Best Known (84, 84+32, s)-Nets in Base 8
(84, 84+32, 562)-Net over F8 — Constructive and digital
Digital (84, 116, 562)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (22, 38, 208)-net over F8, using
- trace code for nets [i] based on digital (3, 19, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 19, 104)-net over F64, using
- digital (46, 78, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 39, 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, 39, 177)-net over F64, using
- digital (22, 38, 208)-net over F8, using
(84, 84+32, 612)-Net in Base 8 — Constructive
(84, 116, 612)-net in base 8, using
- (u, u+v)-construction [i] based on
- (22, 38, 258)-net in base 8, using
- trace code for nets [i] based on (3, 19, 129)-net in base 64, using
- 2 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- base change [i] based on digital (0, 18, 129)-net over F128, using
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 0 and N(F) ≥ 129, using
- the rational function field F128(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 128)-sequence over F128, using
- base change [i] based on digital (0, 18, 129)-net over F128, using
- 2 times m-reduction [i] based on (3, 21, 129)-net in base 64, using
- trace code for nets [i] based on (3, 19, 129)-net in base 64, using
- digital (46, 78, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 39, 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, 39, 177)-net over F64, using
- (22, 38, 258)-net in base 8, using
(84, 84+32, 4310)-Net over F8 — Digital
Digital (84, 116, 4310)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8116, 4310, F8, 32) (dual of [4310, 4194, 33]-code), using
- 210 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 46 times 0, 1, 155 times 0) [i] based on linear OA(8112, 4096, F8, 32) (dual of [4096, 3984, 33]-code), using
- 1 times truncation [i] based on linear OA(8113, 4097, F8, 33) (dual of [4097, 3984, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4097 | 88−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(8113, 4097, F8, 33) (dual of [4097, 3984, 34]-code), using
- 210 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 46 times 0, 1, 155 times 0) [i] based on linear OA(8112, 4096, F8, 32) (dual of [4096, 3984, 33]-code), using
(84, 84+32, 3426429)-Net in Base 8 — Upper bound on s
There is no (84, 116, 3426430)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 573 374947 385622 667841 370374 430160 249061 118657 162465 636965 658440 210178 265855 425356 566746 868668 280102 959937 > 8116 [i]