Best Known (41, 41+11, s)-Nets in Base 9
(41, 41+11, 11826)-Net over F9 — Constructive and digital
Digital (41, 52, 11826)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 16)-net over F9, using
- net from sequence [i] based on digital (1, 15)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 1 and N(F) ≥ 16, using
- net from sequence [i] based on digital (1, 15)-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 (1, 6, 16)-net over F9, using
(41, 41+11, 59075)-Net over F9 — Digital
Digital (41, 52, 59075)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(952, 59075, F9, 11) (dual of [59075, 59023, 12]-code), using
- construction X applied to Ce(10) ⊂ 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(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(96, 26, F9, 4) (dual of [26, 20, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- 1 times truncation [i] based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
(41, 41+11, large)-Net in Base 9 — Upper bound on s
There is no (41, 52, large)-net in base 9, because
- 9 times m-reduction [i] would yield (41, 43, large)-net in base 9, but