Best Known (36−8, 36, s)-Nets in Base 81
(36−8, 36, 2100431)-Net over F81 — Constructive and digital
Digital (28, 36, 2100431)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (3, 7, 3281)-net over F81, using
- net defined by OOA [i] based on linear OOA(817, 3281, F81, 4, 4) (dual of [(3281, 4), 13117, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(817, 6562, F81, 4) (dual of [6562, 6555, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(817, 6563, F81, 4) (dual of [6563, 6556, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(817, 6561, F81, 4) (dual of [6561, 6554, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(815, 6561, F81, 3) (dual of [6561, 6556, 4]-code or 6561-cap in PG(4,81)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(817, 6563, F81, 4) (dual of [6563, 6556, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(817, 6562, F81, 4) (dual of [6562, 6555, 5]-code), using
- net defined by OOA [i] based on linear OOA(817, 3281, F81, 4, 4) (dual of [(3281, 4), 13117, 5]-NRT-code), using
- digital (21, 29, 2097150)-net over F81, using
- net defined by OOA [i] based on linear OOA(8129, 2097150, F81, 8, 8) (dual of [(2097150, 8), 16777171, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(8129, 8388600, F81, 8) (dual of [8388600, 8388571, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(8129, large, F81, 8) (dual of [large, large−29, 9]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(8129, large, F81, 8) (dual of [large, large−29, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(8129, 8388600, F81, 8) (dual of [8388600, 8388571, 9]-code), using
- net defined by OOA [i] based on linear OOA(8129, 2097150, F81, 8, 8) (dual of [(2097150, 8), 16777171, 9]-NRT-code), using
- digital (3, 7, 3281)-net over F81, using
(36−8, 36, large)-Net over F81 — Digital
Digital (28, 36, large)-net over F81, using
- t-expansion [i] based on digital (27, 36, large)-net over F81, using
- 1 times m-reduction [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
- 1 times m-reduction [i] based on digital (27, 37, large)-net over F81, using
(36−8, 36, large)-Net in Base 81 — Upper bound on s
There is no (28, 36, large)-net in base 81, because
- 6 times m-reduction [i] would yield (28, 30, large)-net in base 81, but