On the Definition of Bose-Shimamoto's Cyclical Association Scheme

Constructing cyclic PBIBD(2)s through an optimization approach: Thirty‐two new cyclic designs

“…Hence DðAÞ ¼ ÀDðAÞ and EðAÞ ¼ ÀEðAÞ. This implies that the above relations are symmetrical (see [15]). Proof.…”

“…Hence DðAÞ ¼ ÀDðAÞ and EðAÞ ¼ ÀEðAÞ. This implies that the above relations are symmetrical (see [15]). (iii) Each symbols has exactly n 1 and n 2 first and second associates, respectively.…”

Construction of three-associate class PBIB designs by generating a set of square matrices

“…The above set of n 2 blocks constitutes a PBIB design with the generalized parameters as given below: (15,11,12,13,14,20,25,5,10,16,22,3,9) B16: (16,17,18,19,20,21,1,6,11,22,3,9,15) B17: (17,18,19,20,16,22,2,7,12,23,4,10,11) B18: (18,19,20,16,17,23,3,8,13,24,5,6,12) B19: (...…”

“…Let us illustrate the above construction methodology for n = 6. Now using the matrix S(1) and following the above construction methodology, we will obtain the following set of 12 blocks: B1: (1, 2, 3, 4, 5, 6) B7: (2, 8, 14, 20, 26, 32) B2: (19,20,21,22,23,24) B8: (5,11,17,23,29,35) B3: (1,7,13,19,25,31) B9: (13,14,15,16,17,18) B4: (4,10,16,22,28,34) B10: (31, 32, 33, 34, 35, 36) B5: (7,8,9,10,11,12) B11: (3, 9, 15, 21, 27, 33) B6: (25,26,27,28,29,30) B12: (6, 12, 18, 24, 30, 36) The above set of blocks constitutes a three-associate class PBIB design with the following parameters:…”

“…The first classification of PBIB designs based on their association schemes was performed by [6]. Later on, authors like [7], [8], [9], [10], [11] etc. also studied PBIB designs having twoassociate classes.…”