Best Known (28, 28+9, s)-Nets in Base 81
(28, 28+9, 2097232)-Net over F81 — Constructive and digital
Digital (28, 37, 2097232)-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 (24, 33, 2097150)-net over F81, using
- net defined by OOA [i] based on linear OOA(8133, 2097150, F81, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(8133, 8388601, F81, 9) (dual of [8388601, 8388568, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(8133, large, F81, 9) (dual of [large, large−33, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8133, large, F81, 9) (dual of [large, large−33, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(8133, 8388601, F81, 9) (dual of [8388601, 8388568, 10]-code), using
- net defined by OOA [i] based on linear OOA(8133, 2097150, F81, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
- digital (0, 4, 82)-net over F81, using
(28, 28+9, large)-Net over F81 — Digital
Digital (28, 37, large)-net over F81, using
- t-expansion [i] based on digital (27, 37, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8137, large, F81, 10) (dual of [large, large−37, 11]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8137, large, F81, 10) (dual of [large, large−37, 11]-code), using
(28, 28+9, large)-Net in Base 81 — Upper bound on s
There is no (28, 37, large)-net in base 81, because
- 7 times m-reduction [i] would yield (28, 30, large)-net in base 81, but