Best Known (24−6, 24, s)-Nets in Base 81
(24−6, 24, 2796283)-Net over F81 — Constructive and digital
Digital (18, 24, 2796283)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 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 (15, 21, 2796201)-net over F81, using
- net defined by OOA [i] based on linear OOA(8121, 2796201, F81, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(8121, large, F81, 6) (dual of [large, large−21, 7]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(8121, large, F81, 6) (dual of [large, large−21, 7]-code), using
- net defined by OOA [i] based on linear OOA(8121, 2796201, F81, 6, 6) (dual of [(2796201, 6), 16777185, 7]-NRT-code), using
- digital (0, 3, 82)-net over F81, using
(24−6, 24, large)-Net over F81 — Digital
Digital (18, 24, large)-net over F81, using
- 1 times m-reduction [i] based on digital (18, 25, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8125, large, F81, 7) (dual of [large, large−25, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8125, large, F81, 7) (dual of [large, large−25, 8]-code), using
(24−6, 24, large)-Net in Base 81 — Upper bound on s
There is no (18, 24, large)-net in base 81, because
- 4 times m-reduction [i] would yield (18, 20, large)-net in base 81, but