Best Known (16, 29, s)-Nets in Base 81
(16, 29, 1095)-Net over F81 — Constructive and digital
Digital (16, 29, 1095)-net over F81, using
- 811 times duplication [i] based on digital (15, 28, 1095)-net over F81, using
- net defined by OOA [i] based on linear OOA(8128, 1095, F81, 13, 13) (dual of [(1095, 13), 14207, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8128, 6571, F81, 13) (dual of [6571, 6543, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8128, 6573, F81, 13) (dual of [6573, 6545, 14]-code), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [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(8117, 6562, F81, 9) (dual of [6562, 6545, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(813, 11, F81, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,81) or 11-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to C([0,6]) ⊂ C([0,4]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8128, 6573, F81, 13) (dual of [6573, 6545, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(8128, 6571, F81, 13) (dual of [6571, 6543, 14]-code), using
- net defined by OOA [i] based on linear OOA(8128, 1095, F81, 13, 13) (dual of [(1095, 13), 14207, 14]-NRT-code), using
(16, 29, 4420)-Net over F81 — Digital
Digital (16, 29, 4420)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8129, 4420, F81, 13) (dual of [4420, 4391, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(8129, 6575, F81, 13) (dual of [6575, 6546, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(7) [i] based on
- linear OA(8125, 6561, F81, 13) (dual of [6561, 6536, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(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(12) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(8129, 6575, F81, 13) (dual of [6575, 6546, 14]-code), using
(16, 29, large)-Net in Base 81 — Upper bound on s
There is no (16, 29, large)-net in base 81, because
- 11 times m-reduction [i] would yield (16, 18, large)-net in base 81, but