Recent progress in algebraic design theory

@article{Xiang2005RecentPI,
  title={Recent progress in algebraic design theory},
  author={Qing Xiang},
  journal={Finite Fields Their Appl.},
  year={2005},
  volume={11},
  pages={622-653}
}
  • Qing Xiang
  • Published 2005
  • Computer Science, Mathematics
  • Finite Fields Their Appl.
We survey recent results on difference sets, p-ranks and Smith normal forms of certain set-inclusion matrices and subspace-inclusion matrices. 
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References

SHOWING 1-10 OF 160 REFERENCES
Rank of inclusion matrices and modular representation theory
We use results from the modular representation theory of the groupsSn and GLn (Fq) to determine the rank of inclusion matrices.
Geometry, Combinatorial Designs and Related Structures: Difference sets: an update
In the last few years there has been rapid progress in the theory of difference sets. This is a survey of these fascinating new developments.
Proof of a Conjecture on Hadamard 2-Groups
TLDR
It is proved by construction that every abelian 2-group that meets the exponent bound has a difference set. Expand
Finite Geometry and Character Theory
Preliminaries: Incidence structures with singer groups.- Examples: Existence and non-existence.- Difference sets with classical parameters.- Semiregular relative difference sets.- Projective planesExpand
The invariant factors of some cyclic difference sets
Using the Smith normal forms of the symmetric designs associated with the HKM and Lin difference sets, we show that not only are these two families of difference sets inequivalent, but also that theExpand
On the p-Rank of the Design Matrix of a Difference Set
Abstract : The rank mod p (a prime) of the incidence matrix of the Euclidean and projective hyperplanes is determined. Similar results are obtained for certain other design matrices. (Author)
On a conjecture of Ma
In this paper, we prove a result concerning a conjecture of Ma from diophantine equations, which is connected to an open problem on abelian difference sets of multiplier −1.
A Special Class of Williamson Matrices and Difference Sets
  • R. Turyn
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. A
  • 1984
TLDR
A construction is given of a very special class of Hadamard matrices of the Williamson kind and difference sets of order 4 · 32m. Expand
A Result on Ma's Conjecture
We give a proof for one of the conjectures of S. L. Ma on two Diophantine equations related to abelian difference sets with multiplier ?1.
The Elementary Divisors of the Incidence Matrices of Points and Linear Subspaces in Pn(Fp)
The elementary divisors of the incidence matrices between points and linear subspaces of fixed dimension in Pn(Fp) are computed.
...
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2
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5
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