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) ASM Joint Conference Error-Correction Code 10

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 ASM Joint Conference Error-Correction Code 11

Hsiao-Code Check Matrix Only requires minimum numbers of “1”s in the Check Matrix.  “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 + log 2 ( 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  ASM Joint Conference Error-Correction Code 17

Conditions for Recursively Balanced Matrix  ASM Joint Conference Error-Correction Code 18

Special Cases for Recursively Balanced Matrix  ASM Joint Conference Error-Correction Code 19

Solution for Recursively Balanced Matrix  ASM Joint Conference Error-Correction Code 20

Improved Fast Algorithm for

 ASM Joint Conference Error-Correction Code 21

Improved Fast Algorithm for

 ASM Joint Conference Error-Correction Code 22

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 $q=2^b$, ASM Joint Conference Error-Correction Code 23

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 ASM Joint Conference Error-Correction Code 24

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. 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 ASM Joint Conference Error-Correction Code 26