Best Known (39−18, 39, s)-Nets in Base 81
(39−18, 39, 730)-Net over F81 — Constructive and digital
Digital (21, 39, 730)-net over F81, using
- 1 times m-reduction [i] based on digital (21, 40, 730)-net over F81, using
- net defined by OOA [i] based on linear OOA(8140, 730, F81, 19, 19) (dual of [(730, 19), 13830, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(8140, 6571, F81, 19) (dual of [6571, 6531, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(8140, 6573, F81, 19) (dual of [6573, 6533, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(8137, 6562, F81, 19) (dual of [6562, 6525, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(8129, 6562, F81, 15) (dual of [6562, 6533, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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([0,9]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8140, 6573, F81, 19) (dual of [6573, 6533, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(8140, 6571, F81, 19) (dual of [6571, 6531, 20]-code), using
- net defined by OOA [i] based on linear OOA(8140, 730, F81, 19, 19) (dual of [(730, 19), 13830, 20]-NRT-code), using
(39−18, 39, 3287)-Net over F81 — Digital
Digital (21, 39, 3287)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8139, 3287, F81, 2, 18) (dual of [(3287, 2), 6535, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8139, 6574, F81, 18) (dual of [6574, 6535, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(8139, 6575, F81, 18) (dual of [6575, 6536, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(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(17) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(8139, 6575, F81, 18) (dual of [6575, 6536, 19]-code), using
- OOA 2-folding [i] based on linear OA(8139, 6574, F81, 18) (dual of [6574, 6535, 19]-code), using
(39−18, 39, large)-Net in Base 81 — Upper bound on s
There is no (21, 39, large)-net in base 81, because
- 16 times m-reduction [i] would yield (21, 23, large)-net in base 81, but