Best Known (30−7, 30, s)-Nets in Base 81
(30−7, 30, 2812518)-Net over F81 — Constructive and digital
Digital (23, 30, 2812518)-net over F81, using
- net defined by OOA [i] based on linear OOA(8130, 2812518, F81, 9, 7) (dual of [(2812518, 9), 25312632, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(8130, 2812519, F81, 3, 7) (dual of [(2812519, 3), 8437527, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(815, 16318, F81, 3, 3) (dual of [(16318, 3), 48949, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(815, 16318, F81, 2, 3) (dual of [(16318, 2), 32631, 4]-NRT-code), using
- linear OOA(8125, 2796201, F81, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
- OOA 3-folding [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]
- OOA 3-folding [i] based on linear OA(8125, large, F81, 7) (dual of [large, large−25, 8]-code), using
- linear OOA(815, 16318, F81, 3, 3) (dual of [(16318, 3), 48949, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(8130, 2812519, F81, 3, 7) (dual of [(2812519, 3), 8437527, 8]-NRT-code), using
(30−7, 30, large)-Net over F81 — Digital
Digital (23, 30, large)-net over F81, using
- 811 times duplication [i] based on digital (22, 29, large)-net over F81, using
- t-expansion [i] based on digital (21, 29, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] 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]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8129, large, F81, 8) (dual of [large, large−29, 9]-code), using
- t-expansion [i] based on digital (21, 29, large)-net over F81, using
(30−7, 30, large)-Net in Base 81 — Upper bound on s
There is no (23, 30, large)-net in base 81, because
- 5 times m-reduction [i] would yield (23, 25, large)-net in base 81, but