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Abstract

One of the purposes of the mini-conferences at the meeting is to introduce new areas of mathematics which have found their way into the curriculum at some universities and colleges but deserve more attention. At the same time one can consider the question whether it concerns a development in mathematics that could in some way be incorporated in the high school curriculum. In the present case the topic is algebraic coding theory, an area between information theory, combinatories and applied algebra, which has only been around for about 30 years. More than likely at most of the world’s universities there has never been a course in this subject. On the other hand at a few it has been taught for at least 15 years and it is usually received with enthusiasm by the participating students.

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Annotated References

  1. E.R. Berlekamp, Algebraic Coding Theory ( McGraw-Hill, New York, 1968 ).

    MATH  Google Scholar 

  2. M.R. Best, On the existence of perfect codes, Thesis, University of Amsterdam, 1981.

    Google Scholar 

  3. P.J. Cameron and J.H. van Lint, Graphs, Codes and Designs, Cambridge University Press, 1980.

    MATH  Google Scholar 

  4. D.M. Cvetkovic and J.H. van Lint, An Elementary Proof of Lloyd’s Theorem, Proc. Kon, Ned. Akad. v. Wet. A 80 (1977).

    Google Scholar 

  5. J.H. van Lint, Coding Theory (Springer, New York, I971).

    Google Scholar 

  6. J.H. van Lint (ed.), Inleiding in de Coderingstheorie (M.C. Syllabus 31, Amsterdam 1976 ).

    Google Scholar 

  7. J.H. van Lint, Introduction to Coding Theory (Springer Verlag, 1981).

    Google Scholar 

  8. J.H. van Lint and F.J. MacWilliams, Generalized Quadratic Residue Codes, IEEE Trans. on Inf. Theory II 24 (1978), 730–737.

    Article  Google Scholar 

  9. F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-correcting codes (North Holland, Amsterdam 1977 ).

    Google Scholar 

  10. H.B. Mann (ed.), Error-correcting Codes ( John Wiley & Sons, New YUork, 1968 ).

    Google Scholar 

  11. W.W. Peterson, Error-correcting Codes (M.I.T. Press, Cambridge Press, 1961 ).

    Google Scholar 

  12. N.J.A. Sloane, Binary Codes, Lattices and spherepackings, in Combinatorial Surveys ( P.J. Cameron, ed.), ( Academic Press, London, 1977 ).

    Google Scholar 

  13. H.C.A. van Tilborg, Uniformly packed Codes, Thesis, Eindhoven University of Technology (1976).

    MATH  Google Scholar 

  14. E.F. Assmus, Jr., and H.F. Mattson, Jr., Coding and Combinatorics, SIAM Review 16 (1974), 349–388.

    Article  MathSciNet  MATH  Google Scholar 

  15. E.R. Berlekamp, A Survey of Algebraic Coding Theory, CISM Courses and Lectures 28, Springer Verlag, 1970.

    Google Scholar 

  16. N. Levinson, Coding Theory, A Counterexample to G.H. Hardy’s Conception of Applied Mathematics, Am. Math. Monthly 77 (1970), 249–258.

    Article  MathSciNet  MATH  Google Scholar 

  17. J.H. van Lint, see References (5), (6), (7)

    Google Scholar 

  18. N.J.A. Sloane, A Short Course on Error Correcting Codes, CISM Courses and Lectures 188, Springer Verlag 1975.

    Google Scholar 

  19. N.J.A. Sloane, Error-correcting Codes and Invariant Theory: New Applications of a Nineteenth Century Technique, Am. Math. Monthly 84 (1977) 82–107.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Swoboda, Codierung zur Fehlerkorrektur und Fehlererkennung, Oldenbourg Verlag 1973.

    Google Scholar 

  21. N. Balacheff (1979) Quelques aspects du sens donne a l’explication mathematique par des eleves de 10–11 ans. Seminaire de Recherche Pedagogique, Grenoble USMG-LA7.

    Google Scholar 

  22. C. Berge (1968) Principes de combinatoire. DUNOD.

    Google Scholar 

  23. A. Engel (1971) Geometrical activities for the upper Elementary school. Educational Studies in Mathemtics 3 (1971) 353–394.

    Article  Google Scholar 

  24. J. N. Kappur (1970) Combinatorial analysis and school mathematics. Education Studies in Mathematics 3 (1970)III-127.

    Google Scholar 

  25. J. Kuntsmann (1976) Evolution et etude critique des enseignements. Mathematiques. CEDIC.

    Google Scholar 

  26. P. Mendelshon (1980) Pensee naturelle et logique combinatoire. Laboratoire de psychologie experimentale. Grenoble.

    Google Scholar 

  27. C. Lebosse, C. Hemery, P. Faure. Geometrie et elements de probabilites, classe terminale D. Fernand Nathan 1967.

    Google Scholar 

  28. G. Papelier. Precis d’algebre, d’analyse et de trigonometrie. Vuibert 1916.

    Google Scholar 

  29. M. Queysanne, A. Revuz. Mathematiques T I: Nombres, Probabilites. Terminale CE, Fernand Nathan 1971.

    Google Scholar 

  30. G. Wicker, J.L. Boursin. Mathematiques. Terminale G, Bordas Dunod 1972.

    Google Scholar 

  31. Angevine, J. Solution for the Magic Cube Puzzle. 11 pp, 1979 (Available for $2 and SAE from Logical Games, 4509 Martinwood Drive, Haymarket, Virginia, 22069, USA.)

    Google Scholar 

  32. Berklekamp, E.R., J.H. Conway & R.K. Guy. Winning Ways. (In press ?)

    Google Scholar 

  33. Cairns, C. & D. Griffiths. Teach Yourself Cube-Bashing. 6 pp, Sep 1979. (Available from the authors, ICL Dataskil, 118–128 London St., Reading, RGI 4SU, Berkshire, UK. )

    Google Scholar 

  34. Deledicq, A. & J. -B. Touchard. Le cube Hongrois - Mode d’emploi. 73 pp, 1980. (Available from IREM, Paris VII, T. 56/55, 3eme etage, 2 Place Jussieu, 75005, Paris, France. )

    Google Scholar 

  35. Howlett, G.S. “Magic Cube” - A Guide to the Solution. 2 pp, 1979. (Available from Pentangle, Over Wallop, Stockbridge, Hampshire S020 8Ht, UK.)

    Google Scholar 

  36. Neumann, P.M. The Group Theory of the Hungarian Magic Cube. Chapter 19, pp 299–307 of: Neumann, et al. “Groups and Geometry”. Lecture Notes, April 1980. (Available from Mathematical Institute, 24–29 St. Giles, Oxford, 0X1 3LB, UK, for 2.00 plus 1.22 UK postage or 0.70 overseas postage.)

    Google Scholar 

  37. Ollerenshaw, K. The Hungarian Magic Cube. Bull. Inst. Math. Appl. 16 (Apr 1980) 86–92.

    MathSciNet  Google Scholar 

  38. Singmaster, D. Notes on Rubik’s ‘Magic Cube’. Fifth ed.m 75 pp, Aug 1980. (Available from the author, Polytechnic of the South Bank, London, SE I OAA, UK for 1.50 pounds (= $4.00) in the UK & Europe and 2.00 pounds (= $5.00) elsewhere. Also available from Logical Games - address under ( I. ). A US edition may appear soon. )

    Google Scholar 

  39. Taylor, D.E. The Magic Cube. 18 pp ms, Nov 1978.

    Google Scholar 

  40. Taylor, D.E. Secrets of the Rubik Cube - A Guided Tour of a Permutation Group. In preparation.

    Google Scholar 

  41. Angevine, J. Solution for the Magic Cube Puzzle. II pp, 1979. (Available for $2 and SAE from Logical Games, 4509 Martinwood Drive, Haymarket, Virginia 22069, USA.)

    Google Scholar 

  42. Berlekamp, E.R., J.H. Conway & R.K. Guy. Winning Ways. (In Press).

    Google Scholar 

  43. Cairns, C. & D. Griffiths. Teach Yourself Cube-Bashing. 6 pp, Sep 1979. (Available from the authors, ICL Dataskil, 118–128 London St., Reading, RGI 4SU, Berkshire, UK. )

    Google Scholar 

  44. Deledicq, A. & J.-B. Touchard. Le cube Hongrois - Mode d’emploi. 73 pp, 1980. (Available from IREM, Paris VII, T. 56/55, 3eme etage, 2 Place Jussieu, 75005 Paris, France

    Google Scholar 

  45. Howlett, G.S. “Magic Cube” - A Guide to the Solution. 2 pp, 1979. (Available from Pentangle, Over Wallop, Stockbridge, Hampshire, 5020 8HT, UK.)

    Google Scholar 

  46. Neumann, P.M. The Group Theory of the Hungarian Magic Cube. Chapter 19, pp 299–307 of: Neummann, et al. “Groups and Geometry”. Lecture Notes, April 1980. (Available from Mathematical Institute, 24–29 St. Giles, Oxford, XOI 2LB, UK.)

    Google Scholar 

  47. Ollerenshaw, K. The Hungarian Magic Cube. Bull. Inst. Math. Appl. 16 (Apr 1980) 86–92.

    MathSciNet  Google Scholar 

  48. Singmaster, D. Notes on Rubik’s ‘Magic Cube’. Fifth ed., 75 pp, Aug 1980. (Available from the author, Polytechnic of the South Bank, London, SE I OAA. Also available from Logical Games - address under (1).

    Google Scholar 

  49. Taylor, D.E. The Magic Cube. 18 pp ms, Nov 1978.

    Google Scholar 

  50. Taylor, D.E. Secrets of the Rubik Cube - A Guided Tour of a Permutation Group. In preparation.

    Google Scholar 

  51. Aho, Hoperoft, Ullman, The Design and Analysis of Algorithms. Addison-Wesley 1974.

    Google Scholar 

  52. E.W. Dijkstra, A Discipline of Programming. Prentice-Hall 1976.

    Google Scholar 

  53. A. Engel, Elementarmathematik vom algorithmischen Standpunkt. Ernst Klett 1977.

    Google Scholar 

  54. A. Engel, The role of algorithms and computers in teaching mathematics at school. In “New Trends in Mathematics Teaching IV”. UNESCO, Paris 1979.

    Google Scholar 

  55. A. Engel, Algorithmen fur den Taschenrechner. In “Der Mathematikunterricht”, 1979, Heft 6, pp. 52–77.

    Google Scholar 

  56. A. Engel, Wahrscheinlichkeit und Statistik, Bd 2, Ernst Klett 1976.

    Google Scholar 

  57. A. Engel, The Probabilistic Abacus. Educ. Stud. in Math. 6 (1975), 1–2.

    Article  MATH  Google Scholar 

  58. A. Engel, Why Does the Probabilistic Abacus Work? Educ. Stud. in Math. 7 (1976), 59–69.

    Article  MATH  Google Scholar 

  59. D.E. Knuth, The Art of Computer Programming, vol. I-3. Addison-Wesley.

    Google Scholar 

  60. L. Kronsjo, Algorithms: Their Complexity and Efficiency. Wiley 1979.

    Google Scholar 

  61. Nijenhuis & Wilf, Combinatorial Algorithms. Academic Press 1978.

    Google Scholar 

  62. Reingold, Neivergelt & Deo, Combinatorial Algorithms. Prentice-Hall 19 .

    Google Scholar 

  63. Springer’s Mathematics Calendar, August 1981.

    Google Scholar 

  64. N. Wirth, Systematic Programming. Prentice-Hall 1973.

    Google Scholar 

  65. N. Wirth, Algorithms + Data Structures = Programs. Prentice-Hall 1976.

    Google Scholar 

  66. D.E. Knuth, Fundamental Algorithms, vol. 2, Second Edition 1981. This is one of the most up to date single references on seminumerical algorithms and public key cryptography.

    Google Scholar 

  67. M.O. Rabin, Digitalized signatures and public key functions as intractable as factorization. MIT/LCS/TR-212, Technical Memo MIT (197 ).

    Google Scholar 

  68. A. Engel, Datenschutz durch Chiffrieren. Der Mathematikunterricht, 1979, Heft 6, pp. 30–50. (Rabin’s test explained.)

    Google Scholar 

  69. M.E. Hellman, The Mathematics of Public Key Cryptography. Scientific American, August 1979.

    Google Scholar 

  70. M.O. Rabin, Probabilistic Algorithms. In “Algorithms and Complexity, New Directions and Recent Results”, J.F. Traub (ed.) 1976.

    Google Scholar 

  71. Andrews, D.F., Bickel, P.J., Hampel, F.R., Huber, P.J., Rogers, W.H. and Tukey, J.W. (1972). Robust Estimates of Location-Survey and Advances. Princeton University Press, Princeton, New Jersey. (Extensive Monte Carlo study under a variety of sampling conditions.) (3)

    Google Scholar 

  72. Bickel, P.J. and Doksum, K.A. (1977). Mathematical Statistics: Basic Ideas and Selected Topics. Holden-Day, San Francisco. (Brief introduction to robustness in Section 9.5.) (3)

    Google Scholar 

  73. Daniel, C. and Wood, F.S. (1971). Fitting Equations to Data. Wiley, New York. (Approaches to exploratory analysis and model building presented through the extensive analysis of a limited number of data sets. Includes scatter plots and Q-Q plots.) (2)

    Google Scholar 

  74. Everitt, B. (1974). Cluster Analysis. Wiley, New York. Concise introduction, easy to read.) (4)

    Google Scholar 

  75. Gnanadesikan, R. (1977). Methods for Statistical Data Analysis of Multivariate Observations. Wiley, New York. (Modern treatment of multivariate data analysis; distance metrics; clustering procedures.) (I, 2, 3, 4 )

    Google Scholar 

  76. Hartigan, J.A. (1975). Clustering Algorithms. Wiley, New York. (Authoritative work on clustering; examples.) (4)

    Google Scholar 

  77. Huber, P.J. (1977). Robust Statistical Procedures. S.I.A.M., Philadelphia. (Advanced mathematical treatment.) (3)

    Google Scholar 

  78. Launer, R.L. and Wilkinson, G.N., eds. (1979). Robustness in Statistics. Academic Press, New York. ( Proceedings of a workshop on robustness. )

    Google Scholar 

  79. McNeil, D.R. (1977). Interactive Data Analysis, a Practical Primer. Wiley, New York. (A good introduction, with tutorial examples. Compact. Includes computer programs in Fortran and APL for stem-and-leaf, box plots, etc. Available in paperback.) (2, 3 ).

    Google Scholar 

  80. Mosteller, F. and Tukey, J.W. (1977). Data Analysis and Regression,.a Second Course in Statistics. Addison-Wesley, Reading, Massachusetts. (A data analyst’sview of regression and other more classical tools. In the spirit of Tukey (1977) but more advanced and specialized. Contains a wealth of data sets for exercises.) ( 2, 3 )

    Google Scholar 

  81. Siegel, A.F. and Launer, R.L., eds. (1981). Advances in Data Analysis. Academic Press, New York. (Conference proceedings. Includes regression by repeated medians.) (2, 3)

    Google Scholar 

  82. Sneath, P.H.A. and Sokal, R.C. (1973). Numerical Taxonom. Freeman, San Francisco. (Comprehensive treatment of clustering with extensive bibliography.) (4)

    Google Scholar 

  83. Tukey, J.W. (1977). Exploratory Data Analysis. Addison-Wesley, Reading, Massachusetts. (Wide ranging. Unique. Starts with simple concepts and tools. No probability theory or classical statistics required. Extensive examples.) (I, 2, 3 )

    Google Scholar 

  84. Velleman, P.F. and Hoaglin, D.C. (1980). Applications, Basics, and Computing of Exploratory Data Anal sis. Duxbury Press, North Scituate, Massachusetts. (A companion volume to Tukey, 1977.) (1, 2, 3 )

    Google Scholar 

  85. Beck, B., Denby, L. and Landwehr, J.M. (1976). Statistics in the elementary school. Communications in Statistics - Theory and Methodology. Vol. A5(IO), pp. 883–894. (General comments and perspectives on statistics in the school curriculum.) (I)

    Google Scholar 

  86. Blashfield, R.K. (1976). A consumer report on the versatility and user manuals of cluster analysis software. Proceedings of the Statistical Computing Section of the American Statistical Association, pp. 31–37. (Compares 14 programs using 11 criteria.)

    Google Scholar 

  87. Chen, H.J., Dunn, D.M. and Landwehr, J.M. (1975). Grouping companies based on their operating environment. Proceedings of the Business and Economic Statistics Section of the American Statistical Association, pp. 278–283. (Example of clustering 48 Bell System entities.) (4)

    Google Scholar 

  88. Cleveland, W.S. (1979). Robust locally weigted regression and smoothing scatterplots. Journal of the American Statistical Association, Vol. 74, pp. 829–936. (A new approach to smoothing.) (2, 3 )

    Google Scholar 

  89. Cleveland, W.S. and Kleiner, B. (1975). A graphical technique for enhancing scatterplots with moving statistics. Technometrics, Vol. 17, pp. 447–454. (Moving medians and other smoothers with examples from an air pollution study.) (2, 3 )

    Google Scholar 

  90. Cohen, A., Gnanadesikan, R., Kettenring, J.R. and Landwehr, J.M. (1977). Methodological developments in some applications of clustering. In Applications of Statistics (P.R. Krishnaiah, ed.). North-Holland, Amsterdam, pp. 141–162. (Four real examples, practical problems.) (4)

    Google Scholar 

  91. Cormack, R.M. (1971). A review of classification. Journal of the Royal Statistical Society, Vol. A134, pp. 321–357. (Important review article on cluster analysis.) (4)

    Google Scholar 

  92. Denby, L. and Landwehr, J.M. (1975). Examining one and two sets of data - Parts I, II and III. Prepared for USMES. Available from authors at Bell Laboratories, Murray Hill, New Jersey. (Material developed for secondary school curriculum. Includes histograms, ECDFs, and Q-Q plots.) (2)

    Google Scholar 

  93. Gnanadesikan, R., Kettenring, J.R. and Landwehr, J.M. (1977). Interpreting and assessing the results of cluster analyses. Bulletin of the International Statistical Institute, Vol. 47 (2), pp. 451–463. (Output stage; distance plots.) (4)

    Google Scholar 

  94. Gnanadesikan, R., Kettenring, J.R. and Landwehr, J.M. (1981). Projection plots for displaying clusters. To appear in Statistics and Probability: Essays in Honor of C.R. Rao (G. Kalliandpur, P.R. Krishnaiah, and J.K. Ghosh, eds.), North-Holland, Amsterdam. (Output stage; projection plots.) (4)

    Google Scholar 

  95. Gnanadesikan, R. and Wilk, M.B. (1969). Data analytic methods in multivariate statistical analysis. In Multivariate Analysis Ill (P.R. Krishnaiah, ed.), Academic Press, New York, pp. 593–638. (An expository survey). ( I, 2, 4 )

    Google Scholar 

  96. Kettenring, J.R., Rogers, W.H., Smith, M.E. and Warner, J.L. (1976). Cluster analysis applied to the validation of course objectives. Journal of Educational Statistics, Vol. I, pp. 39–57. (Edcation example.) (4)

    Google Scholar 

  97. Mallows, C.L. and Tukey, J.W. (1980). An overview of techniques of data analysis, emphasizing its exploratory aspects. To appear in a volume in obervance of the 200th anniversary of the Lisbon Academy of Sciences. (A survey of state-of-the-art isues both of the philosophy and the technology of data analysis.) (I)

    Google Scholar 

  98. McGill, R., Tukey, J.W. and Larsen, W.A. (1978). Variations of box plots. The American Statistician. Vol. 32, pp. 12–16. (Details of box plots - - variable widths, notches, etc.) (2)

    Google Scholar 

  99. Siegel, A.F. and Benson, R.H. (1980). Estimating change in animal morphology. Unpublished. (Application of robust methods in fossils and human skulls.) (3)

    Google Scholar 

  100. Tukey, J.W. and Wilk, M.B. (1966). Data analysis and statistics: An expository overview. AFIPS Conf. Proc., Fall Joint comput. Conf. Vol. 29, pp. 695–709. (An early survey of the nature, concerns and needs of data analysis.) (I)

    Google Scholar 

  101. Wilk, M.B. and Gnanadesikan, R. (1968). Probability plotting methods for the analysis of data. Biometrika, Vol. 55, pp. 1–17. (Detailed discussion of Q-Q plots, ECDFs, etc.) (2)

    Google Scholar 

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van Lint, J.H. et al. (1983). Special Mathematical Topics. In: Zweng, M.J., Green, T., Kilpatrick, J., Pollak, H.O., Suydam, M. (eds) Proceedings of the Fourth International Congress on Mathematical Education. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4684-8223-2_10

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