Best Known (58, 58+19, s)-Nets in Base 81
(58, 58+19, 932067)-Net over F81 — Constructive and digital
Digital (58, 77, 932067)-net over F81, using
- 811 times duplication [i] based on digital (57, 76, 932067)-net over F81, using
- net defined by OOA [i] based on linear OOA(8176, 932067, F81, 21, 19) (dual of [(932067, 21), 19573331, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(8176, 2796202, F81, 3, 19) (dual of [(2796202, 3), 8388530, 20]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(8173, 2796201, F81, 3, 19) (dual of [(2796201, 3), 8388530, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8173, large, F81, 19) (dual of [large, large−73, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(8173, large, F81, 19) (dual of [large, large−73, 20]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(8173, 2796201, F81, 3, 19) (dual of [(2796201, 3), 8388530, 20]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(8176, 2796202, F81, 3, 19) (dual of [(2796202, 3), 8388530, 20]-NRT-code), using
- net defined by OOA [i] based on linear OOA(8176, 932067, F81, 21, 19) (dual of [(932067, 21), 19573331, 20]-NRT-code), using
(58, 58+19, large)-Net over F81 — Digital
Digital (58, 77, large)-net over F81, using
- t-expansion [i] based on digital (57, 77, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8177, large, F81, 20) (dual of [large, large−77, 21]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8177, large, F81, 20) (dual of [large, large−77, 21]-code), using
(58, 58+19, large)-Net in Base 81 — Upper bound on s
There is no (58, 77, large)-net in base 81, because
- 17 times m-reduction [i] would yield (58, 60, large)-net in base 81, but