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We introduce the class of skew-circulant lattice rules. These are s-dimensional lattice rules that may be generated by the rows of an s × s skewcirculant matrix. (This is a minor variant of the familiar circulant matrix.) We present briefly some of the underlying theory of these matrices and rules. We are particularly interested in finding rules of specified trigonometric degree d. We describe some of the results of computer-based searches for optimal four-dimensional skew-circulant rules.doi:10.1090/s0025-5718-03-01534-5 fatcat:bxlttbnvabhftcyjkkfnd5oumq