Best Known (53−20, 53, s)-Nets in Base 8
(53−20, 53, 256)-Net over F8 — Constructive and digital
Digital (33, 53, 256)-net over F8, using
- 3 times m-reduction [i] based on digital (33, 56, 256)-net over F8, using
- trace code for nets [i] based on digital (5, 28, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- trace code for nets [i] based on digital (5, 28, 128)-net over F64, using
(53−20, 53, 300)-Net in Base 8 — Constructive
(33, 53, 300)-net in base 8, using
- 3 times m-reduction [i] based on (33, 56, 300)-net in base 8, using
- trace code for nets [i] based on (5, 28, 150)-net in base 64, using
- base change [i] based on digital (1, 24, 150)-net over F128, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 1 and N(F) ≥ 150, using
- net from sequence [i] based on digital (1, 149)-sequence over F128, using
- base change [i] based on digital (1, 24, 150)-net over F128, using
- trace code for nets [i] based on (5, 28, 150)-net in base 64, using
(53−20, 53, 429)-Net over F8 — Digital
Digital (33, 53, 429)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(853, 429, F8, 20) (dual of [429, 376, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(853, 519, F8, 20) (dual of [519, 466, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- linear OA(852, 512, F8, 20) (dual of [512, 460, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(846, 512, F8, 18) (dual of [512, 466, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(19) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(853, 519, F8, 20) (dual of [519, 466, 21]-code), using
(53−20, 53, 39553)-Net in Base 8 — Upper bound on s
There is no (33, 53, 39554)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 730755 887711 979530 399861 063923 416147 180892 102464 > 853 [i]