Best Known (81, 81+21, s)-Nets in Base 9
(81, 81+21, 5921)-Net over F9 — Constructive and digital
Digital (81, 102, 5921)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 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 (70, 91, 5905)-net over F9, using
- net defined by OOA [i] based on linear OOA(991, 5905, F9, 21, 21) (dual of [(5905, 21), 123914, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(991, 59051, F9, 21) (dual of [59051, 58960, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(991, 59054, F9, 21) (dual of [59054, 58963, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(991, 59049, F9, 21) (dual of [59049, 58958, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(986, 59049, F9, 20) (dual of [59049, 58963, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(20) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(991, 59054, F9, 21) (dual of [59054, 58963, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(991, 59051, F9, 21) (dual of [59051, 58960, 22]-code), using
- net defined by OOA [i] based on linear OOA(991, 5905, F9, 21, 21) (dual of [(5905, 21), 123914, 22]-NRT-code), using
- digital (1, 11, 16)-net over F9, using
(81, 81+21, 76368)-Net over F9 — Digital
Digital (81, 102, 76368)-net over F9, using
(81, 81+21, large)-Net in Base 9 — Upper bound on s
There is no (81, 102, large)-net in base 9, because
- 19 times m-reduction [i] would yield (81, 83, large)-net in base 9, but