Best Known (56, 56+18, s)-Nets in Base 8
(56, 56+18, 912)-Net over F8 — Constructive and digital
Digital (56, 74, 912)-net over F8, using
- net defined by OOA [i] based on linear OOA(874, 912, F8, 18, 18) (dual of [(912, 18), 16342, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(874, 8208, F8, 18) (dual of [8208, 8134, 19]-code), using
- trace code [i] based on linear OA(6437, 4104, F64, 18) (dual of [4104, 4067, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(6435, 4096, F64, 18) (dual of [4096, 4061, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(6429, 4096, F64, 15) (dual of [4096, 4067, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(642, 8, F64, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,64)), using
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- Reed–Solomon code RS(62,64) [i]
- discarding factors / shortening the dual code based on linear OA(642, 64, F64, 2) (dual of [64, 62, 3]-code or 64-arc in PG(1,64)), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- trace code [i] based on linear OA(6437, 4104, F64, 18) (dual of [4104, 4067, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(874, 8208, F8, 18) (dual of [8208, 8134, 19]-code), using
(56, 56+18, 1028)-Net in Base 8 — Constructive
(56, 74, 1028)-net in base 8, using
- 82 times duplication [i] based on (54, 72, 1028)-net in base 8, using
- base change [i] based on digital (36, 54, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (9, 18, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 9, 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, 9, 257)-net over F256, using
- digital (18, 36, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 18, 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, 18, 257)-net over F256, using
- digital (9, 18, 514)-net over F16, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (36, 54, 1028)-net over F16, using
(56, 56+18, 8757)-Net over F8 — Digital
Digital (56, 74, 8757)-net over F8, using
(56, 56+18, large)-Net in Base 8 — Upper bound on s
There is no (56, 74, large)-net in base 8, because
- 16 times m-reduction [i] would yield (56, 58, large)-net in base 8, but