Best Known (90, 90+36, s)-Nets in Base 8
(90, 90+36, 514)-Net over F8 — Constructive and digital
Digital (90, 126, 514)-net over F8, using
- t-expansion [i] based on digital (89, 126, 514)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (20, 38, 160)-net over F8, using
- trace code for nets [i] based on digital (1, 19, 80)-net over F64, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 1 and N(F) ≥ 80, using
- net from sequence [i] based on digital (1, 79)-sequence over F64, using
- trace code for nets [i] based on digital (1, 19, 80)-net over F64, using
- digital (51, 88, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 44, 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, 44, 177)-net over F64, using
- digital (20, 38, 160)-net over F8, using
- (u, u+v)-construction [i] based on
(90, 90+36, 593)-Net in Base 8 — Constructive
(90, 126, 593)-net in base 8, using
- (u, u+v)-construction [i] based on
- digital (2, 20, 17)-net over F8, using
- net from sequence [i] based on digital (2, 16)-sequence over F8, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 2 and N(F) ≥ 17, using
- net from sequence [i] based on digital (2, 16)-sequence over F8, using
- (70, 106, 576)-net in base 8, using
- trace code for nets [i] based on (17, 53, 288)-net in base 64, using
- 3 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- base change [i] based on digital (9, 48, 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, 48, 288)-net over F128, using
- 3 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- trace code for nets [i] based on (17, 53, 288)-net in base 64, using
- digital (2, 20, 17)-net over F8, using
(90, 90+36, 4021)-Net over F8 — Digital
Digital (90, 126, 4021)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8126, 4021, F8, 36) (dual of [4021, 3895, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(8126, 4105, F8, 36) (dual of [4105, 3979, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(33) [i] based on
- linear OA(8125, 4096, F8, 36) (dual of [4096, 3971, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(8117, 4096, F8, 34) (dual of [4096, 3979, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(81, 9, F8, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(35) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(8126, 4105, F8, 36) (dual of [4105, 3979, 37]-code), using
(90, 90+36, 2262871)-Net in Base 8 — Upper bound on s
There is no (90, 126, 2262872)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 615661 141138 519018 361925 961638 381775 006753 943990 386851 773915 793867 404599 618204 379398 141139 637757 767091 530184 898701 > 8126 [i]