Best Known (27, 39, s)-Nets in Base 81
(27, 39, 88577)-Net over F81 — Constructive and digital
Digital (27, 39, 88577)-net over F81, using
- net defined by OOA [i] based on linear OOA(8139, 88577, F81, 12, 12) (dual of [(88577, 12), 1062885, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(8139, 531462, F81, 12) (dual of [531462, 531423, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8139, 531464, F81, 12) (dual of [531464, 531425, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
- linear OA(8134, 531441, F81, 12) (dual of [531441, 531407, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8116, 531441, F81, 6) (dual of [531441, 531425, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(815, 23, F81, 5) (dual of [23, 18, 6]-code or 23-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(11) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(8139, 531464, F81, 12) (dual of [531464, 531425, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(8139, 531462, F81, 12) (dual of [531462, 531423, 13]-code), using
(27, 39, 531464)-Net over F81 — Digital
Digital (27, 39, 531464)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8139, 531464, F81, 12) (dual of [531464, 531425, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
- linear OA(8134, 531441, F81, 12) (dual of [531441, 531407, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8116, 531441, F81, 6) (dual of [531441, 531425, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(815, 23, F81, 5) (dual of [23, 18, 6]-code or 23-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(11) ⊂ Ce(5) [i] based on
(27, 39, large)-Net in Base 81 — Upper bound on s
There is no (27, 39, large)-net in base 81, because
- 10 times m-reduction [i] would yield (27, 29, large)-net in base 81, but