Best Known (90−34, 90, s)-Nets in Base 8
(90−34, 90, 354)-Net over F8 — Constructive and digital
Digital (56, 90, 354)-net over F8, using
- 8 times m-reduction [i] based on digital (56, 98, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 49, 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, 49, 177)-net over F64, using
(90−34, 90, 384)-Net in Base 8 — Constructive
(56, 90, 384)-net in base 8, using
- 2 times m-reduction [i] based on (56, 92, 384)-net in base 8, using
- trace code for nets [i] based on (10, 46, 192)-net in base 64, using
- 3 times m-reduction [i] based on (10, 49, 192)-net in base 64, using
- base change [i] based on digital (3, 42, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- base change [i] based on digital (3, 42, 192)-net over F128, using
- 3 times m-reduction [i] based on (10, 49, 192)-net in base 64, using
- trace code for nets [i] based on (10, 46, 192)-net in base 64, using
(90−34, 90, 563)-Net over F8 — Digital
Digital (56, 90, 563)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(890, 563, F8, 34) (dual of [563, 473, 35]-code), using
- 472 step Varšamov–Edel lengthening with (ri) = (6, 2, 2, 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, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 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, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 18 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 24 times 0, 1, 26 times 0, 1, 27 times 0, 1, 29 times 0, 1, 32 times 0) [i] based on linear OA(834, 35, F8, 34) (dual of [35, 1, 35]-code or 35-arc in PG(33,8)), using
- dual of repetition code with length 35 [i]
- 472 step Varšamov–Edel lengthening with (ri) = (6, 2, 2, 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, 0, 1, 0, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 0, 0, 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, 7 times 0, 1, 8 times 0, 1, 8 times 0, 1, 8 times 0, 1, 10 times 0, 1, 10 times 0, 1, 11 times 0, 1, 12 times 0, 1, 12 times 0, 1, 14 times 0, 1, 15 times 0, 1, 15 times 0, 1, 17 times 0, 1, 18 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 times 0, 1, 24 times 0, 1, 26 times 0, 1, 27 times 0, 1, 29 times 0, 1, 32 times 0) [i] based on linear OA(834, 35, F8, 34) (dual of [35, 1, 35]-code or 35-arc in PG(33,8)), using
(90−34, 90, 61921)-Net in Base 8 — Upper bound on s
There is no (56, 90, 61922)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 1897 605702 184328 457534 718763 794029 469339 196816 323489 133450 192232 535840 850586 231126 > 890 [i]