Best Known (19−8, 19, s)-Nets in Base 81
(19−8, 19, 1722)-Net over F81 — Constructive and digital
Digital (11, 19, 1722)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- digital (7, 15, 1640)-net over F81, using
- net defined by OOA [i] based on linear OOA(8115, 1640, F81, 8, 8) (dual of [(1640, 8), 13105, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(8115, 6560, F81, 8) (dual of [6560, 6545, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(8115, 6560, F81, 8) (dual of [6560, 6545, 9]-code), using
- net defined by OOA [i] based on linear OOA(8115, 1640, F81, 8, 8) (dual of [(1640, 8), 13105, 9]-NRT-code), using
- digital (0, 4, 82)-net over F81, using
(19−8, 19, 6645)-Net over F81 — Digital
Digital (11, 19, 6645)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8119, 6645, F81, 8) (dual of [6645, 6626, 9]-code), using
- (u, u+v)-construction [i] based on
- linear OA(814, 82, F81, 4) (dual of [82, 78, 5]-code or 82-arc in PG(3,81)), using
- extended Reed–Solomon code RSe(78,81) [i]
- linear OA(8115, 6563, F81, 8) (dual of [6563, 6548, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(8113, 6561, F81, 7) (dual of [6561, 6548, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(814, 82, F81, 4) (dual of [82, 78, 5]-code or 82-arc in PG(3,81)), using
- (u, u+v)-construction [i] based on
(19−8, 19, large)-Net in Base 81 — Upper bound on s
There is no (11, 19, large)-net in base 81, because
- 6 times m-reduction [i] would yield (11, 13, large)-net in base 81, but