Best Known (67, 67+22, s)-Nets in Base 8
(67, 67+22, 745)-Net over F8 — Constructive and digital
Digital (67, 89, 745)-net over F8, using
- 1 times m-reduction [i] based on digital (67, 90, 745)-net over F8, using
- net defined by OOA [i] based on linear OOA(890, 745, F8, 23, 23) (dual of [(745, 23), 17045, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(890, 8196, F8, 23) (dual of [8196, 8106, 24]-code), using
- trace code [i] based on linear OA(6445, 4098, F64, 23) (dual of [4098, 4053, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(6445, 4096, F64, 23) (dual of [4096, 4051, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(6443, 4096, F64, 22) (dual of [4096, 4053, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- trace code [i] based on linear OA(6445, 4098, F64, 23) (dual of [4098, 4053, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(890, 8196, F8, 23) (dual of [8196, 8106, 24]-code), using
- net defined by OOA [i] based on linear OOA(890, 745, F8, 23, 23) (dual of [(745, 23), 17045, 24]-NRT-code), using
(67, 67+22, 1028)-Net in Base 8 — Constructive
(67, 89, 1028)-net in base 8, using
- 81 times duplication [i] based on (66, 88, 1028)-net in base 8, using
- base change [i] based on digital (44, 66, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (11, 22, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 11, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 11, 257)-net over F256, using
- digital (22, 44, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 22, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256 (see above)
- trace code for nets [i] based on digital (0, 22, 257)-net over F256, using
- digital (11, 22, 514)-net over F16, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (44, 66, 1028)-net over F16, using
(67, 67+22, 8343)-Net over F8 — Digital
Digital (67, 89, 8343)-net over F8, using
(67, 67+22, large)-Net in Base 8 — Upper bound on s
There is no (67, 89, large)-net in base 8, because
- 20 times m-reduction [i] would yield (67, 69, large)-net in base 8, but