Best Known (102, 102+45, s)-Nets in Base 9
(102, 102+45, 768)-Net over F9 — Constructive and digital
Digital (102, 147, 768)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (3, 25, 28)-net over F9, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- the Hermitian function field over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- digital (77, 122, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 61, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 61, 370)-net over F81, using
- digital (3, 25, 28)-net over F9, using
(102, 102+45, 3348)-Net over F9 — Digital
Digital (102, 147, 3348)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9147, 3348, F9, 45) (dual of [3348, 3201, 46]-code), using
- 3200 step Varšamov–Edel lengthening with (ri) = (7, 3, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 15 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 20 times 0, 1, 21 times 0, 1, 23 times 0, 1, 23 times 0, 1, 25 times 0, 1, 26 times 0, 1, 28 times 0, 1, 29 times 0, 1, 31 times 0, 1, 32 times 0, 1, 34 times 0, 1, 36 times 0, 1, 39 times 0, 1, 40 times 0, 1, 42 times 0, 1, 44 times 0, 1, 47 times 0, 1, 50 times 0, 1, 52 times 0, 1, 54 times 0, 1, 58 times 0, 1, 61 times 0, 1, 64 times 0, 1, 67 times 0, 1, 71 times 0, 1, 74 times 0, 1, 79 times 0, 1, 83 times 0, 1, 87 times 0, 1, 91 times 0, 1, 96 times 0, 1, 102 times 0, 1, 106 times 0, 1, 112 times 0, 1, 119 times 0, 1, 124 times 0, 1, 130 times 0, 1, 138 times 0, 1, 144 times 0, 1, 153 times 0, 1, 160 times 0) [i] based on linear OA(945, 46, F9, 45) (dual of [46, 1, 46]-code or 46-arc in PG(44,9)), using
- dual of repetition code with length 46 [i]
- 3200 step Varšamov–Edel lengthening with (ri) = (7, 3, 1, 2, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 4 times 0, 1, 4 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 6 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 9 times 0, 1, 9 times 0, 1, 10 times 0, 1, 11 times 0, 1, 11 times 0, 1, 12 times 0, 1, 13 times 0, 1, 13 times 0, 1, 15 times 0, 1, 15 times 0, 1, 16 times 0, 1, 17 times 0, 1, 18 times 0, 1, 18 times 0, 1, 20 times 0, 1, 21 times 0, 1, 23 times 0, 1, 23 times 0, 1, 25 times 0, 1, 26 times 0, 1, 28 times 0, 1, 29 times 0, 1, 31 times 0, 1, 32 times 0, 1, 34 times 0, 1, 36 times 0, 1, 39 times 0, 1, 40 times 0, 1, 42 times 0, 1, 44 times 0, 1, 47 times 0, 1, 50 times 0, 1, 52 times 0, 1, 54 times 0, 1, 58 times 0, 1, 61 times 0, 1, 64 times 0, 1, 67 times 0, 1, 71 times 0, 1, 74 times 0, 1, 79 times 0, 1, 83 times 0, 1, 87 times 0, 1, 91 times 0, 1, 96 times 0, 1, 102 times 0, 1, 106 times 0, 1, 112 times 0, 1, 119 times 0, 1, 124 times 0, 1, 130 times 0, 1, 138 times 0, 1, 144 times 0, 1, 153 times 0, 1, 160 times 0) [i] based on linear OA(945, 46, F9, 45) (dual of [46, 1, 46]-code or 46-arc in PG(44,9)), using
(102, 102+45, 2434786)-Net in Base 9 — Upper bound on s
There is no (102, 147, 2434787)-net in base 9, because
- 1 times m-reduction [i] would yield (102, 146, 2434787)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 20 864431 224897 424682 387839 849392 767808 346420 952471 137898 844838 370252 173033 714231 853623 851831 192140 478937 096724 195414 969076 911247 553595 855505 > 9146 [i]