Best Known (35, 35+21, s)-Nets in Base 8
(35, 35+21, 354)-Net over F8 — Constructive and digital
Digital (35, 56, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 28, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
(35, 35+21, 456)-Net over F8 — Digital
Digital (35, 56, 456)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(856, 456, F8, 21) (dual of [456, 400, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(856, 520, F8, 21) (dual of [520, 464, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(855, 513, F8, 21) (dual of [513, 458, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 513 | 86−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(849, 513, F8, 19) (dual of [513, 464, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 513 | 86−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [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 C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(856, 520, F8, 21) (dual of [520, 464, 22]-code), using
(35, 35+21, 514)-Net in Base 8 — Constructive
(35, 56, 514)-net in base 8, using
- base change [i] based on digital (21, 42, 514)-net over F16, using
- trace code for nets [i] based on digital (0, 21, 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, 21, 257)-net over F256, using
(35, 35+21, 59955)-Net in Base 8 — Upper bound on s
There is no (35, 56, 59956)-net in base 8, because
- 1 times m-reduction [i] would yield (35, 55, 59956)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 46 770636 549300 773961 975411 796898 972635 679984 770686 > 855 [i]