Best Known (81−20, 81, s)-Nets in Base 8
(81−20, 81, 820)-Net over F8 — Constructive and digital
Digital (61, 81, 820)-net over F8, using
- 81 times duplication [i] based on digital (60, 80, 820)-net over F8, using
- net defined by OOA [i] based on linear OOA(880, 820, F8, 20, 20) (dual of [(820, 20), 16320, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(880, 8200, F8, 20) (dual of [8200, 8120, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(880, 8202, F8, 20) (dual of [8202, 8122, 21]-code), using
- trace code [i] based on linear OA(6440, 4101, F64, 20) (dual of [4101, 4061, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- 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(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(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(19) ⊂ Ce(17) [i] based on
- trace code [i] based on linear OA(6440, 4101, F64, 20) (dual of [4101, 4061, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(880, 8202, F8, 20) (dual of [8202, 8122, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(880, 8200, F8, 20) (dual of [8200, 8120, 21]-code), using
- net defined by OOA [i] based on linear OOA(880, 820, F8, 20, 20) (dual of [(820, 20), 16320, 21]-NRT-code), using
(81−20, 81, 1028)-Net in Base 8 — Constructive
(61, 81, 1028)-net in base 8, using
- 81 times duplication [i] based on (60, 80, 1028)-net in base 8, using
- base change [i] based on digital (40, 60, 1028)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (10, 20, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 10, 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, 10, 257)-net over F256, using
- digital (20, 40, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 20, 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, 20, 257)-net over F256, using
- digital (10, 20, 514)-net over F16, using
- (u, u+v)-construction [i] based on
- base change [i] based on digital (40, 60, 1028)-net over F16, using
(81−20, 81, 8204)-Net over F8 — Digital
Digital (61, 81, 8204)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(881, 8204, F8, 20) (dual of [8204, 8123, 21]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(880, 8202, F8, 20) (dual of [8202, 8122, 21]-code), using
- trace code [i] based on linear OA(6440, 4101, F64, 20) (dual of [4101, 4061, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- 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(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(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(19) ⊂ Ce(17) [i] based on
- trace code [i] based on linear OA(6440, 4101, F64, 20) (dual of [4101, 4061, 21]-code), using
- linear OA(880, 8203, F8, 19) (dual of [8203, 8123, 20]-code), using Gilbert–Varšamov bound and bm = 880 > Vbs−1(k−1) = 7047 074069 270492 436962 796921 310817 287746 933217 522399 873928 533030 426624 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(880, 8202, F8, 20) (dual of [8202, 8122, 21]-code), using
- construction X with Varšamov bound [i] based on
(81−20, 81, large)-Net in Base 8 — Upper bound on s
There is no (61, 81, large)-net in base 8, because
- 18 times m-reduction [i] would yield (61, 63, large)-net in base 8, but