Best Known (62−20, 62, s)-Nets in Base 81
(62−20, 62, 53146)-Net over F81 — Constructive and digital
Digital (42, 62, 53146)-net over F81, using
- net defined by OOA [i] based on linear OOA(8162, 53146, F81, 20, 20) (dual of [(53146, 20), 1062858, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(8162, 531460, F81, 20) (dual of [531460, 531398, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(8158, 531441, F81, 20) (dual of [531441, 531383, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(8143, 531441, F81, 15) (dual of [531441, 531398, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(19) ⊂ Ce(14) [i] based on
- OA 10-folding and stacking [i] based on linear OA(8162, 531460, F81, 20) (dual of [531460, 531398, 21]-code), using
(62−20, 62, 277122)-Net over F81 — Digital
Digital (42, 62, 277122)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8162, 277122, F81, 20) (dual of [277122, 277060, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8162, 531460, F81, 20) (dual of [531460, 531398, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(14) [i] based on
- linear OA(8158, 531441, F81, 20) (dual of [531441, 531383, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(8143, 531441, F81, 15) (dual of [531441, 531398, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(19) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(8162, 531460, F81, 20) (dual of [531460, 531398, 21]-code), using
(62−20, 62, large)-Net in Base 81 — Upper bound on s
There is no (42, 62, large)-net in base 81, because
- 18 times m-reduction [i] would yield (42, 44, large)-net in base 81, but