Best Known (81, 81+14, s)-Nets in Base 9
(81, 81+14, 683310)-Net over F9 — Constructive and digital
Digital (81, 95, 683310)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (3, 10, 28)-net over F9, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- the Hermitian function field over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- digital (71, 85, 683282)-net over F9, using
- net defined by OOA [i] based on linear OOA(985, 683282, F9, 14, 14) (dual of [(683282, 14), 9565863, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(985, 4782974, F9, 14) (dual of [4782974, 4782889, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(985, 4782976, F9, 14) (dual of [4782976, 4782891, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(985, 4782969, F9, 14) (dual of [4782969, 4782884, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(978, 4782969, F9, 13) (dual of [4782969, 4782891, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 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(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(985, 4782976, F9, 14) (dual of [4782976, 4782891, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(985, 4782974, F9, 14) (dual of [4782974, 4782889, 15]-code), using
- net defined by OOA [i] based on linear OOA(985, 683282, F9, 14, 14) (dual of [(683282, 14), 9565863, 15]-NRT-code), using
- digital (3, 10, 28)-net over F9, using
(81, 81+14, 6662347)-Net over F9 — Digital
Digital (81, 95, 6662347)-net over F9, using
(81, 81+14, large)-Net in Base 9 — Upper bound on s
There is no (81, 95, large)-net in base 9, because
- 12 times m-reduction [i] would yield (81, 83, large)-net in base 9, but