Internet DRAFT - draft-lacan-rmt-fec-bb-rs
draft-lacan-rmt-fec-bb-rs
Reliable Multicast Transport J. Lacan
Internet-Draft ENSICA/LAAS-CNRS
Expires: April 20, 2006 V. Roca
INRIA
J. Peltotalo
S. Peltotalo
Tampere University of Technology
October 17, 2005
draft-lacan-rmt-fec-bb-rs-00
Reed Solomon Error Correction Scheme
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Copyright Notice
Copyright (C) The Internet Society (2005).
Abstract
This document describes a Fully-Specified FEC scheme for the Reed-
Solomon forward error correction code and its application to reliable
delivery of data objects on the packet erasure channel.
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The Reed-Solomon codes belong to the class of Maximum Distance
Separable (MDS) codes, i.e, they enable a receiver to recover the k
source symbols from any set of k received symbols.
The implementation described here is compatible with the IPR-free
implementation described in [5].
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 3
3. Definitions Notations and Abbreviations . . . . . . . . . . . 3
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 Abbreviations . . . . . . . . . . . . . . . . . . . . . . 4
4. Formats and Codes . . . . . . . . . . . . . . . . . . . . . . 4
4.1 FEC Payload IDs . . . . . . . . . . . . . . . . . . . . . 4
4.2 FEC Object Transmission Information . . . . . . . . . . . 5
4.2.1 Mandatory Elements . . . . . . . . . . . . . . . . . . 5
4.2.2 Common Elements . . . . . . . . . . . . . . . . . . . 5
4.2.3 Scheme-Specific Elements . . . . . . . . . . . . . . . 6
4.2.4 Encoding Format . . . . . . . . . . . . . . . . . . . 6
5. Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 7
5.1 Determining the Maximum Source Block Length (B) . . . . . 7
5.2 Determining the Number of Encoding Symbols of a Block . . 7
6. Reed-Solomon Codes . . . . . . . . . . . . . . . . . . . . . . 8
6.1 Finite field . . . . . . . . . . . . . . . . . . . . . . . 8
6.2 Reed-Solomon Encoding Algorithm . . . . . . . . . . . . . 9
6.3 Reed-Solomon Decoding Algorithm for the Erasure Channel . 10
6.4 Implementation . . . . . . . . . . . . . . . . . . . . . . 11
6.4.1 Implementation for the Packet Erasure Channel . . . . 11
7. Security Considerations . . . . . . . . . . . . . . . . . . . 12
8. Intellectual Property . . . . . . . . . . . . . . . . . . . . 12
9. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 12
10. References . . . . . . . . . . . . . . . . . . . . . . . . . 13
10.1 Normative References . . . . . . . . . . . . . . . . . . . 13
10.2 Informative References . . . . . . . . . . . . . . . . . . 13
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . 14
Intellectual Property and Copyright Statements . . . . . . . . 15
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1. Introduction
Forward Error Correction (FEC) is one of the most classical solutions
to improve the reliability of multicast or real-time transmissions.
The documents [2] and [3] describe a general framework to use FEC in
the context of data transport. The companion document [4] describes
some applications of FEC codes for content delivery.
Recent FEC schemes [6] or [7] proposed propose erasure codes based
on sparse graphs or matrices. These codes are are efficient in terms
of CPU but not optimal in terms of correction capability, at least
for short lengths.
The FEC scheme presented in this document belongs to the class of
Maximum-Distance Separable codes, i.e. it is optimal in terms of
erasure correction capability. In others words, it enables the
receiver to recover the k source symbols from any set of k encoding
symbols.
Even the coding/decoding complexity is larger than the one of [6] or
[7], this family of codes could be very useful for applications
requiring short length codes (e.g. video and audio streaming).
Actually, many multicast applications already use packet-based Reed-
Solomon codes. Most of these implementations are derived from the
free implementation of Luigi Rizzo [5]. The implementation proposed
in this document is compatible with this one.
2. Terminology
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in RFC 2119 [1].
3. Definitions Notations and Abbreviations
3.1 Definitions
This document uses the same terms and definitions as those specified
in [3]. Additionally, it uses the following definitions:
o Source symbol: unit of data used during the encoding process. The
source symbols are a m-bit vectors considered as an element of a
finite field.
o Encoding symbol: unit of data generated by the encoding process.
The encoding symbols are a m-bit vectors.
o Encoding block: set of encoding symbols generated by an encoding
process.
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o Repair symbol: encoding symbols which are not source symbols.
o Systematic code: a code in which the source symbols are included
as part of the encoding symbols
o Source block: a block of k source symbols which are considered
together for the encoding.
o Encoding Symbol Group: a group of encoding symbols that are sent
together, within the same packet, and whose relationships to the
source object can be derived from a single Encoding Symbol ID.
o Source Packet: a data packet containing source symbols. Note that
the source symbols included in a same packet does not belong to
the same encoding block.
o Repair Packet: a data packet containing repair symbols. Note that
the repair symbols included in a same packet does not belong to
the same encoding block.
o Finite field size parameter, r: this parameter defines the number
of elements in the finite field, q=2^^r.
3.2 Notations
This document uses the following notations:
o L denotes the object transfer length in bytes
o k denotes the number of source symbols
o n denotes the number of repair symbols
o N denotes the number of source blocks into which the object shall
be partitioned
o rate denotes the so-called "code rate", i.e. the k/n ratio
o a ^^ b a raised to the power b
o I_k denotes the k*k identity matrix
o sz denotes the size of the packets
3.3 Abbreviations
This document uses the following abbreviations:
o ESI Encoding Symbol ID
o RS Reed-Solomon
o MDS Maximum Distance separable Code
o F_q finite field with q elements
4. Formats and Codes
4.1 FEC Payload IDs
The FEC Payload ID is composed of the Source Block Number and the
Encoding Symbol ID:
The Source Block Number (20 bit field) identifies from which
source block of the object the encoding symbol(s) in the payload
is(are) generated. There is a maximum of 2^^20 blocks per object.
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The Encoding Symbol ID (12 bit field) identifies which specific
encoding symbol generated from the source block is carried in the
packet payload. There is a maximum of 2^^12 encoding symbols per
block. The first k values (0 to k-1) identify source symbols, the
remaining n-k values identify repair symbols.
There MUST be exactly one FEC Payload ID per packet. In case of en
Encoding Symbol Group, when multiple encoding symbols are sent in the
same packet, the FEC Payload ID refers to the first symbol of the
packet. The other symbols can be deduced from the ESI of the first
symbol by incrementing sequentially the ESI.
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
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Source Block Number (20 bits) | Encoding Symbol ID |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 1: FEC Payload ID encoding format for FEC Encoding ID XX
4.2 FEC Object Transmission Information
4.2.1 Mandatory Elements
o FEC Encoding ID: the Fully-Specified FEC Schemes described in this
document use the FEC Encoding ID XX.
4.2.2 Common Elements
The following elements MUST be used with the present FEC Scheme:
o Transfer-Length (L): a non-negative integer indicating the length
of the object in bytes. There are some restrictions on the
maximum Transfer-Length that can be supported:
maximum transfer length = 2^^20 * B * E
For instance, if B=2^^8 (because the codec operates on a 2^^8
finite field), and if E=1024 bytes, then the maximum transfer
length is 2^^38 bytes (i.e. a bit more than 274 Giga Bytes). It
is expected that other FEC codes (e.g. LDPC codes) or another RS
FEC Scheme be used for larger objects.
o Encoding-Symbol-Length (E): a non-negative integer indicating the
length of each encoding symbol in bytes.
o Maximum-Source-Block-Length (B): a non-negative integer indicating
the maximum number of source symbols in a source block.
o Max-Number-of-Encoding-Symbols (max_n): a non-negative integer
indicating the maximum number of encoding symbols generated for
any source block.
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Section 5 explains how to derive the values of each of these
elements.
4.2.3 Scheme-Specific Elements
o Finite Field size parameter, r (optional): The r parameter defines
the finite field size composed of q=p^^r elements. The r=8 value
is the default. When no finite field size parameter is
communicated to the decoder, then this latter MUST assume that
r=8.
4.2.4 Encoding Format
This section shows two possible encoding formats of the above FEC
OTI. The present document does not specify when or how these
encoding formats should be used.
4.2.4.1 Using the General EXT_FTI Format
The FEC OTI binary format is the following, when the EXT_FTI
mechanism is used.
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
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| HET = 64 | HEL | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +
| Transfer-Length (L) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| 0 (not applicable) | Encoding Symbol Length (E) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Max Source Block Length (B) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Max Nb of Enc. Symbols (max_n) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
. Optional finite field size parameter (r) .
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
The HEL (Header Extension Length) indicates whether the optional
finite field size parameter, r, is present or not.
4.2.4.2 Using the FDT Instance (FLUTE specific)
When it is desired that the FEC OTI be carried in the FDT Instance of
a FLUTE session, the following XML elements must be described for the
associated object:
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o FEC-OTI-Transfer-length
o FEC-OTI-Encoding-Symbol-Length
o FEC-OTI-Maximum-Source-Block-Length
o FEC-OTI-Max-Number-of-Encoding-Symbols
o FEC-OTI-Finite-Field-Size-Parameter (optional)
When no finite field size parameter is to be carried in the FEC OTI,
the sender simply omits the FEC-OTI-Finite-Field-Size-Parameter
element.
5. Procedures
This section defines procedures for FEC Encoding ID XX.
5.1 Determining the Maximum Source Block Length (B)
The B parameter (maximum source block length in symbols) depends on
several parameters: the finite field size parameter, r, the code rate
(rate), as well as possible internal codec limitations.
The finite field size parameter, r, defines the number of elements in
this field, q=2^^r, which is also the maximum number of encoding
symbols for a source block (max_n). When r=8 (default):
max1_B = 2 ^^ 8
Additionally, a codec MAY impose other limitations on the maximum
block size. Yet it is not expected that such limits exist when using
r=8 (default). This decision SHOULD be clarified at implementation
time, when the target use case is known. This results in a max2_B
limitation.
Then, B is given by:
B = min(max1_B, max2_B)
Note that this calculation is only required at the coder, since the B
parameter is communicated to the decoder through the FEC OTI.
5.2 Determining the Number of Encoding Symbols of a Block
The following algorithm, also called "n-algorithm", explains how to
determine the actual number of encoding symbols for a given block.
AT A SENDER:
Input:
B Maximum source block length, for any source block. Section 5.1
explains how to determine its value.
k Current source block length. This parameter is given by the
source blocking algorithm.
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rate FEC code rate, which is given by the user (e.g. when starting
a FLUTE sending application) for a given use case. It is
expressed as a floating point value.
Output:
max_n Maximum number of encoding symbols generated for any source
block
n Number of encoding symbols generated for this source block
Algorithm:
a. max_n = floor(B / R)
b. n = floor(k * max_n / B)
AT A RECEIVER:
Input:
B Extracted from the received FEC OTI
max_n Extracted from the received FEC OTI
k Given by the source blocking algorithm
Output:
n
Algorithm:
a. n = floor(k * max_n / B)
6. Reed-Solomon Codes
Reed-Solomon (RS) codes form a special class of linear block codes,
which offer maximum erasure correction capability. A [n,k]-RS code
encodes a sequence of k source symbols defined over a finite field
F_q into a sequence of n repair symbols, where n is upperbounded by
q-1. The implementation described here is based on a Vandermonde
matrix. The n symbols resulting from the encoding processing do not
include the source symbols. Depending on the application, the
encoding symbols can only be composed of the n repair symbols (non
systematic case) or can also included the source symbols (systematic
case).
6.1 Finite field
A finite field F_q is defined as a finite set of q elements which
have a structure of field. It contains necessarily q=p^r elements,
where p is a prime number. In the practical context of data
networks, p is always set to 2. The elements of the field F_(2^r)
can be represented by polynomials with binary coefficients (i.e. over
F_2) of degree less than r. The polynomials can be associated to
binary vectors of length r. For example, the vector (11001)
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represents the polynomial 1+x+x^^4. This representation is often
called polynomial representation. The addition between two elements
is defined as the addition of binary polynomials in F_2 and the
multiplication is the multiplication modulo a given irreducible (i.e.
non-factorizable) polynomial of degree r with coefficients in F_2.
Since a finite field F_q is completely characterized by the
irreducible polynomial, we propose the following polynomials to
represent the field F_(2^^r), for r varying from 2 to 16 :
r=2, "111" (1+x+x^^2)
r=3, "1101", (1+x+x^^3)
r=4, "11001", (1+x+x^^4)
r=5, "101001", (1+x^^2+x^^5)
r=6, "1100001", (1+x+x^^6)
r=7, "10010001", (1+x^^3+x^^7)
r=8, "101110001", (1+x^^2+x^^3+x^^4+x^^8)
r=9, "1000100001", (1+x^^4+x^^9)
r=10, "10010000001", (1+x^^3+x^^10)
r=11, "101000000001", (1+x^^2+x^^11)
r=12, "1100101000001", (1+x+x^^4+x^^6+x^^12)
r=13, "11011000000001", (1+x+x^^3+x^^4+x^^13)
r=14, "110000100010001", (1+x+x^^6+x^^10+x^^14)
r=15, "1100000000000001", (1+x+x^^15)
r=16, "11010000000010001", (1+x+x^^3+x^^12+x^^16)
For implementation issues, these polynomials are also primitive and
contain the minimum number of monomials.
6.2 Reed-Solomon Encoding Algorithm
The encoding algorithm produces a block of repair symbols c=(c_0,
..., c_(n-1) ) over F_q from an source block of k symbols i=(i_0,
..., i_(k-1) ) over F_q.
The linear codes can be encoded by multiplying the source block by a
generator matrix G of k rows and n columns over F_q. Thus c = i * G.
The definition of the generator matrix completely characterizes the
code.
Let us consider that n = q-1 and 0< k <= n. Let us denote alpha a
primitive element of F_q, i.e. any element of F_q can be expressed as
a power of alpha.
The entry g_{i,j} of the generator matrix G of an RS code is equal to
alpha^^(u*v), where 0<= u <= k-1 and 0<= u <= n-1. This matrix is a
called a Vandermonde matrix. Note that, for practical applications,
the length of the code can be shortened to n'<n by considering the
sub-matrix formed by the n' first columns of G.
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Since the defined generator matrix does not contain the k*k-identity
matrix I_k, the erasure code is not systematic, i.e. the source
symbols do not belong to the set of produced symbols. Since network
applications often require systematic codes, [5] proposes to use
the matrix (I_k | G) as generator matrix. In practical, this is
equivalent to consider that the encoding symbols are composed of the
source and the repair symbols. The number of encoding symbols is
then n+k. In the following, we will use "systematic form" to design
the use of such matrix.
The only drawback of this solution is that the obtained code is not
strictly MDS. Indeed, there exist few patterns of k encoding symbols
which do not allow the recovery of the k source symbols. However,
the proportion of non-recoverable patterns of k received symbols is
extremely low. For example, for q=2**8, k=16 and n=32 (16 source
symbols and 16 repair symbols), this proportion is less than 0.5 per-
cent. Moreover, this proportion decreases when q grows. [9] gives a
theoretical upper-bound of this proportion.
It should be noted that there exist systematic "striclty-MDS" codes.
The non-identity part of the corresponding generator matrix can be
build from a Cauchy matrices [10] or from two distinct Vandermonde
matrices [11]. However, despite this small drawback and due to
their current wide deployment, Vandermonde matrices seems to be the
best candidate to implement MDS codes. The other main advantage is
the encoding/decoding complexity better than the other solutions (see
below).
The encoding complexity is the one of the multiplication i*G, where G
is a k*n-Vandermonde matrix. Thanks to the properties of the
Vandermonde, the complexity of the matrix-vector multiplication,
which is classically k*n (i.e. k operations per repair symbol), can
be reduced to O(log(k)) operations per repair symbol by using Fast
Fourier Transform.
6.3 Reed-Solomon Decoding Algorithm for the Erasure Channel
The Reed-Solomon decoding algorithm for the erasure channel allows
the recovery of the k source symbols from any set of k received
symbols. It is based on the fundamental property of the generator
matrix which is such that any k*k-submatrix is nonsingular (see
[8]). The first step of the decoding consists in extracting the k*k-
submatrix of the generator matrix obtained by considering the columns
corresponding to the received symbols. Indeed, since any encoding
symbol is obtained by multiplying the source block by one column of
the generator matrix, the received block of k encoding symbols can be
considered as the result of the multiplication of the source block by
a k*k-submatrix of the generator matrix. Since this submatrix is
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nonsingular, the second step of the algorithm is to invert this
matrix and to multiply the received block by the obtained matrix to
recover the source block.
As mentioned in the last section, the systematic form admits few non-
recoverable patterns. For these patterns, the corresponding
submatrix is singular. The only solution to perform the decoding is
to use an additional symbol and to insert the corresponding column
into the k*k-submatrix to obtain a matrix of rank k.
The decoding complexity includes the operations for the matrix
inversion and for the matrix-vector multiplication. For the non-
systematic form, the submatrix is a Vandermonde matrix and its
inversion can be performed in O(k^^2) operations. The matrix-vector
multiplication requires O(k^^2) operations (i.e. O(k) operations per
source symbol). Note that there exist some algorithms performing
this matrix-vector multiplication in O(log^^2(k)) operations per
source symbol (see [12]).
6.4 Implementation
6.4.1 Implementation for the Packet Erasure Channel
A packet erasure channel is defined as a channel where the data are
transmitted by packet. Each packet is received correctly or erased.
The location of the erased packets in the sequence of packets is
known. The implementation presented here describes the use of Reed-
Solomon codes for generating redundant packets from k source packets
and to recover the source packets from k received packets.
The k source packets are assumed to be composed of sz m-bit sets
(usually m=8 or 16). Each m-bit set is associated to an element of
the finite field F_(2^^m) through the polynomial representation (see
Section Section 6.1). If some of the source packets contain less
than sz elements, they are virtually padded with zero elements.
The encoding processing produces n packets of sz elements from the k
source packets by encoding sz Reed-Solomon encoding blocks from sz
source blocks (see Figure 3). The j-th source block is composed of
the j-th element of each of the source packets. Similarly, the j-th
encoding block is composed of the j-th element of each encoding
packet. The encoding packets thus contain sz m-bit elements.
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------------ --------------- -------------------
| | | | | | | | | | | |
| | | | | * | generator | = | | | | |
| | | | | | matrix | | | | | |
| | | | | | G | | | | | |
source|--------------| | | |---------------------|
block | | | | | | | --------------- ->| | | | | | |
j |--------------| / |---------------------|
| | | | | / | | | | |
| | | | | encoding | | | | |
| | | | | block | | | | |
| | | | | j | | | | |
| | | | | | | | | |
| | | | | | | | | |
------------ -------------------
k source packets n encoding packets
Figure 3: Packet encoding scheme
It should be noted that the number of generated packets (and
transmitted on the network) can be variable and adapted on demand.
The only constraint is the maximum number depending on the finite
field size (see Section Section 6.1)
The main interest of this scheme is that the losses of some of the
encoding packets produce the same erasure pattern for each of the sz
RS encoding blocks. It follows that the matrix inversion must be
done only once for the sz encoding blocks. For large sz, this
complexity cost of the inversion becomes negligible compared to the
sz matrix-vector multiplications.
7. Security Considerations
The security considerations for this document are the same as they
are for RFC 3452 [2].
8. Intellectual Property
To the best of our knowledge, there is no patent or patent
application identified as being used in the Reed-Solomon FEC scheme.
Yet other flavors of Reed-Solomon codes and associated techniques MAY
be covered by Intellectual Property Rights.
9. Acknowledgments
10. References
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10.1 Normative References
[1] Bradner, S., "Key words for use in RFCs to Indicate Requirement
Levels", RFC 2119.
[2] Luby, M., "Forward Error Correction (FEC) Building Block",
RFC 3452, December 2002.
[3] Watson, M., "Forward Error Correction (FEC) Building Block
(revised)", (draft-ietf-rmt-fec-bb-revised-00 : Work in
progress), April 2005.
[4] Luby, M., "The Use of Forward Error Correction (FEC) in Reliable
Multicast", RFC 3453, December 2002.
10.2 Informative References
[5] Rizzo, L., "Effective Erasure Codes for Reliable Computer
Communication Protocols", ACM SIGCOMM Computer Communication
Review Vol.27, No.2, pp.24-36, April 1997.
[6] Luby, M., "Raptor Forward Error Correction Scheme", Internet
Draft (draft-ietf-rmt-bb-fec-raptor-object-01 : work in
progress), June 2005.
[7] Roca, V., "Low Density Parity Check (LDPC) Forward Error
Correction", Internet Draft (draft-roca-rmt-ldpc-00.txt : work
in progress), June 2005.
[8] Mac Williams, F. and N. Sloane, "The Theory of Error Correcting
Codes", North Holland, 1977 .
[9] Shparlinski, I., "On the singularity of generalised Vandermonde
matrices over finite fields", Finite Fields and Their Appl.,
2005, v.11, 193-199. .
[10] Bloemer, J., "Error Control Coding: Fundamentals and
Applications", An XOR-Based Erasure-Resilient Coding Scheme",
ICSI report TR-95-048, August 1995. .
[11] Lacan, J. and J. Fimes, "Systematic MDS Erasure Codes based on
Vandermonde Matrices", IEEE Communications Letters, pp. 570-
572, Vol. 8, Issue 9, Sept. 2004. .
[12] Gohberg, I. and V. Olshevsky, "Fast algorithms with
preprocessing for matrix-vector multiplication problems",
Journal of Complexity, pp. 411-427, vol. 10, 1994 .
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Authors' Addresses
Jerome Lacan
ENSICA/LAAS-CNRS
1, place Emile Blouin
Toulouse 31056
France
Email: jerome.lacan@ensica.fr
URI:
Vincent Roca
INRIA
655, av. de l'Europe
Zirst; Montbonnot
ST ISMIER cedex 38334
France
Email: vincent.roca@inrialpes.fr
URI:
Tampere University of Technology
P.O. Box 553 (Korkeakoulunkatu 1)
Tampere FIN-33101
Finland
Email: jani.peltotalo@tut.fi
URI:
Tampere University of Technology
P.O. Box 553 (Korkeakoulunkatu 1)
Tampere FIN-33101
Finland
Email: sami.peltotalo@tut.fi
URI:
Lacan, et al. Expires April 20, 2006 [Page 14]
Internet-Draft draft-lacan-rmt-fec-bb-rs-00 October 2005
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Lacan, et al. Expires April 20, 2006 [Page 15]
