Best Known (111−41, 111, s)-Nets in Base 9
(111−41, 111, 448)-Net over F9 — Constructive and digital
Digital (70, 111, 448)-net over F9, using
- 3 times m-reduction [i] based on digital (70, 114, 448)-net over F9, using
- trace code for nets [i] based on digital (13, 57, 224)-net over F81, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 13 and N(F) ≥ 224, using
- net from sequence [i] based on digital (13, 223)-sequence over F81, using
- trace code for nets [i] based on digital (13, 57, 224)-net over F81, using
(111−41, 111, 897)-Net over F9 — Digital
Digital (70, 111, 897)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9111, 897, F9, 41) (dual of [897, 786, 42]-code), using
- 785 step Varšamov–Edel lengthening with (ri) = (6, 3, 2, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 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, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 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, 17 times 0, 1, 17 times 0, 1, 19 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 28 times 0, 1, 30 times 0, 1, 31 times 0, 1, 34 times 0, 1, 36 times 0, 1, 37 times 0, 1, 40 times 0, 1, 43 times 0, 1, 45 times 0) [i] based on linear OA(941, 42, F9, 41) (dual of [42, 1, 42]-code or 42-arc in PG(40,9)), using
- dual of repetition code with length 42 [i]
- 785 step Varšamov–Edel lengthening with (ri) = (6, 3, 2, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 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, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 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, 17 times 0, 1, 17 times 0, 1, 19 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 24 times 0, 1, 25 times 0, 1, 27 times 0, 1, 28 times 0, 1, 30 times 0, 1, 31 times 0, 1, 34 times 0, 1, 36 times 0, 1, 37 times 0, 1, 40 times 0, 1, 43 times 0, 1, 45 times 0) [i] based on linear OA(941, 42, F9, 41) (dual of [42, 1, 42]-code or 42-arc in PG(40,9)), using
(111−41, 111, 183874)-Net in Base 9 — Upper bound on s
There is no (70, 111, 183875)-net in base 9, because
- 1 times m-reduction [i] would yield (70, 110, 183875)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 926 180566 479447 282442 352554 816113 345984 562440 985650 228650 935005 052356 899312 836253 561168 579803 136803 493601 > 9110 [i]