Best Known (72, 72+24, s)-Nets in Base 8
(72, 72+24, 683)-Net over F8 — Constructive and digital
Digital (72, 96, 683)-net over F8, using
- 82 times duplication [i] based on digital (70, 94, 683)-net over F8, using
- net defined by OOA [i] based on linear OOA(894, 683, F8, 24, 24) (dual of [(683, 24), 16298, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(894, 8196, F8, 24) (dual of [8196, 8102, 25]-code), using
- trace code [i] based on linear OA(6447, 4098, F64, 24) (dual of [4098, 4051, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(22) [i] based on
- linear OA(6447, 4096, F64, 24) (dual of [4096, 4049, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- 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(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(23) ⊂ Ce(22) [i] based on
- trace code [i] based on linear OA(6447, 4098, F64, 24) (dual of [4098, 4051, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(894, 8196, F8, 24) (dual of [8196, 8102, 25]-code), using
- net defined by OOA [i] based on linear OOA(894, 683, F8, 24, 24) (dual of [(683, 24), 16298, 25]-NRT-code), using
(72, 72+24, 1028)-Net in Base 8 — Constructive
(72, 96, 1028)-net in base 8, using
- base change [i] based on digital (48, 72, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (12, 24, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 12, 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, 12, 257)-net over F256, using
- digital (24, 48, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 24, 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, 24, 257)-net over F256, using
- digital (12, 24, 514)-net over F16, using
- (u, u+v)-construction [i] based on
(72, 72+24, 8202)-Net over F8 — Digital
Digital (72, 96, 8202)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(896, 8202, F8, 24) (dual of [8202, 8106, 25]-code), using
- trace code [i] based on linear OA(6448, 4101, F64, 24) (dual of [4101, 4053, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(6447, 4096, F64, 24) (dual of [4096, 4049, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [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(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- trace code [i] based on linear OA(6448, 4101, F64, 24) (dual of [4101, 4053, 25]-code), using
(72, 72+24, large)-Net in Base 8 — Upper bound on s
There is no (72, 96, large)-net in base 8, because
- 22 times m-reduction [i] would yield (72, 74, large)-net in base 8, but