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Paper IPM / M / 2311  


Abstract:  
A Hadamard matrix H of order 4n^{2} is called regular if every
row and column sum of H is 2n. A Bushtype Hadamard matrix is
a regular Hadamard matrix of order 4n^{2} with the additional
property of being a block matrix H=[H_{ij}] with blocks of size
2n such that H_{ii}=J_{2n} and H_{ij}J_{2n}=J_{2n}H_{ij}=0,i ≠ j, where J_{m} denotes the allone m by m matrix.
A balanced generalized weighing matrix BGW(v,k,λ)
over a multiplicative group G is a v by v matrix
W=[g_{ij}] with entries from ―G=G∪{0} such that
each row of W contains exactly k nonzero entries, and for
every a, b ∈ { 1,..., v}, a ≠ b, the multiset
{g_{ai}g_{bi}^{−1}  1 ≤ i ≤ v,g_{ai} ≠ 0,g_{bi} ≠ 0}
contains exactly λ/G/ copies of each element of G.
In this paper a Bushtype Hadamard matrix of order 36 is used in a
symmetric BGW(26,25,24) with zero diagonal over a cyclic
group of order 12 to construct a twin strongly regular graph with
parameters (936,375,150,150) whose points can be partitioned in 26
cocliques of size 36. The same Hadamard matrix then is used in a
symmetric BGW(50,49,48) with zero diagonal over a cyclic
group of order 12 to construct a twin strongly regular graph with
parameters (1800,1029,588,588) whose points can be partitioned in
50 cocliques of size 36.
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