Best Known (20, 20+12, s)-Nets in Base 8
(20, 20+12, 208)-Net over F8 — Constructive and digital
Digital (20, 32, 208)-net over F8, using
- 2 times m-reduction [i] based on digital (20, 34, 208)-net over F8, using
- trace code for nets [i] based on digital (3, 17, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 17, 104)-net over F64, using
(20, 20+12, 403)-Net over F8 — Digital
Digital (20, 32, 403)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(832, 403, F8, 12) (dual of [403, 371, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(832, 519, F8, 12) (dual of [519, 487, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(831, 512, F8, 12) (dual of [512, 481, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(825, 512, F8, 10) (dual of [512, 487, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(81, 7, F8, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(832, 519, F8, 12) (dual of [519, 487, 13]-code), using
(20, 20+12, 514)-Net in Base 8 — Constructive
(20, 32, 514)-net in base 8, using
- base change [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
(20, 20+12, 28025)-Net in Base 8 — Upper bound on s
There is no (20, 32, 28026)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 79238 191409 064738 293716 545724 > 832 [i]