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GBRD's: Some new constructions for difference matrices, generalised hadamard matrices and balanced generalised weighing matrices

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Abstract

A new construction is given for difference matrices. The generalized Hadamard matrices GH(q(q − 1)2; EA(q)) are constructed whenq andq − 1 are both prime powers. Other generalised Hadamard matrices are also shown to exist. For example, there exist GH(n; G) forn = 52 ⋅ 2 ⋅ 3, 26 ⋅ 32, 112 ⋅ 22 ⋅ 3, 172 ⋅ 2 ⋅ 32, 532 ⋅ 2 ⋅ 33, 712 ⋅ 22 ⋅ 32, 1072 ⋅ 22 ⋅ 33, 1492 ⋅ 52 ⋅ 2 ⋅ 3,.... Finally, a new construction for the BGW ((q 4 − 1)/(q − 1),q 3,q 2(q − 1);q q-1), and a construction for the new BGW ((q 8 − 1)/(q 2 − 1),q 6,q 4(q 2 − 1);G) are given, wheneverq is a prime power, andG is a group of orderq + 1.

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de Launey, W. GBRD's: Some new constructions for difference matrices, generalised hadamard matrices and balanced generalised weighing matrices. Graphs and Combinatorics 5, 125–135 (1989). https://doi.org/10.1007/BF01788664

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