Collected PD-sets: 1. $C$ the $[28,21,4]_2$ code of the hermitian unital 2-(28,4,1) has $|S| \ge 4$; Aut($C$) is Sp$_6(2)$ and a PD-set of four elements can be found; blox2:= [ {10, 14, 25, 28}, {6, 7, 22, 23}, {5, 12, 14, 22}, {5, 7, 13, 28}, {10, 11, 15, 23}, {2, 12, 23, 26}, {1, 4, 5, 23}, {10, 16, 18, 19}, {4, 17, 18, 28}, {2, 5, 9, 10}, {5, 6, 8, 15}, {12, 13, 16, 17}, {1, 7, 11, 25}, {1, 2, 6, 17}, {3, 15, 17, 21}, {9, 15, 20, 22}, {6, 11, 14, 18}, {3, 14, 24, 26}, {4, 11, 12, 27}, {9, 17, 25, 26}, {3, 16, 23, 28}, {11, 22, 26, 28}, {1, 9, 14, 16}, {13, 19, 23, 24}, {4, 8, 9, 19}, {3, 9, 11, 13}, {8, 21, 23, 25}, {12, 15, 19, 25}, {2, 7, 15, 16}, {3, 7, 8, 18}, {4, 10, 21, 22}, {1, 15, 18, 26}, {2, 8, 14, 27}, {8, 10, 13, 26}, {6, 13, 20, 25}, {3, 6, 10, 12}, {6, 9, 21, 28}, {7, 14, 19, 21}, {7, 9, 12, 24}, {1, 10, 20, 24}, {5, 18, 24, 25}, {4, 6, 16, 24}, {2, 11, 21, 24}, {3, 5, 20, 27}, {16, 22, 25, 27}, {6, 19, 26, 27}, {2, 19, 20, 28}, {1, 13, 21, 27}, {7, 10, 17, 27}, {8, 17, 22, 24}, {2, 13, 18, 22}, {15, 24, 27, 28}, {4, 13, 14, 15}, {12, 18, 20, 21}, {14, 17, 20, 23}, {4, 7, 20, 26}, {5, 16, 21, 26}, {1, 8, 12, 28}, {1, 3, 19, 22}, {2, 3, 4, 25}, {9, 18, 23, 27}, {5, 11, 17, 19}, {8, 11, 16, 20}]; des2:=Design<2,28|blox2>; Permutation group au2 acting on a set of cardinality 28 Order = 12096 = 2^6 * 3^3 * 7 //PDset [(1, 8, 15, 28, 6, 24, 21, 4, 2, 22, 3, 18) (5, 27, 9, 16, 11, 10, 25, 13, 19, 7, 26, 12)(14, 20, 23), (1, 26, 5, 12, 10, 7)(2, 9, 24, 20, 4, 23) (3, 27, 11, 15, 16, 14)(6, 17, 25, 18, 21, 22)(8, 13, 28), Id(au2), (1, 8, 25, 2, 5, 19, 26)(3, 10, 11, 27, 18, 28, 21) (4, 9, 17, 6, 15, 12, 23)(7, 14, 24, 20, 16, 22, 13)] ------------------------------------------------------------------------------- 2. $C^{\perp}$, for $C$ as above, is a $[28,7,12]_2$; here $|S| \ge 10$; we found a PD-set of 30 elements; ------------------------------------------------------------------------------- 3. for $C$ the $[28,19,4]_2$ code of the Ree unital 2-(28,4,1) has $|S| \ge 4$; Aut($C$) is $P\Gamma L_2(8)$ and a PD-set of four elements can be found; [{ 1, 2, 6, 11 }, { 6, 9, 10, 27 }, { 2, 12, 23, 26 }, { 12, 13, 27, 28 }, { 4, 17, 18, 23 }, { 10, 13, 20, 25 }, { 1, 17, 24, 28 }, { 6, 15, 20, 24 }, { 5, 10, 15, 21 }, { 7, 12, 16, 21 }, { 3, 13, 16, 22 }, { 3, 11, 15, 23 }, { 9, 11, 19, 28 }, { 9, 18, 20, 26 }, { 1, 14, 15, 26 }, { 6, 21, 23, 25 }, { 8, 15, 25, 28 }, { 5, 12, 17, 20 }, { 5, 13, 23, 24 }, { 2, 18, 21, 28 }, { 1, 20, 23, 27 }, { 4, 6, 16, 28 }, { 3, 4, 20, 21 }, { 10, 11, 12, 18 }, { 1, 9, 13, 21 }, { 6, 7, 17, 26 }, { 7, 18, 22, 24 }, { 1, 12, 22, 25 }, { 6, 13, 18, 19 }, { 1, 5, 16, 19 }, { 3, 6, 12, 14 }, { 11, 13, 14, 17 },{ 8, 9, 12, 24 }, { 4, 8, 13, 26 }, { 2, 16, 17, 25 }, { 2, 3, 24, 27 }, { 5, 7, 11, 27 }, { 9, 15, 17, 22 }, { 10, 16, 24, 26 }, { 3, 10, 17, 19 }, { 4, 11, 24, 25 }, { 9, 14, 16, 23 }, { 3, 5, 26, 28 }, { 8, 11, 16, 20 }, { 3, 7, 9, 25 }, { 11, 21, 22, 26 }, { 2, 4, 5, 9 }, { 19, 25, 26, 27 }, { 5, 14, 18, 25 }, { 1, 4, 7, 10 }, { 5, 6, 8, 22 }, { 4, 12, 15, 19 }, { 2, 7, 13, 15 }, { 4, 14, 22, 27 }, { 7, 8, 19, 23 }, { 10, 22, 23, 28 },{ 7, 14, 20, 28 }, { 2, 19, 20, 22 }, { 8, 17, 21, 27 }, { 14, 19, 21, 24 }, { 2, 8, 10, 14 }, { 15, 16, 18, 27 },{ 1, 3, 8, 18 }]; Permutation group au acting on a set of cardinality 28 Order = 1512 = 2^3 * 3^3 * 7 \\pdset [ (1, 20, 24)(2, 14, 5)(3, 26, 12)(4, 10, 25)(6, 28, 23)(7, 13, 11) (8, 18, 9)(15, 17, 27)(16, 22, 21), (1, 16, 2, 4, 11, 28)(3, 12, 14)(5, 25, 9, 24, 19, 17) (7, 8, 21, 10, 20, 18)(13, 26, 22, 23, 27, 15), (1, 7, 28, 8, 18, 22, 10, 14, 25)(2, 5, 12, 4, 20, 15, 3, 24, 23) (6, 11, 27, 13, 26, 9, 17, 19, 21), (1, 13, 3, 6, 10, 26)(2, 20, 5, 11, 25, 28)(4, 8, 18, 19, 17, 7) (9, 16, 14)(12, 27, 24, 15, 21, 22)] ------------------------------------------------------------------------------- 4. $C$ the $[31,16,6]_5$ code with $PGL_3(F_5)$ acting (cyclic code), the bound is 7, and a set of 14, inside a cyclic group of order 31 was found; [31, 16, 6] Cyclic Code over GF(5) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 1 1 3 2 1 1 4 0 1 2 3 2 1] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 4 0 3 1 1 0 2 4 4 4 4 1 1] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 3 2 1 0 0 1 2 3 2 1 2 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 3 2 1 0 0 1 2 3 2 1 2] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 4 4 2 4 0 4 2 0 4 3 2 3 4] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 4 2 1 0 2 4 0 1 3 2 1 1 1 4 4] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 4 4 3 2 3 4 0 1 0 3 3 3 4 3 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 4 4 3 2 3 4 0 1 0 3 3 3 4 3] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 3 0 1 1 4 1 0 1 3 1 2 2 4 2 1] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 3 4 0 3 2 0 4 2 3 0 0 4 2 1] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 2 3 2 1 1 4 0 2 2 3 2 2 1] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 4 3 3 4 4 1 0 4 4 2 3 4] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 4 1 2 1 2 0 4 0 3 1 1 1 2 4 4] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 4 4 2 3 4 4 1 0 4 3 2 3 4 4 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 4 4 2 3 4 4 1 0 4 3 2 3 4 4] > au; Permutation group au acting on a set of cardinality 31 Order = 372000 = 2^5 * 3 * 5^3 * 31 > PDset; [ (1, 10, 18, 21, 8, 22, 31, 15, 13, 2, 11, 24, 6, 12, 30, 27, 20, 23, 14, 29, 28, 17, 3, 16, 19, 7, 9, 4, 5, 26, 25), (1, 19, 23, 11, 21, 4, 28, 12, 31, 25, 16, 20, 2, 18, 9, 29, 6, 22, 26, 3, 27, 13, 10, 7, 14, 24, 8, 5, 17, 30, 15), (1, 16, 27, 15, 25, 3, 30, 31, 26, 17, 12, 22, 5, 28, 6, 8, 4, 29, 24, 21, 9, 14, 11, 18, 7, 23, 2, 10, 19, 20, 13), (1, 27, 25, 30, 26, 12, 5, 6, 4, 24, 9, 11, 7, 2, 19, 13, 16, 15, 3, 31, 17, 22, 28, 8, 29, 21, 14, 18, 23, 10, 20), (1, 15, 30, 17, 5, 8, 24, 14, 7, 10, 13, 27, 3, 26, 22, 6, 29, 9, 18, 2, 20, 16, 25, 31, 12, 28, 4, 21, 11, 23, 19), (1, 25, 26, 5, 4, 9, 7, 19, 16, 3, 17, 28, 29, 14, 23, 20, 27, 30, 12, 6, 24, 11, 2, 13, 15, 31, 22, 8, 21, 18, 10), (1, 3, 12, 8, 9, 23, 13, 25, 17, 6, 21, 7, 20, 15, 26, 28, 24, 18, 19, 27, 31, 5, 29, 11, 10, 16, 30, 22, 4, 14, 2), (1, 30, 5, 24, 7, 13, 3, 22, 29, 18, 20, 25, 12, 4, 11, 19, 15, 17, 8, 14, 10, 27, 26, 6, 9, 2, 16, 31, 28, 21, 23), (1, 31, 6, 14, 19, 25, 22, 24, 23, 16, 26, 8, 11, 20, 3, 5, 21, 2, 27, 17, 4, 18, 13, 30, 28, 9, 10, 15, 12, 29, 7), (1, 26, 4, 7, 16, 17, 29, 23, 27, 12, 24, 2, 15, 22, 21, 10, 25, 5, 9, 19, 3, 28, 14, 20, 30, 6, 11, 13, 31, 8, 18), (1, 17, 24, 10, 3, 6, 18, 16, 12, 21, 19, 30, 8, 7, 27, 22, 9, 20, 31, 4, 23, 15, 5, 14, 13, 26, 29, 2, 25, 28, 11), (1, 12, 9, 13, 17, 21, 20, 26, 24, 19, 31, 29, 10, 30, 4, 2, 3, 8, 23, 25, 6, 7, 15, 28, 18, 27, 5, 11, 16, 22, 14), (1, 18, 8, 31, 13, 11, 6, 30, 20, 14, 28, 3, 19, 9, 5, 25, 10, 21, 22, 15, 2, 24, 12, 27, 23, 29, 17, 16, 7, 4, 26), (1, 7, 29, 12, 15, 10, 9, 28, 30, 13, 18, 4, 17, 27, 2, 21, 5, 3, 20, 11, 8, 26, 16, 23, 24, 22, 25, 19, 14, 6, 31) ] ------------------------------------------------------------------------------- 5. the dual of the above is a $[31,15,10]_5$ code, self-orthogonal, the bound is 28, and the normalizer of a Sylow 31-subgroup has order 93. One such group was found to be a PD-set. > C; [31, 15] Linear Code over GF(5) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 4 0 3 1 1 0 2 4 4 4 4 1 1 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 4 0 3 1 1 0 2 4 4 4 4 1 1] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 1 2 4 1 4 2 1 4 1 3 0 0 3 3 1] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 4 0 2 2 1 0 3 2 4 0 2 4 1 4 2 3] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 2 2 3 2 3 3 2 3 1 2 3 0 2 3 1 2] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 3 4 4 3 1 1 1 2 4 3 4 0 2 0 1 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 4 0 4 0 0 0 1 0 0 4 0 1 1 4 1] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 4 0 0 0 1 4 4 0 4 1 1 0 1 0 0 4] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 3 4 0 3 2 0 4 2 3 0 0 4 2 1 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 3 4 0 3 2 0 4 2 3 0 0 4 2 1] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 4 1 2 3 1 4 2 2 3 0 3 4 1 4 3 2] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 3 1 3 2 2 4 2 2 3 0 2 0 1 4 2 3] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 2 1 4 3 3 4 1 2 1 1 3 0 3 3 1 2] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 3 4 3 4 2 1 2 1 3 3 3 0 2 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 4 0 3 1 1 0 2 4 4 4 4 1 1 0 1] Permutation group nsss acting on a set of cardinality 31 (1, 7, 22, 20, 13, 29, 24, 3, 8, 25, 9, 26, 2, 5, 6, 12, 19, 11, 4, 30, 27, 18, 28, 31, 21, 23, 16, 15, 14, 10, 17) (1, 24, 7)(2, 15, 17)(3, 16, 29)(4, 28, 10)(5, 18, 13)(6, 12, 25)(8, 27, 9)(11, 14, 21)(19, 20, 30)(22, 23, 26) > #nsss; 93 -------------------------------------------------------------------------------