Best Known (16, 16+17, s)-Nets in Base 81
(16, 16+17, 820)-Net over F81 — Constructive and digital
Digital (16, 33, 820)-net over F81, using
- net defined by OOA [i] based on linear OOA(8133, 820, F81, 17, 17) (dual of [(820, 17), 13907, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(8133, 6561, F81, 17) (dual of [6561, 6528, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(8133, 6561, F81, 17) (dual of [6561, 6528, 18]-code), using
(16, 16+17, 1791)-Net over F81 — Digital
Digital (16, 33, 1791)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8133, 1791, F81, 3, 17) (dual of [(1791, 3), 5340, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8133, 2187, F81, 3, 17) (dual of [(2187, 3), 6528, 18]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8133, 6561, F81, 17) (dual of [6561, 6528, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- OOA 3-folding [i] based on linear OA(8133, 6561, F81, 17) (dual of [6561, 6528, 18]-code), using
- discarding factors / shortening the dual code based on linear OOA(8133, 2187, F81, 3, 17) (dual of [(2187, 3), 6528, 18]-NRT-code), using
(16, 16+17, 2025533)-Net in Base 81 — Upper bound on s
There is no (16, 33, 2025534)-net in base 81, because
- 1 times m-reduction [i] would yield (16, 32, 2025534)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 11 790218 336436 552113 124114 930076 891407 886335 922759 474709 845761 > 8132 [i]