Several error control coding techniques for reliable satellite communications were investigated to find algorithms for fast decoding of Reed-Solomon codes in terms of dual basis. The decoding of the (255,223) Reed-Solomon code, which is used as the outer code in the concatenated TDRSS decoder, was of particular concern

Lin, S.

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https://ntrs.nasa.gov/search.jsp?R=19850008948 2020-03-20T20:37:13+00:00Z
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NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
Goddard Space Flight Center
Greenbelt, Maryland
CODING FOR RELIABLE SATELLITE COMMUNICATIONS
(NASA —CR-17431;) C0D1N^i FCn BELIAB .LE 	 NE5-17257
SATELLITE CObBUNICATILNS Seaiaanual Report
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Shu Lin
Principal Investigator
Department of Electrical Engineering
University of Hawaii at Manoa
Honolulu, Hawaii 96822
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CODING POP RELIABLE SATELLITE COMMUNICATIONS
ABSTRACT
This research project is set up to investigate several error control
coding techniques for reliable satellite communications. The main objective
of the research is to find algorithms for fast decoding of Reed-Solomon codes
in terms of dual basis. Particularly, we are concerned with the decoding of
the (255,223) Reed-Solomon code which is used as the outer code in the conca-
tenated TDRSS decoder.
Research effort will also be expended in the following problems: (1) to
investigate the performance (error probability and throughput) of a concate-
nated coding scheme using a Reed-Solomon code as the outer code for error
detection only; (2) to investigate the synchronization recovery properties of
Reed-Solomon and extended Reed-Solomon codes; (3) using hybrid coding schemes
(error correction and detection incorporated with retransmission) to improve
system reliability and throughput in satellite communications; and (4) using
coding techniques to resolve collisions in packet satellite communications.
x
I. INTRODUCTION
This research project is set up to investigate several coding techniques
for controlling transmission errors in satellite communications. The main
task of this research is to find algorithms for fast decoding of Reed-Solomon
codes (1-6). Reed-Solomon codes have been applied by NASA for controlling
transmission errors it, satellite communications. For example, the (255,223)
Reed-Solomon code with symbols from GF(2 8 ) has been adopted as the outer
code of a concatenated coding scheme used in NASA's TDRSS System. Recently
Berlekamp (7) devised a new method for encoding Reed-Solomon codes which
greatly reduces the encoding complexity. Berlekamp's encoder is implemented
in terms of dual basis using bit-serial multiplication. In this research, we
will investigate decoding of Reed-Solomon codes in terms of dual basis. Par-
ticularly, we are concerned with the decoding of the (255,223) Reed-Solomon
code used in NASA's TDRSS System. We also intend to suggest possible decoder
architectures which are capable of achieving high speed decoding.
Another task of the proposed research is to investigate the performance
(error probability) of a concatenated coding scheme in which the inner code is
used for both error correction and detection but the outer code is used for
error detection only. A retransmission is requested if the outer code detects
the presence of errors after the inner code decoding. Such a scheme is being
considered for NASA telecommand operation. In this research we will consider
using a Reed-Solomon code as the outer code for error detection.
In random-access packet satellite packet communications, a major cause of
congestion is the channel wastage due to packet collisions. In this research,
we will investigate using coding techniques to resolve packet collisions and
improve the system throughout.
-2-
II. SUMMARY OF RESEARCH PROGRESS
During the first six months of the project period, our research effort
has been divided in the first two areas of the research project, namely
decoding of Reed-Solomon codes in terms of dual basis and performance study of
a concatenated coding scheme for NASA Telecommond System. We will summarize
our research results in this section.
1. Decoding of Reed-Solomon Codes in Dual Basis
Reed-Solomon codes form a class of very powerful cyclic block codes.
They are widely used for controlling transmission errors in data communication
systems as well as data storage systems. Recently Berlekamp [7) devised a new
method for encoding these codes which greatly reduces the encoding com-
plexity. Berlekamp's encoder is implemented in terms of dual basis using
bit-serial multipliers.
During the first six months of the project period, we have investigated
decoding of Reed-Solomon codes using the dual basis. The decoding algorithm
being used is the Peterson-Berlekamp-Chien algorithm. The algorithm consists
of four steps:
1. Compute the syndrome S = (S1,S2, ... S 2t ) from the received
polynomial r(X).
2. Determine the error-location polynomial a(X) from the syndrome 8.
3. Determine the error-value evaluator Z(X) from the syndrome.
4. Evaluate the error-location numbers and error values, and perform
error correction
All the four decoding steps can be carried out in dual basis using bit-serial
multiplications or combinati-,:)n of bit-serial multiplications and parallel
multiplications. The circuits for the four decoding steps are shown in
-3-
Figures 1 to 3. An organization for a Reed-Solomon code decoder is shown in
Figure 4.
A technical report on the decoding of Reed-Solomun codes in the dual form
has been completed and sent to NASA Goddard Space Flight Center.
2. Performance Study of a Concatenated Coding Scheme for
NASA Telecommand System
During the first half of the project period, we have also investigated a
concatenate( coding scheme for error control in data communications. In this
scheme, the inner code is used for both error correction and error detection,
however the outer code is used only for error detection. A retransmission is
requested if the outer code detects the presence of errors after the inner
code decoding. Probability of undetected error is derived and bounded. A
particular scheme proposed for NASA Telecommand system is analyzed.
In the scheme proposed for NASA Telecommand system, both inner code and
outer code are shortened Hamming codes. The inner code is a distance-4
shortened Hamming code with generator polynomial,
g(X) - (X+1)(X6+X+1) - X7+X6+X2+1 .
This code is capable of correcting any single error and detecting any double
errors. The outer code is also a distance-4 shortened Hamming code with
generator polynomial,
g(X)	 X16 + X12 + X5 + 1 .
This code is the X.25 standard for packet-switched data network (8). The 16
parity bits of this code is used for error detection only. The reliability
performance of the above scheme is analyzed. We have shown that, for bit-
error-rate less than 10 -5 , the scheme provides extremely high reliability.
-4-
A technical report on the performance study of the concatenated coding
scheme described above has been completed and sent to NASA Goddard Space
Flight Center.
we have also analyzed the NASA scheme using a shortened Reed-Solomon code
as the outer code. The Reed-Solomon code used has the generator polynomial
g-(X)-(X+l)(X+a) with symbols from the Galois field GF(2 8 ). Our analysis
shows that the scheme with this Reed-Solomon code also gives extremely good
reliability. A report on this analysis is in progress and will be completed
soon.
III. RESEARCH ACTIVITIES
Publications
1. 'On the Probability of Undetected Error for the Maximum Distance
Separable Codes,' IEEE Transactions on Communications., Vol. CON-32, No.
9, September, 1984.
2. "On the Undetected Error Probability for Shortened Hamming Codes,'
accepted for publication in IEEE Transactions on Communications, 1985.
3. "Encoding and Decoding of Reed-Solomon Codes in Dual Basis,' Technical
Report, NASA Grant NAG 5-407, October, 1984.
4. 'Probability of Undetected Error After Decoding for a Concatenated Coding
Scheme For Error Control,' Technical Report NASA Grant 5-407, July, 1984.
Presentations
1. 'Encoding and Decoding of Reed-Solomon Codes in Dual Basis,' Seminar,
Osaka University, October 31, 1984.
2. 'Probability of Undetected Error after Decoding for a Concatenated Coding
Scheme,' 7th Conference on Information Theory and Its Applications,
Kinaugawa, Japan, November 4-7, 1984.
3. 'Future Problems in Coding' (guest speaker), 7th Conference on Informa-
tion Theory and Its Applications, Kinaugawa, Japan, November 4-7, 1984.
4. "Performance Analysis of a Concatenated Coding Scheme for Error Control,'
1984 Globecom, Atlanta, Georgia, November 26-29, 1984.
-5-
Consultation
on November 29, 1984, Dr. Lin visited NA4A Goddard Space Flight Center
and discussed with Dr. James Morakis and Mr. Warner H. Miller on a number of
future research problems.
REFERENCES
1. I.S. Reed and G. Solomon, •Polyno ial Codes over Certain Finite Fields,"
J. Soc. Ind. Appl. Math., 8, pp. 300-304, June, 1960.
2. E.R. Berlekamp, Algebraic Coding Theory, MCGrSW-Hill, 1968.
3. W.W. Peterson and E.J. Weldon, Jr., Error-Correcting Codes, MIT Press,
1972.
4. F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes,
North-Holland, 1977.
5. R.E. Blahut, Theory and Practice of Error Control Coding, Addison Wesley,
1983.
6. S. Lin and D.J. Costello, Jr., Error Control Coding: Fundamentals and
Applications, Prentice-Hall, 1982.
7. E.R. Berlekamp, 'Bit-Serial Reed-Solomon Encoders,' IEEE Transactions on
Information Theory, Vol. IT-28, No. 6, pp. 869-874, November, 1982.
8. CCITT: Recommendation X.25, • interface Between Data Terminal Equipment
and Data Circuit-Terminating Equipment for Terminals Operating in Packet
Mode on Public Data Networks,' with Plenary Assembly, Doc. No. 7, Geneva,
1980.
-6-
Figure 1 A circuit for computing the syndrome component
Si with a bit-serial multiplier
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Coding for reliable satellite communications

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