On μ-resolvable BIB designs

Block Designs: A Randomization Approach

“…Since then, many constructional methods and combinatorial properties of a-resolvable BIB designs have been given, together with relevant tabulations. See, for example, Raghavarao (1963,1964), Sprott (1956), Kageyama (1973aKageyama ( , 1973bKageyama ( , 1976aKageyama ( , 1976b, Mohan (1980), Mohan and Kageyama (1982), Kageyama and Mohan (1983), Rajkundlia (1983). Here, a list of 14 parameter combinations of such a-resolvable BIB designs within the practical range of a ;::: 2, r :::; 10 and v :::; 100 is presented in Table 9.3.…”

“…A large number of a-resolvable BIB designs with a ::::: 1 are available in the literature. For example, refer to Bose (1947), Kageyama (1972bKageyama ( , 1973bKageyama ( , 1976b, Mohan (1980), Kageyama and Mohan (1983), and Shrikhande and Raghavarao (1963). In particular, Kageyama (1972b) and Kageyama and Mohan (1983) have tabulated practical parameter combinations for such BIB designs (along with their solutions) under some restrictions on the range of parameters.…”

“…For the exhaustive list of BIB designs with a prime v, see Takeuchi (1962) who gives difference sets generating BIB designs for general v ::; 100 and r ::; 20, and symmetric BIB designs for general v ::; 100 and r ::; 30 (see also Table 8.2). Furthermore, Kageyama and Mohan (1983) systematically tabulated 237 parameter combinations of a-resolvable BIB designs within the scope of the range as in v ::; 125, b ::; 250,..\ ::; 100,3 ::; k ::; v -3 with a ;::: 2, along with construction methods which are also available in the literature. The reader who is interested in designs with larger values of parameters can refer to the tabulation.…”

“…9* ofTable 9.1; Section 6.0.3;Kageyama and Mohan, 1983), one can get a 3-resolvable semi-regular GD design with parameters v = 12, b = 36, r = 12, k = 4'),1 = 0'),2 = 4, m = 4, n = 3, Le., a 3-resolvable (3; 8; O)-EB design with co = 1, 101 = 3/4, Po = 3, PI = 8, Lo = (I4-4-1141~)03-1131~, Ll = I40(I3-3-1131~).On the other hand, an affine 3-resolvable BIB design with parameters v = 49, b = 56, r = 24, k = 21,)' = 10, ql = 7, q2 = 9, j3 = 7, t = 8(Shrikhande and Raghavarao, 1963) produces a 21-resolvable semi-regular GD design with parameters v = 56, b = 392, r = 168, k = 24'),1 = 56,)'2 = 72, m = 8, n = 7, Le., a 21-resolvable (7; 48; O)-EB design with co = 1,101 = 35/36,po = 7, PI = 48, Lo = (Is-8-11s1~)07-1171~, Ll = Is 0 (17 _7-1171~).However, this design has larger values on parameters.…”

On Construction of Hadamard Rhotrix using a Special Type of Mn- Matrix

“…Using the M n -matrix pattern Vasic and Milenkovic [18] gave a method of construction of Low-Density Parity Check (LDPC) codes. µ-resolvable and affine µ-resolvable Balanced Incomplete Block Designs (BIBD) and Partially Balanced Incomplete Block Designs (PBIBD) were constructed by Kageyama and Mohan [3] using the M n -matrices.…”