# 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} }

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

#### Topics from this paper

#### 30 Citations

Cyclotomy, Gauss Sums, Difference Sets and Strongly Regular Cayley Graphs

- Mathematics, Computer Science
- SETA
- 2012

We survey recent results on constructions of difference sets and strongly regular Cayley graphs by using union of cyclotomic classes of finite fields. Several open problems are raised.

Sequences and Functions Derived from Projective Planes and Their Difference Sets

- Computer Science, Mathematics
- WAIFI
- 2012

This paper surveys some of these objects which are related to difference set representations of projective planes related to the interplay between the additive and the multiplicative structure of the finite field. Expand

Partial geometric designs and difference families

- Mathematics
- 2015

We examine the designs produced by different types of difference families. Difference families have long been known to produce designs with well behaved automorphism groups. These designs provide the… Expand

The Elementary Divisors of the Incidence Matrices of Skew Lines in PG(3,p)

- Mathematics
- 2010

The elementary divisors of the incidence matrices of lines in $PG(3,p)$ are computed, where two lines are incident if and only if they are skew.

Partial difference sets from quadratic forms and $p$-ary weakly regular bent functions

- Mathematics
- 2010

We generalize the construction of affine polar graphs in two different ways to obtain new partial difference sets and amorphic association schemes. The first generalization uses a combination of… Expand

Partial difference sets from $p$-ary weakly regular bent functions and quadratic forms

- Mathematics
- 2010

Fe b 20 10 PARTIAL DIFFERENCE SETS FROM pARY WEAKLY REGULAR BENT FUNCTIONS AND QUADRATIC FORMS

- 2010

We generalize the construction of affine polar graphs in two different ways to obtain partial difference sets and amorphic association schemes. In the first generalization we replace the quadratic… Expand

Cyclotomic constructions of skew Hadamard difference sets

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 2012

Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields by using union of cyclotomic classes of F"q of order N=2p"1^m, where p"1 is a prime and m a positive integer. Expand

Tiling Groups with Difference Sets

- Mathematics, Computer Science
- Electron. J. Comb.
- 2015

In the case of tilings with two difference sets, the equivalence to skew Hadamard difference sets is shown, and it is proved that they must be normalized if the group is abelian. Expand

Efficient matrix rank computation with application to the study of strongly regular graphs

- Mathematics, Computer Science
- ISSAC '07
- 2007

A new heuristic approach is given that is notably effective in practice as applied to the strongly regular graph adjacency matrices, which involves projection to a matrix of order slightly above the rank. Expand

#### References

SHOWING 1-10 OF 160 REFERENCES

Rank of inclusion matrices and modular representation theory

- Mathematics
- 1990

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

- Mathematics
- 1997

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.

Finite Geometry and Character Theory

- Mathematics
- 1995

Preliminaries: Incidence structures with singer groups.- Examples: Existence and non-existence.- Difference sets with classical parameters.- Semiregular relative difference sets.- Projective planes… Expand

Symmetric Designs: An Algebraic Approach

- Mathematics
- 1983

1. Symetric Designs 2. An Algebraic Approach 3. Automorphisms 4. Difference Sets 5. Multiplier Theorems 6. Open Questions Appendices.

Proof of a Conjecture on Hadamard 2-Groups

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 1993

It is proved by construction that every abelian 2-group that meets the exponent bound has a difference set. Expand

The invariant factors of some cyclic difference sets

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 2003

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 the… Expand

On the p-Rank of the Design Matrix of a Difference Set

- Mathematics, Computer Science
- Inf. Control.
- 1968

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

- Mathematics
- 2005

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

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 1984

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

- Mathematics, Computer Science
- J. Comb. Theory, Ser. A
- 1996

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.