Coding Theory and Design Theory

1. Front Matter

   Pages i-xiv

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2. Baer Subplanes, Ovals and Unitals

   • E. F. Assmus Jr., J. D. Key

   Pages 1-8

3. On the Length of Codes with a Given Covering Radius

   • Richard A. Brualdi, Vera S. Pless

   Pages 9-15

4. The Differential Encoding of Coset Codes by Algebraic Methods

   • A. R. Calderbank

   Pages 16-34

5. Families of Codes with Few Distinct Weights from Singular and Non-Singular Hermitian Varieties and Quadrics in Projective Geometries and Hadamard Difference Sets and Designs Associated with Two-Weight Codes

   • I. M. Chakravarti

   Pages 35-50

6. Perfect Multiple Coverings in Metric Schemes

   • Richard Clayton

   Pages 51-64

7. Nonlinear Feedforward Sequences of m-Sequences II

   • Zongduo Dai, Xuning Feng, Mulan Liu, Zhe-Xian Wan

   Pages 65-71

8. Loops of Clutters

   • Michel Marie Deza, Komei Fukuda

   Pages 72-92

9. Positive Independence and Enumeration of Codes with a Given Distance Pattern

   • M. Deza, D. K. Ray-Chaudhuri, N. M. Singhi

   Pages 93-101

10. Bounds on the Number of Pairs of Unjoined Points in a Partial Plane

    • David A. Drake, Paul Erdös

    Pages 102-112

11. Inside Euclid’s Algorithm

    • Willard L. Eastman

    Pages 113-127

12. Construction of Designs

    • Marshall Hall Jr.

    Pages 128-136

13. Algebraic Geometric Codes

    • Jacobus H. Van Lint

    Pages 137-162

14. Combinatorial Characters of Quasigroups

    • Jonathan D. H. Smith

    Pages 163-187

15. Self-Dual Codes and Self-Dual Designs

    • Patrick Solé

    Pages 188-192

16. The Incidence Algebra of a Uniform Poset

    • Paul Terwilliger

    Pages 193-212

17. Some Recent Results on Signed Graphs with Least Eigenvalues ≥ -2

    • G. R. Vijayakumar, N. M. Singhi

    Pages 213-218

18. Self-Orthogonal Codes and the Topology of Spinor Groups

    • Jay A. Wood

    Pages 219-239

This IMA Volume in Mathematics and its Applications Coding Theory and Design Theory Part I: Coding Theory is based on the proceedings of a workshop which was an integral part of the 1987-88 IMA program on APPLIED COMBINATORICS. We are grateful to the Scientific Committee: Victor Klee (Chairman), Daniel Kleitman, Dijen Ray-Chaudhuri and Dennis Stanton for planning and implementing an exciting and stimulating year­ long program. We especially thank the Workshop Organizer, Dijen Ray-Chaudhuri, for organizing a workshop which brought together many of the major figures in a variety of research fields in which coding theory and design theory are used. A vner Friedman Willard Miller, Jr. PREFACE Coding Theory and Design Theory are areas of Combinatorics which found rich applications of algebraic structures. Combinatorial designs are generalizations of finite geometries. Probably, the history of Design Theory begins with the 1847 pa­ per of Reverand T. P. Kirkman "On a problem of Combinatorics", Cambridge and Dublin Math. Journal. The great Statistician R. A. Fisher reinvented the concept of combinatorial 2-design in the twentieth century. Extensive application of alge­ braic structures for construction of 2-designs (balanced incomplete block designs) can be found in R. C. Bose's 1939 Annals of Eugenics paper, "On the construction of balanced incomplete block designs". Coding Theory and Design Theory are closely interconnected. Hamming codes can be found (in disguise) in R. C. Bose's 1947 Sankhya paper "Mathematical theory of the symmetrical factorial designs".

• Combinatorics
• algorithms
• coding theory
• communication
• graphs

• Department of Mathematics, Ohio State University, Columbus, USA

  Dijen Ray-Chaudhuri

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