The solution of the Waterloo problem

Circulant weighing matrices

“…De ne A = f g i j W(1 i ) = 1 i = 0 1 : : : n ; 1g and B = f g i j W(1 i ) = ; 1 i = 0 1 : : : n ; 1g (1) It is easy to see that j A j + j B j= k.…”

Cyclic Relative Difference Sets with Classical Parameters

“…Our construction can be regarded as the affine analogue of the construction in [1]. Our proof, like the one in [1], exploits an F 2 -quadratic form; but instead of a geometric argument we give here a very simple algebraic argument based on the Hadamard transform.…”

“…In particular, when q=4 and d=3 our construction yields a cyclic (21, 6, 16, 2)-relative difference set as found by Lam. Our strategy is to combine the affine hyperplane construction of ((q d &1)Â(q&1), q&1, q d&1 , q d&2 )-relative difference sets and the construction of ((q d &1)Â(q&1), 2, q d&1 , q d&2 (q&1)Â2)-relative difference sets in [1]. Our construction can be regarded as the affine analogue of the construction in [1].…”

“…Our strategy is to combine the affine hyperplane construction of ((q d &1)Â(q&1), q&1, q d&1 , q d&2 )-relative difference sets and the construction of ((q d &1)Â(q&1), 2, q d&1 , q d&2 (q&1)Â2)-relative difference sets in [1]. Our construction can be regarded as the affine analogue of the construction in [1]. Our proof, like the one in [1], exploits an F 2 -quadratic form; but instead of a geometric argument we give here a very simple algebraic argument based on the Hadamard transform.…”

Circulant weighing designs

“…Strassler [14] has recently announced new results giving CW(31, 16), CW(71, 25), and CW(127, 64). CW(31, 16) and CW(127, 64) are already constructed in [6] and [1], but the CW(71, 25) of Strassler is new. For more on weighing designs and related topics refer to [6].…”