Best Known (72, 111, s)-Nets in Base 8
(72, 111, 363)-Net over F8 — Constructive and digital
Digital (72, 111, 363)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 19, 9)-net over F8, using
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 0 and N(F) ≥ 9, using
- the rational function field F8(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- digital (53, 92, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 46, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 46, 177)-net over F64, using
- digital (0, 19, 9)-net over F8, using
(72, 111, 518)-Net in Base 8 — Constructive
(72, 111, 518)-net in base 8, using
- 1 times m-reduction [i] based on (72, 112, 518)-net in base 8, using
- base change [i] based on digital (44, 84, 518)-net over F16, using
- trace code for nets [i] based on digital (2, 42, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- trace code for nets [i] based on digital (2, 42, 259)-net over F256, using
- base change [i] based on digital (44, 84, 518)-net over F16, using
(72, 111, 952)-Net over F8 — Digital
Digital (72, 111, 952)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8111, 952, F8, 39) (dual of [952, 841, 40]-code), using
- 840 step Varšamov–Edel lengthening with (ri) = (6, 3, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 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, 5 times 0, 1, 7 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 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, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 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, 32 times 0, 1, 34 times 0, 1, 35 times 0, 1, 38 times 0, 1, 40 times 0, 1, 43 times 0, 1, 45 times 0, 1, 48 times 0) [i] based on linear OA(839, 40, F8, 39) (dual of [40, 1, 40]-code or 40-arc in PG(38,8)), using
- dual of repetition code with length 40 [i]
- 840 step Varšamov–Edel lengthening with (ri) = (6, 3, 2, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 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, 5 times 0, 1, 7 times 0, 1, 6 times 0, 1, 7 times 0, 1, 8 times 0, 1, 8 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, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 16 times 0, 1, 18 times 0, 1, 19 times 0, 1, 20 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, 32 times 0, 1, 34 times 0, 1, 35 times 0, 1, 38 times 0, 1, 40 times 0, 1, 43 times 0, 1, 45 times 0, 1, 48 times 0) [i] based on linear OA(839, 40, F8, 39) (dual of [40, 1, 40]-code or 40-arc in PG(38,8)), using
(72, 111, 191649)-Net in Base 8 — Upper bound on s
There is no (72, 111, 191650)-net in base 8, because
- 1 times m-reduction [i] would yield (72, 110, 191650)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 2187 460034 936533 705016 166323 791888 600362 679308 193309 405194 269582 138655 130413 608939 456262 776403 140816 > 8110 [i]