Recent progress in algebraic design theory

We survey recent results on difference sets, p-ranks and Smith normal forms of certain set-inclusion matrices and subspace-inclusion matrices.

[1]  K. T. Arasu,et al.  Some New Difference Sets , 1995, J. Comb. Theory, Ser. A.

[2]  Andreas Stein,et al.  High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams , 2004 .

[3]  M. Muzychuk,et al.  Difference Sets with n = 2pm , 1998 .

[4]  Qing Xiang,et al.  Cyclic Relative Difference Sets and their p-Ranks , 2003, Des. Codes Cryptogr..

[5]  Richard M. Wilson,et al.  A course in combinatorics , 1992 .

[6]  Marshall Hall,et al.  A survey of difference sets , 1956 .

[7]  Alexander Pott,et al.  Finite Geometry and Character Theory , 1995 .

[8]  Qing Xiang On Balanced Binary Sequences with Two-Level Autocorrelation Functions , 1998, IEEE Trans. Inf. Theory.

[9]  Siu Lun Ma A Family of Difference Sets Having -1 as an Invariant , 1989, Eur. J. Comb..

[10]  H. Kharaghani On the Siamese Twin Designs , 2001 .

[11]  H. Ryser Combinatorial Mathematics: THE PRINCIPLE OF INCLUSION AND EXCLUSION , 1963 .

[12]  Siu Lun Ma,et al.  Difference sets in dihedral groups , 1991, Des. Codes Cryptogr..

[13]  Vladimir D. Tonchev,et al.  Spreads in Strongly Regular Graphs , 1996, Des. Codes Cryptogr..

[14]  Tanja Vu i i New symmetric designs and nonabelian difference sets with parameters(100, 45, 20) , 2000 .

[15]  L. D. Baumert,et al.  On the existence of cyclic difference sets with small parameters , 2003, math/0304502.

[16]  David B. Chandler,et al.  The invariant factors of the incidence matrices of points and subspaces in $\operatorname{PG}(n,q)$ and $\operatorname{AG}(n,q)$ , 2006 .

[17]  ROBERT E. KIBLER,et al.  A Summary of Noncyclic Difference Sets, k<20 , 1978, J. Comb. Theory, Ser. A.

[18]  T. Helleseth,et al.  A New Family of Ternary Sequences with Ideal Two-level Autocorrelation Function , 2000, 2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060).

[19]  Bernhard Schmidt,et al.  The Field Descent Method , 2005, Des. Codes Cryptogr..

[20]  Hadi Kharaghani,et al.  A Recursive Construction for New Symmetric Designs , 2005, Des. Codes Cryptogr..

[21]  Henk D. L. Hollmann,et al.  Maximal Arcs in Projective Three-Spaces and Double-Error-Correcting Cyclic Codes , 2001, J. Comb. Theory, Ser. A.

[22]  Jr. Hall,et al.  Cyclic projective planes , 1947 .

[23]  Robert L. McFarland,et al.  Translates and multipliers of abelian difference sets , 1978 .

[24]  James A. Davis,et al.  Difference sets in abelian 2-groups , 1991, J. Comb. Theory, Ser. A.

[25]  Peter Sin,et al.  THE PERMUTATION MODULES FOR GL( n +1, [ ] q ) ACTING ON ℙ n ([ ] q ) AND [ ] n +1 q , 2000 .

[26]  Richard M. Wilson,et al.  Constructions of Hadamard Difference Sets , 1997, J. Comb. Theory, Ser. A.

[27]  David B. Chandler,et al.  The invariant factors of the incidence matrices of points and subspaces in PG(n,q) and AG(n,q) , 2003, math/0312506.

[29]  Siu Lun Ma McFarland's conjecture on abelian difference sets with multiplier −1 , 1991, Des. Codes Cryptogr..

[30]  Qing Xiang,et al.  A Result on Ma's Conjecture , 1996, J. Comb. Theory, Ser. A.

[31]  Zhenlei Jia,et al.  New Necessary Conditions for the Existence of Difference Sets without Self-Conjugacy , 2002, J. Comb. Theory, Ser. A.

[32]  Jack E. Graver,et al.  The Module Structure of Integral Designs , 1973, J. Comb. Theory, Ser. A.

[33]  Robert L. McFarland,et al.  A Family of Difference Sets in Non-cyclic Groups , 1973, J. Comb. Theory A.

[34]  P. Dembowski,et al.  Planes of ordern with collineation groups of ordern2 , 1968 .

[35]  D. Jungnickel,et al.  Another case of the prime power conjecture for finite projective planes. , 2002 .

[36]  Richard M. Wilson,et al.  Signed Hypergraph Designs and Diagonal Forms for Some Incidence Matrices , 1999, Des. Codes Cryptogr..

[37]  Qing Xiang,et al.  An exponent bound on skew Hadamard abelian difference sets , 1994, Des. Codes Cryptogr..

[38]  Antonio Maschietti Difference Sets and Hyperovals , 1998, Des. Codes Cryptogr..

[39]  Robert S. Coulter,et al.  Planar Functions and Planes of Lenz-Barlotti Class II , 1997, Des. Codes Cryptogr..

[40]  Gary L. Mullen,et al.  Finite Fields and Applications , 2007, Student mathematical library.

[41]  Hadi Kharaghani On the Twin Designs with the Ionin-type Parameters , 2000, Electron. J. Comb..

[42]  David B. Chandler THE SMITH NORMAL FORMS OF DESIGNS WITH CLASSICAL PARAMETERS , 2004 .

[43]  S. Lang Number Theory III , 1991 .

[44]  James A. Davis,et al.  A Unifying Construction for Difference Sets , 1997, J. Comb. Theory, Ser. A.

[45]  Tor Helleseth,et al.  A New Family of Ternary Sequences with Ideal Two-level Autocorrelation Function , 2001, Des. Codes Cryptogr..

[46]  Ken W. Smith,et al.  Non-Abelian Hadamard Difference Sets , 1995, J. Comb. Theory, Ser. A.

[47]  D. Gottlieb A certain class of incidence matrices , 1966 .

[48]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[49]  Anka Golemac,et al.  New difference sets in nonabelian groups of order 100 , 2001 .

[50]  D. Jungnickel,et al.  Balanced generalized weighing matrices and their applications , 2006 .

[51]  David G. Glynn,et al.  On the classification of geometric codes by polynomial functions , 1995, Des. Codes Cryptogr..

[52]  Paul Camion,et al.  Antisymmetric difference sets , 1972 .

[53]  Arieh Yakir,et al.  Rank of inclusion matrices and modular representation theory , 1990 .

[54]  Yury J. Ionin New Symmetric Designs from Regular Hadamard Matrices , 1997, Electron. J. Comb..

[55]  Jong-Seon No,et al.  Binary Pseudorandom Sequences of Period with Ideal Autocorrelation Generated by the Polynomial , 1998 .

[56]  Vladimir D. Tonchev,et al.  Bush‐type Hadamard matrices and symmetric designs , 2001 .

[57]  Siu Lun Ma,et al.  Nonexistence of Abelian difference sets: Lander's conjecture for prime power orders , 2003 .

[58]  Rudolf Lide,et al.  Finite fields , 1983 .

[59]  Chris D. Godsil,et al.  Problems in Algebraic Combinatorics , 1995, Electron. J. Comb..

[60]  Bernhard Schmidt,et al.  Asymptotic Nonexistence of Difference Sets in Dihedral Groups , 2002, J. Comb. Theory, Ser. A.

[61]  Jennifer Wallis,et al.  On Hadamard matrices , 1975 .

[62]  A. S. Hedayat,et al.  On Theory and Applications of BIB Designs with Repeated Blocks. , 1977 .

[63]  Jong-Seon No,et al.  On the p-Ranks and Characteristic Polynomials of Cyclic Difference Sets , 2004, Proceedings IEEE International Symposium on Information Theory,.

[64]  Qing Xiang,et al.  Symmetric Bush-type Hadamard matrices of order $4m^4$ exist for all odd $m$ , 2005 .

[65]  Complete settling of the multiplier conjecture for the case of n =3pr , 2002 .

[66]  J. H. van Lint,et al.  A Course in Combinatorics: Trees , 2001 .

[67]  Navin M. Singhi,et al.  Tags on Subsets , 2003, Electron. Notes Discret. Math..

[68]  Dieter Jungnickel,et al.  Geometry, Combinatorial Designs and Related Structures: Difference sets: an update , 1997 .

[69]  Robert A. Liebler Constructive Representation Theoretic Methods and Non-Abelian Difference Sets , 1999 .

[70]  Wan Zhaoze Difference Sets in Groups of Order 4p~4 , 2006 .

[71]  R. Tennant Algebra , 1941, Nature.

[72]  Richard M. Wilson,et al.  A Diagonal Form for the Incidence Matrices of t-Subsets vs.k-Subsets , 1990, Eur. J. Comb..

[73]  A. Blokhuis,et al.  Proof of the prime power conjecture for projective planes of order $n$ with Abelian collineation groups of order $n^2$ , 2001 .

[74]  Zvonimir Janko The Existence of a Bush-type Hadamard Matrix of Order 36 and Two New Infinite Classes of Symmetric Designs , 2001, J. Comb. Theory, Ser. A.

[75]  Hans Dobbertin,et al.  New cyclic difference sets with Singer parameters , 2004, Finite Fields Their Appl..

[76]  Robert G. Kraemer,et al.  Proof of a Conjecture on Hadamard 2-Groups , 1993, J. Comb. Theory, Ser. A.

[77]  Yu Qing Chen On the Existence of Abelian Hadamard Difference Sets and a New Family of Difference Sets , 1997 .

[78]  N. Hamada,et al.  On the $p$-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its applications to error correcting codes , 1973 .

[79]  E. Lander Symmetric Designs: An Algebraic Approach , 1983 .

[80]  Peter Sin,et al.  The Elementary Divisors of the Incidence Matrices of Points and Linear Subspaces in Pn(Fp) , 2000 .

[81]  Qing Xiang Recent Results on Difference Sets with Classical Parameters , 1999 .

[82]  S. Chowla,et al.  On Difference Sets. , 1949, Proceedings of the National Academy of Sciences of the United States of America.

[83]  D. Wan,et al.  A CHEVALLEY-WARNING APPROACH TO p-ADIC ESTIMATES OF CHARACTER SUMS , 1995 .

[84]  Bhaskar Bagchi,et al.  Projective Geometric Codes , 2002, J. Comb. Theory, Ser. A.

[85]  Joel E. Iiams On Difference Sets in Groups of Order 4p2 , 1995, J. Comb. Theory, Ser. A.

[86]  Shuo-Yen Robert Li,et al.  On the Structure of t-Designs , 1980, SIAM J. Algebraic Discret. Methods.

[87]  D. Garling,et al.  Algebra, Volume 1 , 1969, Mathematical Gazette.

[88]  Richard J. Turyn,et al.  Hadamard Matrices, Baumert-Hall Units, Four-Symbol Sequences, Pulse Compression, and Surface Wave Encodings , 1974, J. Comb. Theory A.

[89]  Mikhail H. Klin,et al.  A Root Graph that is Locally the Line Graph of the Peterson Graph , 2003, Discret. Math..

[90]  Peter Sin,et al.  The Permutation Modules for GL(n+1, Fq) Acting on Pn(Fq) and Fqn+1 , 2000 .

[91]  Henk D. L. Hollmann,et al.  Gauss Sums, Jacobi Sums, and p-Ranks of Cyclic Difference Sets , 1999, J. Comb. Theory, Ser. A.

[92]  K. T. Arasu,et al.  A New Family of Cyclic Difference Sets with Singer Parameters in Characteristic Three , 2003, Des. Codes Cryptogr..

[93]  Solomon W. Golomb,et al.  Binary Pseudorandom Sequences of Period 2n-1 with Ideal Autocorrelation , 1998, IEEE Trans. Inf. Theory.

[94]  F. Luca,et al.  On a conjecture of Ma , 2005 .

[95]  William M. Kantor,et al.  Note on GMW Designs , 2001, Eur. J. Comb..

[96]  R. Turyn Character sums and difference sets. , 1965 .

[97]  Vladimir D. Tonchev,et al.  The Existence of a Bush-Type Hadamard Matrix of Order 324 and Two New Infinite Classes of Symmetric Designs , 2001, Des. Codes Cryptogr..

[98]  Hanfried Lenz,et al.  Design theory , 1985 .

[99]  Behruz Tayfeh-Rezaie,et al.  A Hadamard matrix of order 428 , 2005 .

[100]  K. A. Bush Unbalanced Hadamard Matrices and Finite Projective Planes of Even Order , 1971, J. Comb. Theory, Ser. A.

[101]  N. S. Barnett,et al.  Private communication , 1969 .

[102]  N Hamada,et al.  On the BIB Design Having the Minimum p-Rank , 1975, J. Comb. Theory A.

[103]  F. Jessie MacWilliams,et al.  On the p-Rank of the Design Matrix of a Difference Set , 1968, Inf. Control..

[104]  J. W. P. Hlrschfeld SYMMETRIC DESIGNS An Algebraic Approach (London Mathematical Society Lecture Note Series, 74) , 1984 .

[105]  Richard J. Turyn,et al.  A Special Class of Williamson Matrices and Difference Sets , 1984, J. Comb. Theory, Ser. A.

[106]  Philippe Delsarte,et al.  On cyclic codes that are invariant under the general linear group , 1970, IEEE Trans. Inf. Theory.

[107]  Qing Xiang,et al.  The invariant factors of some cyclic difference sets , 2003, J. Comb. Theory, Ser. A.

[108]  Jong-Seon No,et al.  New Cyclic Difference Sets with Singer Parameters Constructed from d-Homogeneous Functions , 2004, Des. Codes Cryptogr..

[109]  Mikhail H. Klin,et al.  Switching of Edges in Strongly Regular Graphs I: A Family of Partial Difference Sets on 100 Vertices , 2003, Electron. J. Comb..

[110]  E. C Johnsen,et al.  Skew-hadamard abelian group difference sets , 1966 .

[111]  K. T. Arasu,et al.  On abelian (22m+1(2m-1+1), 2m(2m+1), 2m)-difference sets , 2006, J. Comb. Theory, Ser. A.

[112]  John F. Dillon,et al.  Multiplicative Difference Sets via Additive Characters , 1999, Des. Codes Cryptogr..