Best Known (106−30, 106, s)-Nets in Base 8
(106−30, 106, 514)-Net over F8 — Constructive and digital
Digital (76, 106, 514)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (17, 32, 160)-net over F8, using
- trace code for nets [i] based on digital (1, 16, 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, 16, 80)-net over F64, using
- digital (44, 74, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 37, 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, 37, 177)-net over F64, using
- digital (17, 32, 160)-net over F8, using
(106−30, 106, 576)-Net in Base 8 — Constructive
(76, 106, 576)-net in base 8, using
- 10 times m-reduction [i] based on (76, 116, 576)-net in base 8, using
- trace code for nets [i] based on (18, 58, 288)-net in base 64, using
- 5 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
- base change [i] based on digital (9, 54, 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, 54, 288)-net over F128, using
- 5 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
- trace code for nets [i] based on (18, 58, 288)-net in base 64, using
(106−30, 106, 3914)-Net over F8 — Digital
Digital (76, 106, 3914)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8106, 3914, F8, 30) (dual of [3914, 3808, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(8106, 4105, F8, 30) (dual of [4105, 3999, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(27) [i] based on
- linear OA(8105, 4096, F8, 30) (dual of [4096, 3991, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(897, 4096, F8, 28) (dual of [4096, 3999, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(29) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(8106, 4105, F8, 30) (dual of [4105, 3999, 31]-code), using
(106−30, 106, 2210562)-Net in Base 8 — Upper bound on s
There is no (76, 106, 2210563)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 534000 153195 823572 145527 809131 911012 806082 614575 164369 884060 006368 383124 330981 576792 506932 498176 > 8106 [i]