Best Known (40, 40+19, s)-Nets in Base 81
(40, 40+19, 59051)-Net over F81 — Constructive and digital
Digital (40, 59, 59051)-net over F81, using
- net defined by OOA [i] based on linear OOA(8159, 59051, F81, 19, 19) (dual of [(59051, 19), 1121910, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(8159, 531460, F81, 19) (dual of [531460, 531401, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(8155, 531441, F81, 19) (dual of [531441, 531386, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(8140, 531441, F81, 14) (dual of [531441, 531401, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(814, 19, F81, 4) (dual of [19, 15, 5]-code or 19-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(18) ⊂ Ce(13) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(8159, 531460, F81, 19) (dual of [531460, 531401, 20]-code), using
(40, 40+19, 291169)-Net over F81 — Digital
Digital (40, 59, 291169)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8159, 291169, F81, 19) (dual of [291169, 291110, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(8159, 531460, F81, 19) (dual of [531460, 531401, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(8155, 531441, F81, 19) (dual of [531441, 531386, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(8140, 531441, F81, 14) (dual of [531441, 531401, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(814, 19, F81, 4) (dual of [19, 15, 5]-code or 19-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(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(8159, 531460, F81, 19) (dual of [531460, 531401, 20]-code), using
(40, 40+19, large)-Net in Base 81 — Upper bound on s
There is no (40, 59, large)-net in base 81, because
- 17 times m-reduction [i] would yield (40, 42, large)-net in base 81, but