We computed the Smith normal form of Sylvester-Hadamard designs and its complement, their derived and residual Sylvester-Hadamard designs and their complementary derived and residual Sylvester-Hadamard designs. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 398 – 403 ¨ ITAK ˙ c TUB doi: Derived and residual Sylvester-Hadamard designs and the Smith normal form ∗ ˙ ˘ Ilhan HACIOGLU C ¸ anakkale Onsekiz Mart University, Department of Mathematics, C ¸ anakkale, Turkey Received: • • Accepted: Published Online: • Printed: Abstract: We computed the Smith normal form of Sylvester-Hadamard designs and its complement, their derived and residual Sylvester-Hadamard designs and their complementary derived and residual Sylvester-Hadamard designs. Key words: Sylvester-Hadamard design, Smith normal form, derived, residual designs 1. Preliminaries The p-ranks of Sylvester-Hadamard designs play an important role in shift registers and pseudo-noise matrices [1]. In this article by finding the Smith normal form we completely solve this problem, give formulas for their derived and residual Sylvester-Hadamard designs and their complementary derived and residual SylvesterHadamard designs. By a balanced incomplete block design (BIBD ) with parameters (v, b, r, k, λ) we mean an arrangement of v treatments into b subsets of these treatments called “blocks”, such that (i) each block consists of k distinct treatments; (ii) each treatment occurs in r different blocks; (iii) each pair of distinct treatments occur together in λ different blocks. The following equations are satisfied by any BIBD : vr = bk and λ(v − 1) = r(k − 1) A BIBD is said to be symmetric if v = b and in consequence r = k . We call such a design a (v, k, λ) design. Existence of a (v, k, λ) design implies the existence of its derived and residual design with parameters (k, b − 1, r − 1, λ, λ − 1) and (v − k, b − 1, r, k − λ, λ), respectively. They are obtained, respectively, by deleting a block of the (v, k, λ) design retaining all the treatments in b − 1 blocks that appear (or do not appear) in .
