Best Known (41−21, 41, s)-Nets in Base 81
(41−21, 41, 656)-Net over F81 — Constructive and digital
Digital (20, 41, 656)-net over F81, using
- net defined by OOA [i] based on linear OOA(8141, 656, F81, 21, 21) (dual of [(656, 21), 13735, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8141, 6561, F81, 21) (dual of [6561, 6520, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- OOA 10-folding and stacking with additional row [i] based on linear OA(8141, 6561, F81, 21) (dual of [6561, 6520, 22]-code), using
(41−21, 41, 1648)-Net over F81 — Digital
Digital (20, 41, 1648)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8141, 1648, F81, 3, 21) (dual of [(1648, 3), 4903, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8141, 2187, F81, 3, 21) (dual of [(2187, 3), 6520, 22]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8141, 6561, F81, 21) (dual of [6561, 6520, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- OOA 3-folding [i] based on linear OA(8141, 6561, F81, 21) (dual of [6561, 6520, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(8141, 2187, F81, 3, 21) (dual of [(2187, 3), 6520, 22]-NRT-code), using
(41−21, 41, 2436831)-Net in Base 81 — Upper bound on s
There is no (20, 41, 2436832)-net in base 81, because
- 1 times m-reduction [i] would yield (20, 40, 2436832)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 21847 460898 613321 785746 202527 718153 409011 708869 349950 806650 771924 500241 689601 > 8140 [i]