Best Known (86, 86+41, s)-Nets in Base 8
(86, 86+41, 402)-Net over F8 — Constructive and digital
Digital (86, 127, 402)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (11, 31, 48)-net over F8, using
- net from sequence [i] based on digital (11, 47)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 11 and N(F) ≥ 48, using
- net from sequence [i] based on digital (11, 47)-sequence over F8, using
- digital (55, 96, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 48, 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, 48, 177)-net over F64, using
- digital (11, 31, 48)-net over F8, using
(86, 86+41, 576)-Net in Base 8 — Constructive
(86, 127, 576)-net in base 8, using
- 7 times m-reduction [i] based on (86, 134, 576)-net in base 8, using
- trace code for nets [i] based on (19, 67, 288)-net in base 64, using
- 3 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
- base change [i] based on digital (9, 60, 288)-net over F128, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 9 and N(F) ≥ 288, using
- net from sequence [i] based on digital (9, 287)-sequence over F128, using
- base change [i] based on digital (9, 60, 288)-net over F128, using
- 3 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
- trace code for nets [i] based on (19, 67, 288)-net in base 64, using
(86, 86+41, 1680)-Net over F8 — Digital
Digital (86, 127, 1680)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8127, 1680, F8, 41) (dual of [1680, 1553, 42]-code), using
- 1552 step Varšamov–Edel lengthening with (ri) = (7, 3, 2, 1, 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, 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, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 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, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 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, 33 times 0, 1, 34 times 0, 1, 37 times 0, 1, 39 times 0, 1, 40 times 0, 1, 43 times 0, 1, 46 times 0, 1, 48 times 0, 1, 50 times 0, 1, 54 times 0, 1, 56 times 0, 1, 60 times 0, 1, 63 times 0, 1, 66 times 0, 1, 70 times 0, 1, 74 times 0, 1, 78 times 0, 1, 82 times 0) [i] based on linear OA(841, 42, F8, 41) (dual of [42, 1, 42]-code or 42-arc in PG(40,8)), using
- dual of repetition code with length 42 [i]
- 1552 step Varšamov–Edel lengthening with (ri) = (7, 3, 2, 1, 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, 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, 4 times 0, 1, 5 times 0, 1, 6 times 0, 1, 5 times 0, 1, 6 times 0, 1, 7 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, 11 times 0, 1, 11 times 0, 1, 11 times 0, 1, 13 times 0, 1, 13 times 0, 1, 14 times 0, 1, 14 times 0, 1, 16 times 0, 1, 17 times 0, 1, 17 times 0, 1, 19 times 0, 1, 20 times 0, 1, 21 times 0, 1, 22 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, 33 times 0, 1, 34 times 0, 1, 37 times 0, 1, 39 times 0, 1, 40 times 0, 1, 43 times 0, 1, 46 times 0, 1, 48 times 0, 1, 50 times 0, 1, 54 times 0, 1, 56 times 0, 1, 60 times 0, 1, 63 times 0, 1, 66 times 0, 1, 70 times 0, 1, 74 times 0, 1, 78 times 0, 1, 82 times 0) [i] based on linear OA(841, 42, F8, 41) (dual of [42, 1, 42]-code or 42-arc in PG(40,8)), using
(86, 86+41, 580317)-Net in Base 8 — Upper bound on s
There is no (86, 127, 580318)-net in base 8, because
- 1 times m-reduction [i] would yield (86, 126, 580318)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 615661 456131 785883 942630 175165 313568 197445 894266 701999 015611 228569 939491 535556 991577 727129 116809 979415 485241 165553 > 8126 [i]