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“School of Mathematics”

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Paper   IPM / M / 2311
School of Mathematics
  Title:   A block negacyclic Bush-type Hadamard matrix and two strongly regular graphs
  Author(s): 
1.  Z. Janko
2.  H. Kharaghani
  Status:   Published
  Journal: J. Combin. Theory Ser. A
  Vol.:  98
  Year:  2002
  Pages:   118-126
  Supported by:  IPM
  Abstract:
A Hadamard matrix H of order 4n2 is called regular if every row and column sum of H is 2n. A Bush-type Hadamard matrix is a regular Hadamard matrix of order 4n2 with the additional property of being a block matrix H=[Hij] with blocks of size 2n such that Hii=J2n and HijJ2n=J2nHij=0,ij, where Jm denotes the all-one 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=[gij] with entries from G=G∪{0} such that each row of W contains exactly k nonzero entries, and for every a, b ∈ { 1,..., v}, ab, the multi-set {gaigbi−1 | 1 ≤ iv,gai ≠ 0,gbi ≠ 0} contains exactly λ/|G|/ copies of each element of G.
In this paper a Bush-type 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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