Skip to main content Accessibility help
×
Home
Designs and their Codes
  • Cited by 255
  • Export citation
  • Recommend to librarian
  • Buy the print book

Book description

Algebraic coding theory has in recent years been increasingly applied to the study of combinatorial designs. This book gives an account of many of those applications together with a thorough general introduction to both design theory and coding theory - developing the relationship between the two areas. The first half of the book contains background material in design theory, including symmetric designs and designs from affine and projective geometry, and in coding theory, coverage of most of the important classes of linear codes. In particular, the authors provide a new treatment of the Reed-Muller and generalized Reed-Muller codes. The last three chapters treat the applications of coding theory to some important classes of designs, namely finite planes, Hadamard designs and Steiner systems, in particular the Witt systems. The book is aimed at mathematicians working in either coding theory or combinatorics - or related areas of algebra. The book is, however, designed to be used by non-specialists and can be used by those graduate students or computer scientists who may be working in these areas.

Reviews

"...a valuable resource for researchers in either finite geometries or coding theory as well as for algebraists who want to learn about this lively, growing area." Vera Pless, Mathematical Reviews

"...the relationship between the two subjects is very much a two-way channel, and the book is a mine of useful information from whichever direction one approaches it....a useful compilation of material which, together with the extensive bibliography, will prove useful to anyone whose research impinges on these topics." N.L. Biggs

"...speaks to the tremendous influence the plane of order ten has subsequently had on the analysis and classification of designs in a much broader context than projective planes...a welcome addition to a very exciting and relatively new application of an established discipline to combinatorics...a truly fascinating and useful book. It belongs on the shelves of all those who wish to be current on the state of design theory and who are seeking interesting problems in the field to pursue." M.A. Wertheimer, Bulletin of the American Mathematical Society

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
×

Contents

Bibliography
[1] R. W., Ahrens and G., Szekeres. On a combinatorial generalization of 27 lines associated with a cubic surface. J. Austral. Math. Soc, 10:485–492, 1969.
[2] W. O., Alltop. An infinite class of 5–designs. J. Combin. Theory, 12:390–395, 1972.
[3] J., André. Über nicht–Desarguessche Ebenen mit transitiver Translationsgruppe. Math. Z., 60:156–186, 1954.
[4] B. R., Andriamanalimanana. Ovals, Unitals and Codes. PhD thesis, Lehigh University, 1979.
[5] E., Artin. Geometric Algebra. New York: Wiley Interscience, 1957.
[6] E. F. Assmus, Jr. The binary code arising from a 2–design with a nice collection of ovals. IEEE Trans. Inform. Theory, 29:367–369, 1983.
[7] E. F. Assmus, Jr.On the theory of designs. In J., Siemons, editor, Surveys in Combinatorics, 1989, pages 1–21. Cambridge: Cambridge University Press, 1989. London Mathematical Society Lecture Note Series 141.
[8] E. F. Assmus, Jr.On the Reed–Muller codes. Discrete Math., 106/107:25–33, 1992.
[9] E. F. Assmus, Jr. and J. D., Key. On an infinite class of Steiner systems with t = 3 and k = 6. J. Combin. Theory, Ser. A, 42:55–60, 1986.
[10] E. F. Assmus, Jr. and J. D., Key. Arcs and ovals in the hermitian and Ree unitals. European J. Combin., 10:297–308, 1989.
[11] E. F. Assmus, Jr. and J. D., Key. Affine and projective planes. Discrete Math., 83:161–187, 1990.
[12] E. F. Assmus, Jr. and J. D., Key. Baer subplanes, ovals and unitals.In Dijen, Ray–Chaudhuri, editor, Coding Theory and Design Theory, Part I, pages 1–8. New York: Springer–Verlag, 1990. IMA Volumesin Mathematics and its Applications, 20.
[13] E. F., Assmus|Jr. and J. D., Key. Translation planes and derivation sets. J. Geom., 37:3–16, 1990.
[14] E. F. Assmus, Jr. and J. D., Key. Hadamard matrices and their designs: a coding–theoretic approach. Trans. Amer. Math. Soc, 330:269–293, 1992.
[15] E. F. Assmus, Jr. and J. H. van, Lint.Ovals in projective designs. J. Combin. Theory, Ser. A, 27:307–324, 1979.
[16] E. F. Assmus, Jr. and H. F. Mattson, Jr.Perfect codes and the Mathieu groups. Arch. Math., 17:122–135, 1966.
[17] E. F. Assmus, Jr. and H. F. Mattson, Jr.On tactical configurations and error–correcting codes. J. Combin. Theory, 2:243–257, 1967.
[18] E. F. Assmus, Jr. and H. F. Mattson, Jr.New 5–designs. J. Combin. Theory, 6:122–151, 1969.
[19] E. F. Assmus, Jr. and H. F. Mattson, Jr.Coding and combinatorics. SIAM Review, 16:349–388, 1974.
[20] E. F. Assmus, Jr., H. F., Mattson|Jr., and R. J., Turyn. Research to Develop the Algebraic Theory of Codes. Applied Research Laboratory, Sylvania Electronic Systems, June 1967. No. AFCRL–67–0365. Contract No. AF19(628)–5998.
[21] E. F. Assmus, Jr. and A. R., Prince. Biplanes and near biplanes. J. Geom., 40:1–14, 1991.
[22] E. F. Assmus, Jr. and H. E., Sachar. Ovals from the point of view of coding theory. In M., Aigner, editor, Higher Combinatorics, pages 213–216. Dordrecht: D. Reidel, 1977. Proceedings of the NATO Conference, Berlin 1976.
[23] E. F. Assmus, Jr. and C. J., Salwach.The (16,6,2) designs. Intemat. J. Math. & Math. Sci., 2:261–281, 1979.
[24] R., Baer. Homogeneity of projective planes. Amer. J. Math, 64:137–152, 1942.
[25] R., Baer. Projectivities of finite projective planes. Amer. J. Math, 69:653–684, 1947.
[26] R., Baer. Linear Algebra and Projective Geometry. New York: Academic Press. 1952.
[27] B., Bagchi and N. S. N., Sastry. Even order inversive planes, generalized quadrangles and codes. Geom. Dedicata, 22:137–147, 1987.
[28] S., Bagchi and B., Bagchi. Designs from pairs of finite fields: I. A cyclic unital [/(6) and other regular Steiner 2–designs. J. Combin. Theory, Ser. A, 52:51–61, 1989.
[29] R. D., Baker and G. L., Ebert. Intersection of unitals in the desarguesian plane. In Proceedings of the S.E. Conference on Combinatorics, Graph Theory and Computing, 1989.
[30] A., Barlotti. Un'estensione del teorema di Segre–Kustaanheimo. Boll. Un. Mat Ital, Gruppo IV, Serie III, 10:498–506, 1955.
[31] L. M., Batten. Combinatorics of Finite Geometries. Cambridge: Cambridge University Press, 1986.
[32] L., Bénéteau. Topics about 3–Moufang loops and Hall triple systems. Simon Stevin, 54(2): 107–128, 1980.
[33] T., Berger and P., Charpin. The automorphism group of the generalized Reed–Muller codes. Paris: INRIA Rapports de Recherche No. 1363, 1991.
[34] E. R., Berlekamp. Factoring polynomials over finite fields. Bell System Tech. J., 46:1853–1859, 1967.
[35] E. R., Berlekamp and L. R., Welch. Weight distributions of the cosets of the (32,6) Reed–Muller code. IEEE Trans. Inform. Theory, 18:203– 207, 1972.
[36] S. D., Berman. On the theory of group codes. Kibemetika, 3(l):31–39, 1967.
[37] Th., Beth, D., Jungnickel, and H., Lenz. Design Theory. Mannheim, Wien, Zürich: Bibliographisches Institut Wissenschaftsverlag, 1985.
[38] V. N., Bhat and S. S., Shrikhande. Non–isomorphic solutions of some balanced incomplete block designs. I. J. Combin. Theory, 9:174–191, 1970.
[39] N. L., Biggs and A. T., White. Permutation Groups and Combinatorial Structures. Cambridge: Cambridge University Press, 1979. London Mathematical Society Lecture Notes Series 33.
[40] R. E., Blahut. Transform techniques for error control codes. IBM J. Res. Develop., 23:299–315, 1979.
[41] R. E., Blahut. Theory and Practice of Error Control Codes. New York: Addison–Wesley, 1983.
[42] I. F., Blake and R. C., Mullin. The Mathematical Theory of Coding. New York: Academic Press, 1975.
[43] R. E., Block. On the orbits of collineation groups. Math. Z., 96:33–49, 1967.
[44] A., Blokhuis, A., Brouwer, and H., Wilbrink. Hermitian unitals are codewords. Discrete Math., 97:63–68, 1991.
[45] R. C., Bose and D. K., Ray–Chaudhuri. On a class of error correcting binary group codes. Inform, and Control, 3:68–79, 1960.
[46] R. C., Bose and S. S., Shrikhande. A note on a result in the theory of code construction. Inform, and Control, 2:183–194, 1959.
[47] R. C., Bose and S. S., Shrikhande. On the construction of sets of mutually orthogonal latin squares and the falsity of a conjecture of Euler. Trans. Amer. Math. Soc, 95:191–209, 1960.
[48] R., Brauer. On the connection between the ordinary and the modular characters of groups of finite order. Ann. Math., 42:926–935, 1941.
[49] W. G., Bridges. Algebraic duality theorems with combinatorial applications. Linear Algebra Appl, 22:157–162, 1978.
[50] A. E., Brouwer. Some unitals on 28 points and their embeddings in projective planes of order 9. In M., Aigner and D., Jungnickel, editors, Geometries and Groups, pages 183–188. Berlin: Springer–Verlag, 1981. Lecture Notes in Mathematics, 893.
[51] A. E., Brouwer, A. M., Cohen, and A.|Neumaier. Distance–Regular Graphs. Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge 3, Band 18. Berlin, New York: Springer–Verlag, 1989.
[52] R. H., Bruck. Construction problems in finite projective spaces. In A., Barlotti, editor, Finite Geometric Structures and their Applications, pages 105–188. C.I.M.E., Edizioni Cremonese, Roma, 1973. Corso tenuto a Bressannone dal 18 al 27 Giugno 1972.
[53] R. H., Bruck and R. C., Bose. The construction of translation planes from projective spaces. J. Algebra, 1:85–102, 1964.
[54] A. A., Bruen. Blocking sets in finite projective planes. SIAM J. Appl. Math., 21:380–392, 1971.
[55] A. A., Bruen and J. W. P., Hirschfeld. Intersections in projective space I: Combinatorics. Math. Z., 193:215–225, 1986.
[56] A. A., Bruen and U., Ott. On the p–rank of incidence matrices and a question of E. S. Lander. Contemp. Math., 111:39–45, 1990.
[57] F., Buekenhout. Existence of unitals in finite translation planes of order q2 with a kernel of order q. Geom. Dedicata, 5:189–194, 1976.
[58] F., Buekenhout, A., Delandtsheer, and J.|Doyen. Finite linear spaces with flag–transitive groups. J. Combin. Theory, Ser. A, 49:268–293, 1988.
[59] F., Buekenhout, A., Delandtsheer, J., Doyen, P. B., Kleidman, M. W., Liebeck, and J., Saxl. Linear spaces with flag–transitive automorphism groups. Preprint.
[60] W., Burau. Uber die zur Kummerkonfiguration analogen Schematavon 16 Punkten und 16 Blocken und ihre Gruppen. Abh. Math. Sem. Univ. Hamburg, 26:129–144, 1963.
[61] A. R., Calderbank, P., Delsarte, and N. J. A., Sloane. A strengthening of the Assmus–Mattson theorem. IEEE Trans. Inform. Theory, 37:1261–1268, 1991.
[62] P. J., Cameron. Biplanes. Math. Z, 131:85–101, 1973.
[63] P. J., Cameron. Parallelisms of Complete Designs. Cambridge: Cambridge University Press, 1976. London Mathematical Society Lecture Notes Series 23.
[64] P. J., Cameron and J. H. van, Lint. Graphs, Codes and Designs. Cambridge: Cambridge University Press, 1980. London Mathematical Society Lecture Notes Series 43.
[65] P., Camion. Difference Sets in Elementary Abelian Groups. Les Presses de l'Univérsite de Montreal, 1979. Seminaire de Mathématiques Supérieures, Département de Mathématiques et de Statistique, Université de Montréal.
[66] J., Cannon. Cayley: A Language for Group Theory. Department of Mathematics, University of Sydney, July 1982.
[67] P., Charpin. Codés cycliques étendus invariants sous le groupe affine. Thése de Doctorat d'État, Université Paris VII, 1987.
[68] P., Charpin. A new description of some polynomial codes: the primitive generalized Reed–Muller code. Technical report, Université Paris VII, 1985.
[69] P., Charpin. Une généralisation de la construction de Berman des codes de Reed et Muller p–aires. Communications in Algebra, 16:2231–2246, 1988.
[70] P., Charpin. Codes cycliques étendus affines–invariants et antichaines d'un ensemble partiellement ordonné. Discrete Math., 80:229–247, 1990.
[71] W. E., Cherowitzo. Hyper ovals in desarguesian planes of even order. Ann. of Discrete Math., 37:87–94, 1988.
[72] W. E.|Cherowitzo. Hyper ovals in the translation planes of order 16. J. Combin. Math. & Combin. Comput., 9:39–55, 1991.
[73] J. H., Conway. Three lectures on exceptional groups. In M. B., Powell and G., Higman, editors, Finite Simple Groups, pages 215–247. New York, London: Academic Press, 1971. Proceedings of an Instructional Conference Organized by the London Mathematical Society — a NATO Advanced Study Institute.
[74] J. H., Conway and V., Pless. On the enumeration of self–dual codes. J. Combin. Theory, Ser. A, 28:26–53, 1980.
[75] J. H., Conway and N. J. A., Sloane. Sphere Packings, Lattices and Groups. Grundlehren der mathematischen Wissenschaften 290. New York: Springer–Verlag, 1988.
[76] R. T., Curtis. The regular dodecahedron and the binary Golay code. Ars Combin., 29B:55–64, 1990.
[77] R. T., Curtis. On graphs and codes. Geom. Dedicata, 41:127–134, 1992.
[78] T., Czerwinski and D. J., Oakden. The translation planes of order twenty–five. J. Combin. Theory, Ser. A, 59:193–217, 1992.
[79] M., Dehon. Ranks of incidence matrices of t–designs S (t,t + 1, A). European J. Combin., 1:97–100, 1980.
[80] P., Delsarte. A geometric approach to a class of cyclic codes. J. Combin. Theory, 6:340–358, 1969.
[81] P., Delsarte. On cyclic codes that are invariant under the general linear group. IEEE Trans. Inform. Theory, 16:760–769, 1970.
[82] P., Delsarte. Majority logic decodable codes derived from finite inversive planes. Inform, and Control, 18:319–325, 1971.
[83] P., Delsarte, J. M., Goethals, and F. J., MacWilliams. On generalized Reed–Muller codes and their relatives. Inform, and Control, 16:403– 442, 1970.
[84] P., Delsarte, J. M., Goethals, and J. J., Seidel. Spherical codes and designs. Geom. Dedicata, 6:363–388, 1977.
[85] P., Dembowski. Finite Geometries. Ergebnisse der Mathematik undihrer Grenzbegiete, Band 44. Berlin, Heidelberg, New York: Springer–Verlag, 1968.
[86] P., Dembowski and A., Wagner. Some characterizations of finite projective spaces. Arch. Math., 11:465–469, 1960.
[87] U., Dempwolff and A., Reifart. The classification of the translation planes of order 16, I. Geom. Dedicata, 15:137–153, 1983.
[88] R. H. F., Denniston. Some new 5–designs. Bull. London Math. Soc, 8:263–267, 1976.
[89] L. E., Dickson. Linear Groups with an Exposition of the Galois Field Theory. New York: Dover Publications, 1958. (With an introduction by Wilhelm Magnus).
[90] J., Dieudonné. La Géométrie des Groupes Classiques. Ergebnisseder Mathematik und ihrer Grenzgebiete, Neue Folge, Band 5. Berlin, Gottingen, Heidelberg: Springer–Verlag, second edition, 1963.
[91] J., Dieudonné. Sur les Groupes Classiques. Actualites scientifiques et industrielles 1040. Paris: Hermann, third edition, 1973.
[92] J. F., Dillon. Private communication.
[93] J. F., Dillon. A survey of bent functions. NSAL–S–203,092.
[94] J. F., Dillon. Elementary Hadamard Difference Sets. PhD thesis, University of Maryland, 1974.
[95] J. F., Dillon. Elementary Hadamard difference sets. Congressus Numerantium, 14:237–249, 1975.
[96] J. F., Dillon and J. R., Schatz. Block designs with the symmetric difference property. In Robert L., Ward, editor, Proceedings of the NSA Mathematical Sciences Meetings, pages 159–164. The United States Government, 1987.
[97] S. T., Dougherty. Nets and their codes. PhD thesis, Lehigh University, 1992.
[98] J., Doyen. Linear spaces and Steiner systems. In M.|Aigner and D., Jungnickel, editors, Geometries and Groups, pages 30–42. Berlin: Springer–Verlag, 1981. Lecture Notes in Mathematics, 893.
[99] J., Doyen, X., Hubaut, and M., Vandensavel. Ranks of incidence matrices of Steiner triple systems. Math. Z., 163:251–259, 1978.
[100] G. L., Ebert. Translation planes of order q2: asymptotic estimates. Trans. Amer. Math. Soc, 238:301–308, 1978.
[101] J. C., Fisher, J. W. P., Hirschfeld, and J. A., Thas. Complete arcs in planes of square order. Ann. Discrete Math., 30:243–250, 1986.
[102] M. J., Ganley. A class of unitary block designs. Math. Z., 128:34–42, 1972.
[103] D., Ghinelli–Smit. Functions on symmetric designs. Ars Combin., 24B:217–230, 1987.
[104] D., Gluck. Affine planes and permutation polynomials. In Dijen, Ray–Chaudhuri, editor, Coding Theory and Design Theory, Part II, pages 99–100. Springer–Verlag, 1990. IMA Volumes in Mathematics and its Applications, 21.
[105] J. M., Goethals and P., Delsarte. On a class of majority–logic decodable cyclic codes. IEEE Trans. Inform. Theory, 14:182–188, 1968.
[106] J. M., Goethals and J. J., Seidel. Strongly regular graphs derived from combinatorial designs. Canad. J. Math., 22:597–614, 1970.
[107] M. J. E., Golay. Notes on digital coding. Proc. IRE, 37:657, 1949.
[108] M. J. E., Golay. Anent codes, priorities, patents, etc. Proc. IEEE, 64:572, 1976.
[109] R. L., Graham and F. J., MacWilliams. On the number of information symbols in difference–set cyclic codes. Bell System Tech. J., 45:1057–1070, 1966.
[110] K., Grey. Further results on designs carried by a code. Ars Combin., 26B:133–152, 1988.
[111] B. H., Gross. Intersection triangles and block intersection numbers of Steiner systems. Math. Z., 139:87–104, 1974.
[112] K. W., Gruenberg and A. J., Weir. Linear Geometry. Graduate Texts in Mathematics: 49. New York: Springer Verlag, second edition, 1977.
[113] J., Hadamard. Résolution d'une question relative aux determinants. Bull. Sci. Math., 2:240–246, 1893.
[114] A. J., Hahn and O. T., O'Meara. The Classical Groups and K–Theory. Grundlehren der mathematischen Wissenschaften 291. New York: Springer–Verlag, 1989.
[115] M. Hall, Jr.Projective planes. Trans. Amer. Math. Soc, 54:229–277, 1943.
[116] M. Hall, Jr.Cyclic projective planes. Duke Math. J., pages 1079–1090. 1947.
[117] M. Hall, Jr.A survey of difference sets. Proc. Amer. Math. Soc, 7:975–986, 1956.
[118] M. Hall, Jr.Automorphisms of Steiner triple systems. Proc. Sympos. Pure Math., 6:47–66, 1962.
[119] M. Hall, Jr.Note on the Mathieu group M12. Arch. Math., 13:334–340, 1962.
[120] M. Hall, Jr.Ovals in the desarguesian plane of order 16. Ann. Mat. Pura Appl. (4), 102:159–176, 1975. CLXXIV della Raccolta sotto gli auspici del Consiglio Nazionale delle Ricerche.
[121] M. Hall, Jr.Semi–automorphisms of Hadamard matrices. Math. Proc. Camb. Phil. Soc, 77:459–473, 1975.
[122] M. Hall, Jr.Combinatorial Theory. New York: Wiley, second edition, 1986.
[123] M. Hall, Jr. and H. J., Ryser.Cyclic incidence matrices. Canad. Math. J., 3:495–502, 1951.
[124] N., Hamada. The rank of the incidence matrix of points and d–flats in finite geometries. J. Sci. Hiroshima Univ. Ser. A–I, 32:381–396, 1968.
[125] N., Hamada. On the p–rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error correcting codes. Hiroshima Math. J., 3:153–226, 1973.
[126] N., Hamada. The geometric structure and the p–rank of an affine triple system derived from a nonassociative Moufang loop with the maximum associative center. J. Combin. Theory, Ser. A, 30:285–297, 1981.
[127] N., Hamada and H., Ohmori. On the BIB design having the minimum p–rank. J. Combin. Theory, Ser. A, 18:131–140, 1975.
[128] R. W., Hamming. Error detecting and error correcting codes. Bell System Tech. J., 29:147–160, 1950.
[129] R. W., Hamming. Coding and Information Theory. Englewood Cliffs, N.J.: Prentice Hall, 1980.
[130] H., Hanani.On quadruple systems. Canad. J. Math, 12:145–157, 1960.
[131] H., Hanani. The existence and construction of balanced incomplete block designs. Ann. Math. Statist, 32:361–386, 1961.
[132] H., Hanani. A class of three–designs. J. Combin. Theory, Ser. A, 26:1–19, 1979.
[133] C., Hering. On codes and projective designs. Technical Report 344, Kyoto University Mathematics Research Institute Seminar Notes, 1979.
[134] R., Hill. A First Course in Coding Theory. Oxford Applied Mathematics and Computing Science Series. Oxford: Oxford University Press, 1986.
[135] G., Hillebrandt. The p –rank of (0, l)–matrices. J. Combin. Theory, Ser. A, 60:131–139, 1992.
[136] J. W. P., Hirschfeld. Projective Geometries over Finite Fields. Oxford: Oxford University Press, 1979.
[137] J. W. P., Hirschfeld. Finite Projective Spaces of Three Dimensions. Oxford: Oxford University Press, 1985.
[138] G., Hiss. On the incidence matrix of the Ree unital. Preprint.
[139] A., Hocquenghem. Codes correcteurs d'erreurs. Chiffres, 2:147–158, 1959.
[140] G., Hölz. Construction of designs which contain a unital. Arch. Math., 37:179–183, 1981.
[141] D. A., Huffman. The synthesis of linear sequential coding networks. In Colin, Cherry, editor, Information Theory. London: Butterworths Scientific Publishers, 1956. Papers read at a Symposium on ‘Information Theory’ held in London in 1955.
[142] D. R., Hughes. On t–designs and groups. Amer. J. Math., 87:761–778, 1965.
[143] D. R., Hughes and F. C., Piper. Projective Planes. Graduate Texts in Mathematics 6. New York: Springer–Verlag, 1973.
[144] D. R., Hughes and F. C., Piper. Design Theory. Cambridge: Cambridge University Press, 1985.
[145] B., Huppert. Endliche Gruppen I. Berlin, Heidelberg: Springer Verlag, 1967.
[146] Q. M., Hussain. On the totality of the solutions for the symmetrical incomplete block designs A = 2, k = 5 or 6. Sankhya, 7:204–208, 1945.
[147] N., Ito, J. S., Leon, and J. Q., Longyear. Classification of 3–(24,12,5) designs and 24–dimensional Hadamard matrices. J. Combin. Theory, Ser. A, 31:66–93, 1981.
[148] D., Jungnickel and V. D., Tonchev. On symmetric and quasi–symmetric designs with the symmetric difference property and their codes. J. Combin. Theory, Ser. A, 59:40–50, 1992.
[149] W. M., Kantor. Plane geometries associated with certain 2–transitive groups. J. Algebra, 37:489–521, 1975.
[150] W. M., Kantor. Symplectic groups, symmetric designs and line ovals. J. Algebra, 33:43–58, 1975.
[151] W. M., Kantor. Homogeneous designs and geometric lattices. J. Combin. Theory, Ser. A, 38:66–74, 1985.
[152] I., Kaplansky. Linear Algebra and Geometry—A Second Course. Boston: Allyn and Bacon, 1969.
[153] T., Kasami, S., Lin, and W. W., Peterson. Some results on cyclic codes which are invariant under the affine group and their applications. Inform, and Control, 11:475–496, 1967.
[154] T., Kasami, S., Lin, and W. W., Peterson. New generalizations of the Reed–Muller codes. Part I: Primitive codes. IEEE Trans. Inform. Theory, 14:189–199, 1968.
[155] T., Kasami, S., Lin, and W. W., Peterson. Polynomial codes. IEEE Trans. Inform. Theory, 14:807–814, 1968.
[156] B. C., Kestenband. Unital intersections in finite projective planes. Geom. Dedicata, 11:107–117, 1981.
[157] J. D., Key. A class of 1–designs. European J. Combin., 14:37–41, 1993.
[158] J. D., Key. Hermitian varieties as codewords. Des. Codes Cryptogr., 1:255–259, 1991.
[159] J. D.|Key. Extendable Steiner systems. Geom. Dedicata, 41:201–205, 1992.
[160] J. D., Key and K., Mackenzie. An upper bound for the p–rank of a translation plane. J. Combin. Theory, Ser. A, 56:297–302, 1991.
[161] J. D., Key and K., Mackenzie. Ovals in the designs W (2m). Ars Combin., 33:113–117, 1992.
[162] J. D., Key and E. E., Shult. Steiner triple systems with doubly transitive automorphism groups: a corollary to the classification theorem for finite simple groups. J. Combin. Theory, Ser. A, 36:105–110, 1984.
[163] J. D., Key and A., Wagner. On an infinite class of Steiner systems constructed from affine spaces. Arch. Math., 47:376–378, 1986.
[164] R. E., Kibler. A summary of noncyclic difference sets, k < 20. J. Combin. Theory, Ser. A, 25:62–67, 1978.
[165] H., Kimura. Classification of Hadamard matrices of order 28 with Hall sets. Preprint.
[166] H., Kimura. On equivalence of Hadamard matrices. Hokkaido Math. J., 17:139–146, 1988.
[167] H., Kimura. New Hadamard matrix of order 24. Graphs and Combin., 5:235–242, 1989.
[168] H., Kimura and H., Ohmori. Construction of Hadamard matrices of order 28. Graphs Combin., 2:247–257, 1986.
[169] T. P., Kirkman. On a problem in combinations. Cambridge and Dublin Math. J., 2:191–204, 1847.
[170] E., Kleinfeld. Techniques for enumerating Veblen–Wedderburn systems. J. Assoc. Comput. Mach., 7:330–337, 1960.
[171] M., Klemm. Über die Reduktion von Permutationsmoduln. Math. Z., 143:113–117, 1975.
[172] M., Klemm. Über den p–Rang von Inzidenzmatrizen. J. Combin. Theory, Ser. A, 43:138–139, 1986.
[173] H., Koch. On self–dual, doubly even codes of length 32. J. Combin. Theory, Ser. A, 51:63–76, 1989.
[174] H., Koch. On self–dual doubly–even extremal codes. Discrete Math., 83:291–300, 1990.
[175] G., Korchmáros. Old and new results on ovals in finite projective planes. In A. D., Keedwell, editor, Surveys in Combinatorics, 1991, pages 41–72. Cambridge: Cambridge University Press, 1991. London Mathematical Society Lecture Note Series 166.
[176] C. W. H., Lam. The search for a finite projective plane of order 10. Amer. Math. Monthly, 98:305–318, 1991.
[177] C. W. H., Lam, G., Kolesova, and L., Thiel. A computer search for finite projective planes of order 9. Discrete Math., 92:187–195, 1991.
[178] C. W. H., Lam, L., Thiel, and A., Pautasso. On self–dual ternary codes generated by the inequivalent Hadamard matrices of order 24. Preprint.
[179] C. W. H., Lam, L., Thiel, and S., Swiercz. The non–existence of finite projective planes of order 10. Canad. J. Math., 41:1117–1123, 1989.
[180] C. W. H., Lam, L., Thiel, S., Swiercz, and J., McKay. The non–existence of ovals in a projective plane of order 10. Discrete Math., 45:319–321, 1983.
[181] E. S., Lander. Symmetric Designs: an Algebraic Approach. Cambridge: Cambridge University Press, 1983. London Mathematical Society Lecture Notes Series 74.
[182] P., Landrock and O., Manz. Classical codes as ideals in group algebras. Des. Codes Cryptogr., 2:273–285, 1992.
[183] J. S., Leon, V., Pless, and N. J. A., Sloane. Self–dual codes over GF(5). J. Combin. Theory, Ser. A, 32:178–194, 1982.
[184] J. S., Leon, V., Pless, and N. J. A., Sloane. On ternary self–dual codes of length 24. IEEE Trans. Inform. Theory, 27:176–180, 1981.
[185] R., Lidl and H., Niederreiter. Introduction to Finite Fields and their Applications. Cambridge: Cambridge University Press, 1986.
[186] J. H. van, Lint. Coding Theory. Lecture Notes in Mathematics, 201. Berlin: Springer–Verlag, 1970.
[187] J. H. van, Lint, A survey of perfect codes. Rocky Mountain J. Math., 5:199–224, 1975.
[188] J. H. van, Lint. Introduction to Coding Theory. Graduate Texts in Mathematics 86. New York: Springer–Verlag, 1982.
[189] J. H. van, Lint. Algebraic geometric codes. In Dijen, Ray–Chauddhuri, editor, Coding Theory and Design Theory, Part I, pages 137–162. New York: Springer–Verlag, 1990. IMA Volumes in Mathematics and its Applications, 20.
[190] J. H. van, Lint. Codes and combinatorial designs. In D., Jungnickel et al., editors, Design Theory, Coding Theory and Group Theory. New York: Wiley Inter science.
[191] J. H. van, Lint and G. van der, Geer. Introduction to Coding Theory and Algebraic Geometry. DMV Seminar Band 12. Basel: Birkhauser Verlag, 1988.
[192] P., Lorimer. A projective plane of order 16. J. Combin. Theory, 16:334–347, 1974.
[193] H., Lüneburg. Charakterisierungen der endlichen desarguesschen projektiven Ebenen. Math. Z., 85:419–450, 1964.
[194] H., Lüneburg. Some remarks concerning the Ree group of type (G2). J. Algebra, 3:256–259, 1966.
[195] H., Lüneburg. Lectures on projective planes. Technical report, University of Illinois at Chicago Circle, 1968/69.
[196] H., Lüneburg. Transitive Erweiterungen endlicher Permutationsgruppen. Lecture Notes in Mathematics, 84. Berlin: Springer–Verlag, 1969.
[197] H., Lüneburg. Translation Planes. New York: Springer–Verlag, 1980.
[198] L., Lunelli and M., See. k –Archi completi nei piani proietivi desarguesiani di rango 8 e 16. Technical report, Centro Calcoli Numerici,Politecnico di Milano, 1958.
[199] R. J., McEliece. The reliability of computer memories. Scientific American, 252:2–7, 1985.
[200] K., Mackenzie. Codes of Designs. PhD thesis, University of Birmingham, 1989.
[201] S., MacLane and G., Birkoff. Algebra: Second Edition. New York: Collier Macmillan, 1979.
[202] F. J., MacWilliams and H. B., Mann. On the p–rank of the design matrix of a difference set. Inform, and Control, 12:474–489, 1968.
[203] F. J., MacWilliams and N. J. A., Sloane. The Theory of Error– Correcting Codes. Amsterdam: North–Holland, 1983.
[204] F. J., MacWilliams, N. J. A., Sloane, and J. G., Thompson. Good self–dual codes exist. Discrete Math., 3:153–162, 1972.
[205] S. S., Magliveras and D. M., Leavitt. Simple 6–(33,8,36) designs from PFL2(32). In Computational Group Theory, pages 337–352. New York: Academic Press, 1984.
[206] J. A., Maiorana. A classification of the cosets of the Reed–Muller code R (l,6). Math. Comp., 57:403–414, 1991.
[207] H. B., Mann. Addition Theorems: The Addition Theorems of Group Theory and Number theory. Interscience Tracts in Pure and Applied Mathematics: 18. New York: Interscience Publishers, 1965.
[208] A., Maschietti. Hyperovals and Hadamard designs. J. Geom., 44:107– 116, 1992.
[209] J. L., Massey. Book Review: Theory and Practice of Error Control Codes, by R. E., Blahut. IEEE Trans. Inform. Theory, 31:553–554, 1985.
[210] R., Mathon. Constructions of cyclic Steiner 2–designs. Ann. Discrete Math.. 34:353–362, 1987.
[211] H. F., Mattson|Jr. Book Review: The Theory of Error–Correcting Codes, by F. J. Mac, Williams and N. J. A., Sloane. SI AM Review, 22:513–519, 1980.
[212] H. F. Mattson, Jr. and G., Solomon. A new treatment of Bose–Chaudhuri codes. J. Soc. Indust. Appl. Math., 9:654–669, 1961.
[213] N. S., Mendelsohn and B., Wolk. A search for a non–desarguesian plane of prime order. In C. A., Baker and L. M., Batten, editors, Finite Geometries, pages 199–208. New York: Marcel Dekker, 1985. Lecture Notes in Pure and Applied Mathematics, 103.
[214] P. K., Menon. Difference sets in abelian groups. Proc. Amer. Math. Soc, 11:368–376, 1960.
[215] R., Metz. On a class of unitals. Geom. Dedicata, 8:125–126, 1979.
[216] E. H., Moore. Concerning triple systems. Math. Ann., 43:271–285, 1893.
[217] E. H., Moore. Tactical memoranda I–III. Amer. J. Math., 18:264–303, 1896.
[218] G. E. Moor, house. Bruck nets, codes, and characters of loops. Des. Codes Cryptogr., 1:7–29, 1991.
[219] B., Mortimer. The modular permutation representations of the known doubly transitive groups. Proc. London Math. Soc. (3), 41:1–20, 1980.
[220] D. W., Newhart. On minimum weight codewords in QR codes. J. Combin. Theory, Ser. A, 48:104–119, 1988.
[221] C. W., Norman. Nonisomorphic Hadamard designs. J. Combin. Theory, Ser. A, 21:336–344, 1976.
[222] M. E., O'Nan. Automorphisms of unitary block designs. J. Algebra, 20:495–511, 1972.
[223] T. G., Ostrom. Semi–translation planes. Trans. Amer. Math. Soc, 111:1–18, 1964.
[224] T. G., Ostrom. Finite Translation Planes. Lecture Notes in Mathematics, 158. Berlin: Springer–Verlag, 1970.
[225] T. G., Ostrom and A., Wagner. On projective and affine planes with transitive collineation groups. Math. Z., 71:186–199, 1959.
[226] U., Ott. Endliche zyklische Ebenen. Math. Z., 144:195–215, 1975.
[227] U., Ott. Some remarks on representation theory in finite geometry. In M., Aigner and D., Jungnickel, editors, Geometries and Groups, pages 68–110. Berlin: Springer–Verlag, 1981. Lecture Notes in Mathematics, 893.
[228] U., Ott. An elementary introduction to algebraic methods for finite projective planes. Technical Report 50, Universita di Roma La Sapienza, Marzo 1984. Seminario di Geometrie Combinatorie, diretto da G. Tallini.
[229] L. J., Paige. A note on the Mathieu groups. Canad. J. Math., 9:15–18, 1957.
[230] R. E. A. C., Paley. On orthogonal matrices. J. Math. Phys., 12:311– 320, 1933.
[231] G., Panella. Caratterizzazione delle quadriche di uno spazio (tridimensionale) lineare sopra un corpo finito. Boll. Uni. Mat. Ital, Gruppo IV, Serie III, 10:507–513, 1955.
[232] D. S., Passman. Permutation Groups. New York: W.A. Benjamin Inc., 1968.
[233] S. E., Payne and J. A., Thas. Finite Generalized Quadrangles. Research Notes in Mathematics 110. Boston: Pitman, 1984.
[234] T., Penttila and I., Pinneri. Private communication.
[235] W. W., Peterson. Error–correcting codes. Scientific American, 206:96– 108, 1962.
[236] F., Piper. Unitary block designs. In R.M., Wilson, editor, Graph Theory and Combinatorics, pages 98–105. Pitman, 1979. Research Notes in Math., 34.
[237] V., Pless. Symmetry codes over GF(3) and new 5–designs. J. Combin. Theory, 12:119–142, 1972.
[238] V., Pless. The Theory of Error–Correcting Codes. New York: John Wiley and Sons, 1989. Second Edition.
[239] V., Pless and N. J. A., Sloane. Binary self–dual codes of length 24. Bull. Amer. Math. Soc, 80:1173–1178, 1974.
[240] A., Pott. Applications of the DFT to abelian difference sets. Arch. Math., 51:283–288, 1988.
[241] E., Prange. Cyclic error–correcting codes in two symbols. Electronics Research Directorate, Air Force Cambridge Research Center, September 1957. AFCRC–TN–57–103. ASTIA Document AD133749.
[242] E., Prange. An algorism for factoring xn — 1 over a finite field. Electronics Research Directorate, Air Force Cambridge Research Center, October 1959. AFCRC–TN–59–775.
[243] E., Prange. The use of coset equivalence in the analysis and decoding of group codes. Electronics Research Directorate, Air Force Cambridge Research Center, June 1959. AFCRC–TN–59–164.
[244] B., Qvist. Some remarks concerning curves of second degree in a finite plane. Ann. Acad. Sci. Fenn. Ser. AI, (134), 1952.
[245] D. K., Ray–Chaudhuri and Richard M., Wilson. On t–designs. Osaka J. Math., 12:737–744, 1975.
[246] R., Ree. A family of simple groups associated with the simple Lie algebra of type (G2). Amer. J. Math., 83:432–462, 1961.
[247] K. J., Rose. Generalized Reed–Muller codes and finite geometries. PhD thesis, Lehigh University, 1993.
[248] O. S., Rothaus. On “bent” functions. J. Combin. Theory, Ser. A, 20:300–305, 1976.
[249] L. D., Rudolph. A class of majority logic decodable codes. IEEE Trans. Inform. Theory, 13:305–307, 1967.
[250] H. J., Ryser. Combinatorial Mathematics. Mathematical Association of America, Wiley, 1963.
[251] H., Sachar. The Fp span of the incidence matrix of a finite projective plane. Geom. Dedicata, 8:407–415, 1979.
[252] R., Safavi–Naini and I. F., Blake. Generalized t–designs and weighted majority decoding. Inform, and Control, 42:261–282, 1979.
[253] R., Safavi–Naini and I. F., Blake. On designs from codes. Utilitas Math., 14:49–63, 1979.
[254] U., Scarpis. Sui determinant! di valore massimo. Rendiconti Reale Istituto Lombardo di Scienze e Lettere (Milan Rendiconti), 31:1441– 1446, 1898.
[255] T., Schaub. A linear complexity approach to cyclic codes. PhD thesis, Swiss Federal Institute of Technology, Zurich, 1988. Diss. ETH No. 8730.
[256] W. M., Schmidt. Equations over Finite Fields: An elementary approach. Berlin: Springer–Verlag, 1976.
[257] R., Schoof and M. van der, Vlugt. Hecke operators and the weight distributions of certain codes. J. Combin. Theory, Ser. A, 57:163–186, 1991.
[258] M. P., Schiitzenberger. A non–existence theorem for an infinite family of symmetrical block designs. Ann. Eugenics, pages 286–287, 1949.
[259] B., Segre. Ovals in a finite projective plane. Canad. J. Math., 7:414–416, 1955.
[260] C. E., Shannon. A mathematical theory of communication. Bell System Tech. J., 27:379–423,623–656, 1948.
[261] E. P., Shaughnessy. Associated t–designs and automorpism groups of certain linear codes. PhD thesis, Lehigh University, 1969.
[262] M. S., Shrikhande and S. S., Sane. Quasi–Symmetric Designs. Cambridge: Cambridge University Press, 1991. London Mathematical Society Lecture Notes Series, 164.
[263] S. S., Shrikhande and N. K., Singh. On a method of constructing incomplete block designs. Sankhya, A, 24:25–32, 1962.
[264] J., Siemons. Orbits in finite incidence structures. Geom. Dedicata, 14:87–94, 1983.
[265] J., Singer. A theorem in finite projective geometry and some applications to number theory. Trans. Amer. Math. Soc, 43:377–385, 1938.
[266] D., Slepian. Some further theory of group codes. Bell System Tech. J., 39:1219–1252, 1960.
[267] K. J. C., Smith. On the p–rank of the incidence matrix of points and hyperplanes in a finite projective geometry. J. Combin. Theory, 7:122–129, 1969.
[268] J., Steiner. Combinatorische Aufgabe. Crelle's Journal, XLV:181–182, 1853.
[269] J. J., Sylvester. Thoughts on Inverse Orthogonal Matrices, simultaneous Sign–succession, and Tessellated Pavements in two or more colours, with applications to Newton's Rule, Ornamental Tile–work, and the Theory of Numbers. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 34:461–475, December 1867.
[270] L., Teirlinck. On projective and affine hyperplanes. J. Combin. Theory, Ser. A, 28:290–306, 1980.
[271] L., Teirlinck. Non–trivial t–designs without repeated blocks exist for all t. Discrete Math., 65:301–311, 1987.
[272] J. A., Thas. Extensions of finite generalized quadrangles. Symposia Mathematica, 28:127–143, 1986. Published by Istituto Nazionale di Alta Matematica Francesco Severi and distributed by Academic Press.
[273] J. A., Thas. Solution of a classical problem on finite inversive planes. In W.M., Kantor, R.A., Liebler, S.E., Payne, and E.E., Shult, editors, Finite Geometries, Buildings, and Related Topics, pages 145–159. Oxford: Oxford University Press, 1990.
[274] J. G., Thompson. Fixed point free involutions and finite projective planes. In Michael J., Collins, editor, Finite Simple Groups II, pages 321–337. New York: Academic Press, 1980.
[275] A., Tietavainen. On the nonexistence of perfect codes over finite fields. SI AM J. Appl. Math., 24:88–96, 1973.
[276] J., Tits. Les groupes simples de Suzuki et de Ree. Seminaire Bourbaki, 13, 210:1–18, 1960/61.
[277] J. A., Todd. A combinatorial problem. J. Math. Phys., 12:321–333, 1933.
[278] J. A., Todd. Projective and Analytic Geometry. Pitman, 1947.
[279] V. D., Tonchev. Hadamard matrices of order 28 with automorphisms of order 7. J. Combin. Theory, Ser. A, 40:62–81, 1985.
[280] V. D., Tonchev. Quasi–symmetric 2–(31,7,7) designs and a revision of Hamada's conjecture. J. Combin. Theory, Ser. A, 42:104–110, 1986.
[281] V. D., Tonchev. Combinatorial Configurations. Pitman Monographs and Surveys in Pure and Applied Mathematics, 40. New York: Longman, 1988.
[282] V. D., Tonchev. Unitals in the Holz design on 28 points. Geom. Dedicata, 38:357–363, 1991.
[283] R. J., Turyn. Character sums and difference sets. Pacific J. Math., 15:319–346, 1965.
[284] Ju. L., Vasil'ev. On nongroup close–packed codes. Probl. Kibemet, 8:337–339, 1962. Translated from the Russian in Probleme der Kybernetik, 8(1965), 375–378.
[285] O., Veblen and J. H., Maclaglan–Wedderburn. Non–desarguesian and non–pascalian geometries. Trans. Amer. Math. Soc, 8:379–388, 1907.
[286] O., Veblen and J. W., Young. Projective Geometry: Volumes I and II. Boston: Ginn and Co., 1918.
[287] H. L. de, Vries. Some Steiner Quadruple Systems 5(3,4,16) such that all 16 derived Steiner Triple Systems 5(2,3,15) are isomorphic. Ars Combin., 24A: 107–129, 1987.
[288] A., Wagner. Orbits on finite incidence structures. Symposia Mathematica, 28:219–229, 1986.
[289] H. N., Ward. On Ree's series of simple groups. Trans. Amer. Math. Soc, 121:62–89, 1966.
[290] H. N., Ward. Quadratic residue codes and symplectic groups. J. Algebra, 29:150–171, 1974.
[291] H. N., Ward. Quadratic residue codes in their prime. J. Algebra,150:87–100, 1992.
[292] E. J. Weldon, Jr.New generalizations of the Reed–Muller codes. Part II: Nonprimitive codes. IEEE Trans. Inform. Theory, 14:199–205, 1968.
[293] M. A., Wertheimer. Designs in Quadrics. PhD thesis, University of Pennsylvania, 1986.
[294] M. A., Wertheimer. Oval designs in quadrics. Contemp. Math., 111:287–297, 1990.
[295] H., Weyl. The Classical Groups. Princeton: Princeton University Press, 1946.
[296] H., Wielandt. Finite Permutation Groups. New York: Academic Press, 1964.
[297] H., Wilbrink. A characterization of the classical unitals. In N. L., Johnson, M. J., Kallaher, and C. T., Long, editors, Finite Geometries, pages 445–454. Marcel Dekker,Inc, 1983. Lecture Notes in Pure and Applied Mathematics, 82.
[298] H. S., Wilf. The ‘Snake Oil’ method for proving combinatorial identities. In J., Siemons, editor, Surveys in Combinatorics, 1989, pages 208–217. Cambridge: Cambridge University Press, 1989. London Mathematical Society Lecture Note Series 141.
[299] R. M., Wilson. Inequalities in S(t,k,v). Lecture notes, IMA, Minnesota, 1988.
[300] E., Witt. Uber Steinersche Systeme. Abh. Math. Sem. Univ. Hamburg, 12:265–275, 1938.
[301] H. P., Young. Affine triple systems and matroid designs. Math. Z., 132:343–359. 1973.

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.