Best Known (74−24, 74, s)-Nets in Base 81
(74−24, 74, 44288)-Net over F81 — Constructive and digital
Digital (50, 74, 44288)-net over F81, using
- 811 times duplication [i] based on digital (49, 73, 44288)-net over F81, using
- net defined by OOA [i] based on linear OOA(8173, 44288, F81, 24, 24) (dual of [(44288, 24), 1062839, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(8173, 531456, F81, 24) (dual of [531456, 531383, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(19) [i] based on
- linear OA(8170, 531441, F81, 24) (dual of [531441, 531371, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- 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(813, 15, F81, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,81) or 15-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 Ce(23) ⊂ Ce(19) [i] based on
- OA 12-folding and stacking [i] based on linear OA(8173, 531456, F81, 24) (dual of [531456, 531383, 25]-code), using
- net defined by OOA [i] based on linear OOA(8173, 44288, F81, 24, 24) (dual of [(44288, 24), 1062839, 25]-NRT-code), using
(74−24, 74, 265730)-Net over F81 — Digital
Digital (50, 74, 265730)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8174, 265730, F81, 2, 24) (dual of [(265730, 2), 531386, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8174, 531460, F81, 24) (dual of [531460, 531386, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
- linear OA(8170, 531441, F81, 24) (dual of [531441, 531371, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- 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(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(23) ⊂ Ce(18) [i] based on
- OOA 2-folding [i] based on linear OA(8174, 531460, F81, 24) (dual of [531460, 531386, 25]-code), using
(74−24, 74, large)-Net in Base 81 — Upper bound on s
There is no (50, 74, large)-net in base 81, because
- 22 times m-reduction [i] would yield (50, 52, large)-net in base 81, but