Infinite Families of Non-Embeddable Quasi-Residual Menon Designs Tariq Alraqad, Mohan Shrikhande Department of Mathematics, Central Michigan University, Mt. Pleasant, MI 48859, USA, E-mail: alraq1ta@cmich.edu; shrik1m@cmich.edu Received October 29, 2007; revised February 26, 2008 Published online 22 April 2008 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.20192 Abstract: A Menon design of order h 2 is a symmetric (4h 2 , 2h 2 - h, h 2 - h)-design. Quasi- residual and quasi-derived designs of a Menon design have parameters 2 - (2h 2 + h, h 2 ,h 2 - h) and 2 - (2h 2 - h, h 2 - h, h 2 - h - 1), respectively. In this article, regular Hadamard matrices are used to construct non-embeddable quasi-residual and quasi-derived Menon designs. As applications, we construct the first two new infinite families of non-embeddable quasi-residual and quasi-derived Menon designs. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 53–62, 2009 Keywords: quasi-residual design; Menon design; regular Hadamard matrix 1. INTRODUCTION We refer to [3] as a general reference on designs and to [8] for symmetric designs and related combinatorial structures. A 2-(v, k, λ) design is a pair D = (X, B), where X is a v-set, B is a collection of k-subsets of X, and every pair of points is contained in exactly λ blocks. In a 2-(v, k, λ) design every point is contained in exactly r = λ(v−1) k−1 blocks and the number of blocks in the design is b = vr k . Alternatively, the notation (v, b, r, k, λ) design is used to represent a 2-(v, k, λ) design. If X ={x 1 ,x 2 , ··· ,x v } and B ={B 1 ,B 2 , ··· ,B b }, then the incidence matrix of D is a v × b matrix A = [a ij ], where a ij is 1 if x i ∈ B j and 0 otherwise. A (0, 1) matrix A of size v × b is an incidence matrix of a (v, b, r, k, λ) de- sign if and only if J v A = kJ v×b , and AA T = (r − λ)I v + λJ v , where I v , J v , and J v×b are the identity matrix of order v, the v × v all one matrix, and the v × b all one matrix, respectively. A(v, b, r, k, λ) design with v = b (equivalently r = k), is called a symmetric (v, k, λ)- design. The complementary design D ′ of a design D is obtained by replacing every block of D by its complement. The complementary design of a (v, b, r, k, λ) design is a (v, b, b − r, v − k, b − 2r + λ) design. Journal of Combinatorial Designs © 2008 Wiley Periodicals, Inc. 53