Best Known (78−27, 78, s)-Nets in Base 81
(78−27, 78, 1009)-Net over F81 — Constructive and digital
Digital (51, 78, 1009)-net over F81, using
- net defined by OOA [i] based on linear OOA(8178, 1009, F81, 27, 27) (dual of [(1009, 27), 27165, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8178, 13118, F81, 27) (dual of [13118, 13040, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8178, 13124, F81, 27) (dual of [13124, 13046, 28]-code), using
- (u, u+v)-construction [i] based on
- linear OA(8125, 6562, F81, 13) (dual of [6562, 6537, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(8153, 6562, F81, 27) (dual of [6562, 6509, 28]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(8125, 6562, F81, 13) (dual of [6562, 6537, 14]-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(8178, 13124, F81, 27) (dual of [13124, 13046, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8178, 13118, F81, 27) (dual of [13118, 13040, 28]-code), using
(78−27, 78, 70103)-Net over F81 — Digital
Digital (51, 78, 70103)-net over F81, using
(78−27, 78, large)-Net in Base 81 — Upper bound on s
There is no (51, 78, large)-net in base 81, because
- 25 times m-reduction [i] would yield (51, 53, large)-net in base 81, but