Best Known (51−11, 51, s)-Nets in Base 9
(51−11, 51, 11820)-Net over F9 — Constructive and digital
Digital (40, 51, 11820)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 10)-net over F9, using
- net from sequence [i] based on digital (0, 9)-sequence over F9, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 0 and N(F) ≥ 10, using
- the rational function field F9(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 9)-sequence over F9, using
- digital (35, 46, 11810)-net over F9, using
- net defined by OOA [i] based on linear OOA(946, 11810, F9, 11, 11) (dual of [(11810, 11), 129864, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(946, 59051, F9, 11) (dual of [59051, 59005, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(946, 59054, F9, 11) (dual of [59054, 59008, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(946, 59049, F9, 11) (dual of [59049, 59003, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(941, 59049, F9, 10) (dual of [59049, 59008, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(90, 5, F9, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(946, 59054, F9, 11) (dual of [59054, 59008, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(946, 59051, F9, 11) (dual of [59051, 59005, 12]-code), using
- net defined by OOA [i] based on linear OOA(946, 11810, F9, 11, 11) (dual of [(11810, 11), 129864, 12]-NRT-code), using
- digital (0, 5, 10)-net over F9, using
(51−11, 51, 59070)-Net over F9 — Digital
Digital (40, 51, 59070)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(951, 59070, F9, 11) (dual of [59070, 59019, 12]-code), using
- construction XX applied to Ce(10) ⊂ Ce(6) ⊂ Ce(5) [i] based on
- linear OA(946, 59049, F9, 11) (dual of [59049, 59003, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(931, 59049, F9, 7) (dual of [59049, 59018, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(926, 59049, F9, 6) (dual of [59049, 59023, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(94, 20, F9, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,9)), using
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(10) ⊂ Ce(6) ⊂ Ce(5) [i] based on
(51−11, 51, large)-Net in Base 9 — Upper bound on s
There is no (40, 51, large)-net in base 9, because
- 9 times m-reduction [i] would yield (40, 42, large)-net in base 9, but