Best Known (30, 30+9, s)-Nets in Base 81
(30, 30+9, 2100390)-Net over F81 — Constructive and digital
Digital (30, 39, 2100390)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (2, 6, 3240)-net over F81, using
- net defined by OOA [i] based on linear OOA(816, 3240, F81, 4, 4) (dual of [(3240, 4), 12954, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(816, 6480, F81, 4) (dual of [6480, 6474, 5]-code), using
- 1 times truncation [i] based on linear OA(817, 6481, F81, 5) (dual of [6481, 6474, 6]-code), using
- OA 2-folding and stacking [i] based on linear OA(816, 6480, F81, 4) (dual of [6480, 6474, 5]-code), using
- net defined by OOA [i] based on linear OOA(816, 3240, F81, 4, 4) (dual of [(3240, 4), 12954, 5]-NRT-code), 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 (2, 6, 3240)-net over F81, using
(30, 30+9, large)-Net over F81 — Digital
Digital (30, 39, large)-net over F81, using
- 2 times m-reduction [i] based on digital (30, 41, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
(30, 30+9, large)-Net in Base 81 — Upper bound on s
There is no (30, 39, large)-net in base 81, because
- 7 times m-reduction [i] would yield (30, 32, large)-net in base 81, but