Best Known (58−19, 58, s)-Nets in Base 9
(58−19, 58, 364)-Net over F9 — Constructive and digital
Digital (39, 58, 364)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (9, 18, 164)-net over F9, using
- trace code for nets [i] based on digital (0, 9, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 9, 82)-net over F81, using
- digital (21, 40, 200)-net over F9, using
- trace code for nets [i] based on digital (1, 20, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 20, 100)-net over F81, using
- digital (9, 18, 164)-net over F9, using
(58−19, 58, 1131)-Net over F9 — Digital
Digital (39, 58, 1131)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(958, 1131, F9, 19) (dual of [1131, 1073, 20]-code), using
- 1072 step Varšamov–Edel lengthening with (ri) = (3, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 13 times 0, 1, 14 times 0, 1, 16 times 0, 1, 19 times 0, 1, 21 times 0, 1, 24 times 0, 1, 28 times 0, 1, 32 times 0, 1, 36 times 0, 1, 41 times 0, 1, 46 times 0, 1, 53 times 0, 1, 60 times 0, 1, 68 times 0, 1, 77 times 0, 1, 88 times 0, 1, 99 times 0, 1, 112 times 0, 1, 127 times 0) [i] based on linear OA(919, 20, F9, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,9)), using
- dual of repetition code with length 20 [i]
- 1072 step Varšamov–Edel lengthening with (ri) = (3, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 9 times 0, 1, 10 times 0, 1, 13 times 0, 1, 14 times 0, 1, 16 times 0, 1, 19 times 0, 1, 21 times 0, 1, 24 times 0, 1, 28 times 0, 1, 32 times 0, 1, 36 times 0, 1, 41 times 0, 1, 46 times 0, 1, 53 times 0, 1, 60 times 0, 1, 68 times 0, 1, 77 times 0, 1, 88 times 0, 1, 99 times 0, 1, 112 times 0, 1, 127 times 0) [i] based on linear OA(919, 20, F9, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,9)), using
(58−19, 58, 573051)-Net in Base 9 — Upper bound on s
There is no (39, 58, 573052)-net in base 9, because
- 1 times m-reduction [i] would yield (39, 57, 573052)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 2 465063 433874 085196 166936 550624 064774 953147 544100 817121 > 957 [i]