Multipliers and generalized multipliers of cyclic objects and cyclic codes

Classification of Cyclic Steiner Quadruple Systems

“…Theorem 8 handles the case of v = 26 in the current work, as well as corroborates the earlier result [13] that CSQS (22)s are isomorphic iff they are multiplier equivalent.…”

“…Unfortunately, from (1) and the fact that φ(v) is even for v ≥ 3, it follows that Pálfy's result is not applicable in the current work. Neither is the result by Huffman et al [22] that isomorphism of cyclic objects over a set of size p 2 , where p is a odd prime, can be checked using at most φ(p 2 ) permutations. Consider cyclic objects over {0, 1, .…”

“…Phelps [36] proved that there are 29 isomorphism classes of CSQS (20)s. Finally, Frenz and Kreher [13] showed that there are 114 isomorphism classes of CSQS (22)s, thereby correcting an earlier erroneous result in [8]. One aim of the current paper is to extend these classification results.…”

“…By [36], there is no affine-invariant CSQS (20). By the discussion after Theorem 3, there are exactly 4 isomorphism classes of affine-invariant CSQS(26)s. Number = 51 Order = 392 Generators: (1,8,14,20,25,4,10,15,22,27,5,12,17,23,2,7,13,19,26,3,9,16,21,28,6,11,18,24 (20,24), (1,11,7,4,14,10,6,2,12,8,3,13,9,5), (1,21,14,24,12,25,9,28,7,15,6,17,3,20) (2,22,13,23,11,26,10,27,8,16,5,18,4,19 (9,20,…”

“…Number = 249 Order = 3528 Generators: (1,5)(2,6)(3,4) (7,23,20,27,12,16) (8,24,19,28,11,15)(9,26)(10,25)(13,21)(14,22)(17,18), (1,24,14,19,25,15,10,11,22,7,5,4,17,28,2,23,13,20,26,16,9,12,21,8,6,3,18,27). Number = 326 Order = 7056 Generators: (1,13)(2,14)(3,11)(4,12)(5,9)(6,10)(7,8) (17,21,20,28,24,25) (18,22,19,27,23,26), (1,22,7,26,14,16,6,19,12,23,3,27,9,18) (2,21,8,25,13,15,5,20,11,24,4,28,10,17), (1,…”