Joint Mathematics Meetings
Washington, DC, January 5-8, 2009 (Monday - Thursday)
Error-Correction Coding Using
Combinatorial Representation
Matrices
Li Chen, Ph.D.
Department of Computer Science and Information Technology
University of the District of Columbia
4200 Connecticut Avenue, N.W.
Washington, DC 20008
Combinatorial Representation
Matrices (CRM)
CRM is to use matrices to represent the
combinatorial problem to provide an intuitive
visualization and simple understanding. Then to
find a relatively easier solution for the problem.
Combinatorial Matrix Theory is
Different from CRM
Richard A. Brualdi : “ Combinatorial Matrix Theory (CMT) is
the name generally ascribed to the very successful
partnership between Matrix Theory (MT) and
Combinatorics & Graph Theory (CGT).” “ The key to the
partnership of MT and CGT is the adjacency matrix of a
graph. A graph with n vertices has an adjacency matrix A of
order n which is a symmetric (0,1)-matrix.”
More information about MMT, please see R. Brualdi, H. Ryser,
Combinatorial Matrix Theory, Cambridge University Press,
1991
Basic Combinatorial Representation
Matrices
1) CRM of Permutation problem: Give a set S={1,2,...,n}, its CRM is
Basic CRMs
2) CRM of the Combination problem: Give a set S={1,2,...,n},
select k items but the order does not count. Its CRM is
Basic CRMs
3) CRM of k-Permutation problem: Give a set S={1,2,...,n}, select k
items but the order does count. Its CRM is
Basic CRMs
4) CRM of k-Permutation problem for multi-sets: Give a multi-set
M={1,..,1,2,...,2,...,m,...,m}, select k items but the order does
count. M has n(i) i's in the set. and n=\Sigma_{i}^m n_{i}. Its
CRM is
Basic CRMs
5) CRM of finite set mapping: N={1,2,...,n}, M={1,2,...,m}, list all
different mapping N M. Its CRM is
Hsiao Code
The optimal SEC-DED code, or Hamming code
SEC-DED codes : single error
correction and double-error detection codes.
Brief History of Hsiao Codes
SEC-DED code is widely used in
Computer Memory
M.Y. Hsiao. A Class of Optimal Minimum
Odd-weight-column SEC-DED Codes.
IBM J. of Res. and Develop., vol. 14, no.
4, pp. 395-401 (1970)
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Check Matrix
To determine if a binary string is a
codeword
To determine if the string contains one
bit error to a codeword or two bit error.
The Key for error-correction and
detection.
a Hardware Component in Computer
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Hsiao-Code Check Matrix
Only requires minimum numbers of “1”s
in the Check Matrix.
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“1” means a unit circuit.
minimum numbers of “1”s means minimal
power required.
the optimal DEC-DED code or Hamming
code.
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Definition of Hsiao-Code Check
Matrix
Every column contains an odd number
of 1's.
The total number of 1's reaches the
minimum.
The difference of the number of 1's in
any two rows is not greater than 1
No two columns are the same.
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Information Bit k and Check bit R
R  1 + log2( k + R )
(R, J, m) = a {0,1}-type (R x m) matrix with
column weight J, 0  J  R. No two columns are
the same.
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Check Matrix H
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(R,J,m)
Is the Problem of generating  a Polynomial
problem?
Yes!
Why it is a Problem? Because few papers used
genetic algorithms to solve this problem and
they do not know Li Chen’s original work in
1986.
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Recursively Balanced Matrix 
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Conditions for Recursively Balanced
Matrix 
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Special Cases for Recursively
Balanced Matrix 
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Solution for Recursively Balanced
Matrix 
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Improved Fast Algorithm for 
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Improved Fast Algorithm for 
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k-Linearly Independent Vectors
on GF(2^b)
The set of $k$-Linearly Independent
Vectors on $GF(2^{b})$ has a lot of
applications in error-correction codes.
Assume $q=2^b$,
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k-Linearly Independent Vectors
on GF(2^b)
Let $A(R,k)$ is a sub matrix of $I(R,m)$
and every $k$ columns are linearly
independent. Then
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References
This paper: http://arxiv.org/abs/0803.1217
M.Y. Hsiao. A Class of Optimal Minimum Odd-weight-column
SEC-DED Codes. IBM J. of Res. and Develop., vol. 14, no. 4, pp.
395-401 (1970)
L. Chen, An optimal generating algorithm for a matrix of equalweight columns and quasi-equal-weight rows. Journal of
Nanjing Inst. Technol. 16, No.2, 33-39 (1986).
S. Ghosh, S. Basu, N.A. Touba, Reducing Power Consumption in
Memory ECC Checkers, Proceedings of IEEE International Test
Conference, 2004. pp 1322-1331
S. Ghosh, S. Basu, N.A. Touba, Selecting Error Correcting Codes
to Minimize Power in Memory Checker Circuits, J. Low Power
Electronics 1, pp.63-72(2005)
W. Stallings, Computer Organization and Architecture, 7ed,
Prentice Hall, Upper Saddle River, NJ, 2006.
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About the Author
 Fast Algorithm for Optimal SEC-DED Code (Hsiao-code), 1981,
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published in Chinese in 1986. Unrecognized???
Polynomial Algorithm for basis of finite Abelian Groups, 1982, published
in Chinese in 1986. The actual origin of the famous hidden subgroup
problem in author view. International did not know until 2006
according to P. Shor’s Quantum Computing Report in 2004.
A Solving algorithm for fuzzy relation equations, 1982, Unpublished
Proceeding printing 1987. Published in 2002 with P. Wang. Cited by two
books and many research papers.
Gradually varied surface fitting, Published in 1989. Merged with P. Hell’s
Absolute Retracts in Graph Homomorphism in 2006 published in
Discrete Math (G. Agnarsson and L. Chen).
Digital Manifolds, Published in 1993. Cited by a paper in 2008 in IEEE
PAMI.
Monograph: Discrete Surfaces and Manifolds, 2004 self published. Cited
by few publications.
Current focus: Discrete Geometry Relates to Differential Geometry and
Topology
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