Error-Correction Coding Using Combinatorial Representation Matrices

1. Joint Mathematics MeetingsWashington, 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

2. 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.

3. 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

4. Basic Combinatorial Representation Matrices 1) CRM of Permutation problem: Give a set S={1,2,...,n}, its CRM is

5. 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

6. 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

7. 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

8. 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

9. Hsiao Code • The optimal SEC-DED code, or Hamming code • SEC-DED codes : single error • correction and double-error detection codes.

10. 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) Error-Correction Code

11. 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 Error-Correction Code

12. 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. Error-Correction Code

13. 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. Error-Correction Code

14. 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. Error-Correction Code

15. Check Matrix H Error-Correction Code

16. (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. Error-Correction Code

17. Recursively Balanced Matrix Error-Correction Code

18. Conditions for Recursively Balanced Matrix Error-Correction Code

19. Special Cases for Recursively Balanced Matrix Error-Correction Code

20. Solution for Recursively Balanced Matrix Error-Correction Code

21. Improved Fast Algorithm for  Error-Correction Code

22. Improved Fast Algorithm for  Error-Correction Code

23. 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 $q=2^b$, Error-Correction Code

24. 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 Error-Correction Code

25. 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. Error-Correction Code

26. 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 Error-Correction Code