Best Known (66, 88, s)-Nets in Base 8
(66, 88, 745)-Net over F8 — Constructive and digital
Digital (66, 88, 745)-net over F8, using
- 82 times duplication [i] based on digital (64, 86, 745)-net over F8, using
- net defined by OOA [i] based on linear OOA(886, 745, F8, 22, 22) (dual of [(745, 22), 16304, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(886, 8195, F8, 22) (dual of [8195, 8109, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(886, 8196, F8, 22) (dual of [8196, 8110, 23]-code), using
- trace code [i] based on linear OA(6443, 4098, F64, 22) (dual of [4098, 4055, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- 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(6441, 4096, F64, 21) (dual of [4096, 4055, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- trace code [i] based on linear OA(6443, 4098, F64, 22) (dual of [4098, 4055, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(886, 8196, F8, 22) (dual of [8196, 8110, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(886, 8195, F8, 22) (dual of [8195, 8109, 23]-code), using
- net defined by OOA [i] based on linear OOA(886, 745, F8, 22, 22) (dual of [(745, 22), 16304, 23]-NRT-code), using
(66, 88, 1028)-Net in Base 8 — Constructive
(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
(66, 88, 8202)-Net over F8 — Digital
Digital (66, 88, 8202)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(888, 8202, F8, 22) (dual of [8202, 8114, 23]-code), using
- trace code [i] based on linear OA(6444, 4101, F64, 22) (dual of [4101, 4057, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- 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(6439, 4096, F64, 20) (dual of [4096, 4057, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(21) ⊂ Ce(19) [i] based on
- trace code [i] based on linear OA(6444, 4101, F64, 22) (dual of [4101, 4057, 23]-code), using
(66, 88, large)-Net in Base 8 — Upper bound on s
There is no (66, 88, large)-net in base 8, because
- 20 times m-reduction [i] would yield (66, 68, large)-net in base 8, but