Structure of group invariant weighing matrices of small weight

@article{Leung2018StructureOG,
  title={Structure of group invariant weighing matrices of small weight},
  author={Ka Hin Leung and Bernhard Schmidt},
  journal={J. Comb. Theory, Ser. A},
  year={2018},
  volume={154},
  pages={114-128}
}
Abstract We show that every weighing matrix of weight n invariant under a finite abelian group G can be generated from a subgroup H of G with | H | ≤ 2 n − 1 . Furthermore, if n is an odd prime power and a proper circulant weighing matrix of weight n and order v exists, then v ≤ 2 n − 1 . We also obtain a lower bound on the weight of group invariant matrices depending on the invariant factors of the underlying group. These results are obtained by investigating the structure of subsets of finite… Expand

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References

SHOWING 1-10 OF 38 REFERENCES
Finiteness of circulant weighing matrices of fixed weight
TLDR
It is shown that, for every odd prime power q, there are at most finitely many proper circulant weighing matrices of weight q. Expand
Symmetric Weighing Matrices Constructed using Group Matrices
TLDR
It is proved that there is no symmetric abelian group weighing matrices of order 2pr and weight p2 where p is a prime and p≥ 5. Expand
Group developed weighing matrices
TLDR
The question of existence for 318 weighing matrices of order and weight both below 100 is answered, and some of the new results provide insight into the existence of matrices with larger weights and orders. Expand
Some results on weighing matrices
It is shown that if q is a prime power then there exists a circulant weighing matrix of order q 2 + q + 1 with q 2 nonzero elements per row and column. This result allows the bound N to be lowered inExpand
Finitely Generated Abelian Groups and Similarity of Matrices over a Field
Part 1 :Finitely Generated Abelian Groups: Matrices with Integer Entries: The Smith Normal Form.- Basic Theory of Additive Abelian Groups.- Decomposition of Finitely Generated Z-Modules. Part 2:Expand
Some New Results on Circulant Weighing Matrices
We obtain a few structural theorems for circulant weighing matrices whose weight is the square of a prime number. Our results provide new schemes to search for these objects. We also establish theExpand
Circulant weighing matrices of weight 22t
TLDR
A complete computer search is made for all circulant weighing matrices of order 16 such that MMT = kIn for some positive integer t and new structural results are obtained. Expand
Characters and Cyclotomic Fields in Finite Geometry
1. Introduction: The nature of the problems.- The combinatorial structures in question.- Group rings, characters, Fourier analysis.- Number theoretic tools.- Algebraic-combinatorial tools. 2. TheExpand
Perfect Ternary Arrays
A perfect ternary array is an r-dimensional array with entries 0, +1 and —1 such that all of its out-of-phase periodic autocorrelation coefficients are zero. Such an array is equivalent to a groupExpand
Determination of all possible orders of weight 16 circulant weighing matrices
We show that a circulant weighing matrix of order n and weight 16 exists if and only if n>=21 and n is a multiple of 14,21 or 31.
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