Best Known (18, 18+14, s)-Nets in Base 81
(18, 18+14, 939)-Net over F81 — Constructive and digital
Digital (18, 32, 939)-net over F81, using
- 1 times m-reduction [i] based on digital (18, 33, 939)-net over F81, using
- net defined by OOA [i] based on linear OOA(8133, 939, F81, 15, 15) (dual of [(939, 15), 14052, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8133, 6574, F81, 15) (dual of [6574, 6541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(8133, 6575, F81, 15) (dual of [6575, 6542, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- linear OA(8129, 6561, F81, 15) (dual of [6561, 6532, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(814, 14, F81, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,81)), using
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- Reed–Solomon code RS(77,81) [i]
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(8133, 6575, F81, 15) (dual of [6575, 6542, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(8133, 6574, F81, 15) (dual of [6574, 6541, 16]-code), using
- net defined by OOA [i] based on linear OOA(8133, 939, F81, 15, 15) (dual of [(939, 15), 14052, 16]-NRT-code), using
(18, 18+14, 5625)-Net over F81 — Digital
Digital (18, 32, 5625)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8132, 5625, F81, 14) (dual of [5625, 5593, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(8132, 6578, F81, 14) (dual of [6578, 6546, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(7) [i] based on
- linear OA(8127, 6561, F81, 14) (dual of [6561, 6534, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(815, 17, F81, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,81)), using
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- Reed–Solomon code RS(76,81) [i]
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- construction X applied to Ce(13) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(8132, 6578, F81, 14) (dual of [6578, 6546, 15]-code), using
(18, 18+14, large)-Net in Base 81 — Upper bound on s
There is no (18, 32, large)-net in base 81, because
- 12 times m-reduction [i] would yield (18, 20, large)-net in base 81, but