely identified by a
block number (when the object is large enough to be partitioned into
more than one block) and an encoding symbol ID. The block numbers,
if they are used, are generally numbered consecutively starting from
zero within the object. The range of possible values for an encoding
symbol ID is orders of magnitude larger than the number of source
symbols in a block, i.e., on the range of possible values is gen-
erally in the billions. The block number and the encoding symbol ID
are both chosen uniformly and randomly from their range when an
encoding symbol is to be generated and transmitted. For example,
suppose the number of source symbols is 64,000 and the number of
blocks is 2. Then, each packet generated by the LT encoder could be
of the form (b, x: y). In this example, the first two fields iden-
tify the encoding symbol, where the first field is the block number b
= 0 or 1 and the second field is the randomly chosen encoding symbol
ID x. The value y after the colon is the value of the encoding sym-
bol.
3. Passing FEC coding information to receivers
There are two basic methods for passing FEC coding information to
receivers in order to decode an object: within the IP multicast
packet headers or through out of band methods. A description of the
variety of out of band methods is outside the scope of this document.
The FEC coding information can be classified as three types. FEC ses-
sion information is information needed by the FEC decoder that may
remain fixed for the transmission of many objects. FEC object
transmission information is information particular to the object
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transmission session needed by the FEC decoder. The FEC payload ID
identifies the symbols in the payload of the ALC packet.
FEC coding information include the FEC codec type, the source block
length, the symbol length, the object length, the encoding block
number, the encoding symbol ID, and an encoding flag indicating
whether the encoding symbol is a source symbol or a redundant symbol.
The FEC codec type, the source block length and the symbol length are
often FEC session information, although they may classified as FEC
object transmission information for some protocols. Thus, sometimes
this information is passed to the receiver out of band, although they
can equally well be included in each IP multicast packet header as
long as the amount of space they take within each packet is minimal.
The object length is part of FEC object transmission information.
Depending on the protocol, the object length is passed to the
receiver out of band or included within each IP multicast packet
header. The FEC payload ID consists of the encoding block number (if
used), the encoding symbol ID and the encoding flag. The FEC payload
ID must be contained within each IP multicast packet header.
4. Security Considerations
The use of FEC, in and of itself, imposes no additional security con-
siderations versus sending the same information without FEC. How-
ever, just like for any transmission system, a malicious sender may
intentionally transmit bad symbols. If a receiver accepts one or more
bad symbols in place of authentic ones then such a receiver will have
its entire object down-load corrupted by the bad symbol.
Application-level transmission object authentication can detect the
corrupted transfer, but the receiver must then discard the
transferred object. Thus, transmitting false symbols is at least an
effective denial of service attack. At worst, a malicious sender
could add, delete, or replace arbitrary data within the transmitted
object.
In light of this possibility, FEC receivers may screen the source
address of a received symbol against a list of authentic transmitter
addresses. Since source addresses may be spoofed, FEC transport pro-
tocols may provide mechanisms for robust source authentication of
each encoded symbol. Multicast routers along the path of a FEC
transfer may provide the capability of discarding multicast packets
that originated on that subnet, and whose source IP address does not
correspond with that subnet.
5. Intellectual Property Disclosure
Both Tornado codes and LT codes have patents pending.
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6. Acknowledgments
Thanks to Vincent Roca and Hayder Radha for their detailed comments
on this draft.
7. References
[AFZ95] Acharya, S., Franklin, M., and Zdonik, S., ``Dissemination-
Based Data Delivery Using Broadcast Disks'', IEEE Personal
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[BLA94] Blahut, R.E., ``Theory and Practice of Error Control Codes'',
Addison Wesley, MA 1984.
[BYE98] Byers, J.W., Luby, M., Mitzenmacher, M., and Rege, A., ``A
Digital Fountain Approach to Reliable Distribution of Bulk Data'',
Proceedings ACM SIGCOMM '98, Vancouver, Canada, Sept 1998.
[DEE88] Deering, S., ``Host Extensions for IP Multicasting'', RFC
1058, Stanford University, Stanford, CA, 1988.
[GEM99] Gemmell, J., Schooler, E., and Gray, J., ``ALC Scalable
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1999.
[HAN98] Handley, M., and Jacobson, V., ``SDP: Session Description
Protocol'', RFC 2327, April 1998.
[HAN96] Handley, M., ``SAP: Session Announcement Protocol'', Internet
Draft, IETF MMUSIC Working Group, Nov 1996.
[LUB97] Luby, M., Mitzenmacher, M., Shokrollahi, A., Spielman, D.,
Stemann, V., ``Practical Loss-Resilient Codes'' 29th STOC'97.
[LUB99] Luby, M., Vicisano, L., Speakman, T. ``Heterogeneous multicast
congestion control based on router packet filtering'', presented at
RMT meeting in Pisa, March 1999.
[R2068] Fielding, R., Gettys, J., Mogul, J. Frystyk, H., Berners-Lee,
T., Hypertext Transfer Protocol HTTP/1.1 (IETF RFC2068)
http://www.rfc-editor.org/rfc/rfc2068.txt
[R2119] Bradner, S., Key words for use in RFCs to Indicate Requirement
Levels (IETF RFC 2119) http://www.rfc-editor.org/rfc/rfc2119.txt
[RIZ97a] Rizzo, L, and Vicisano, L., ``Reliable Multicast Data
Distribution protocol based on software FEC techniques'', Proceedings
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of the Fourth IEEES Workshop on the Architecture and Implementation of
High Performance Communication Systems, HPCS-97, Chalkidiki, Greece,
June 1997.
[RIZ97b] Rizzo, L., and Vicisano, L., ``Effective Erasure Codes for
Reliable Computer Communication Protocols'', ACM SIGCOMM Computer
Communication Review, Vol.27, No.2, pp.24-36, Apr 1997.
[RIZ97c] Rizzo, L., ``On the Feasibility of Software FEC'', DEIT Tech
Report, http://www.iet.unipi.it/~luigi/softfec.ps, Jan 1997.
[RUB99] Rubenstein, Dan, Kurose, Jim and Towsley, Don, ``The Impact of
Multicast Layering on Network Fairness'', Proceedings of ACM SIGCOMM'99.
[VIC98A] L.Vicisano, L.Rizzo, J.Crowcroft, ``TCP-like Congestion
Control for Layered Multicast Data Transfer'', IEEE Infocom '98, San
Francisco, CA, Mar.28-Apr.1 1998.
[VIC98B] Vicisano, L., ``Notes On a Cumulative Layered Organization
of Data Packets Across Multiple Streams with Different Rates'',
University College London Computer Science Research Note RN/98/25,
Work in Progress (May 1998).
8. Authors' Addresses
Michael Luby
luby@dfountain.com
Digital Fountain
600 Alabama Street
San Francisco, CA, USA, 94110
Lorenzo Vicisano
lorenzo@cisco.com
cisco Systems, Inc.
170 West Tasman Dr.,
San Jose, CA, USA, 95134
Luigi Rizzo
luigi@iet.unipi.it
Dip. di Ing. dell'Informazione
Universita` di Pisa
via Diotisalvi 2, 56126 Pisa, Italy
Jim Gemmell
jgemmell@microsoft.com
Microsoft Research
301 Howard St., #830
San Francisco, CA, USA, 94105
Luby, Vicisano, Rizzo, Gemmell, Crowcroft, Lueckenhoff [Page 12]
Internet Draft RMT BB, Forward Error Correction Codes March 2000
Jon Crowcroft
J.Crowcroft@cs.ucl.ac.uk
Department of Computer Science
University College London
Gower Street,
London WC1E 6BT, UK
Bruce Lueckenhoff
brucelu@cadence.com
Cadence Design Systems, Inc.
120 Cremona Drive, Suite C
Santa Barbara, CA 93117
Luby, Vicisano, Rizzo, Gemmell, Crowcroft, Lueckenhoff [Page 13]
Internet Draft RMT BB, Forward Error Correction Codes March 2000
Luby, Vicisano, Rizzo, Gemmell, Crowcroft, Lueckenhoff [Page 14]
