ErrorCorrection Coding Using Combinatorial Representation Matrices

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Error-Correction Coding Using Combinatorial
Representation Matrices
Joint Mathematics MeetingsWashington, DC,
January 5-8, 2009 (Monday - Thursday)

• 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

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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.
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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

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Basic Combinatorial Representation Matrices
1) CRM of Permutation problem Give a set
S1,2,...,n, its CRM is
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Basic CRMs
2) CRM of the Combination problem Give a set
S1,2,...,n, select k items but the order does
not count. Its CRM is
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Basic CRMs
3) CRM of k-Permutation problem Give a set
S1,2,...,n, select k items but the order does
count. Its CRM is
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Basic CRMs
4) CRM of k-Permutation problem for multi-sets
Give a multi-set M1,..,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_im n_i.
Its CRM is
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Basic CRMs
5) CRM of finite set mapping N1,2,...,n,
M1,2,...,m, list all different mapping N? M.
Its CRM is
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Hsiao Code
The optimal SEC-DED code, or Hamming code SEC-DED
codes single error correction and double-error
detection codes.
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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 1s in the
  Check Matrix.
• 1 means a unit circuit.
• minimum numbers of 1s 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 Chens 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(2b)

• The set of k-Linearly Independent Vectors on
  GF(2b) has a lot of applications in
  error-correction codes. Assume q2b,

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k-Linearly Independent Vectors on GF(2b)

• 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 equal-weight 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, 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. Shors 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. Hells 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