Best Known (37−12, 37, s)-Nets in Base 81
(37−12, 37, 88576)-Net over F81 — Constructive and digital
Digital (25, 37, 88576)-net over F81, using
- net defined by OOA [i] based on linear OOA(8137, 88576, F81, 12, 12) (dual of [(88576, 12), 1062875, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(8137, 531456, F81, 12) (dual of [531456, 531419, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [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(8122, 531441, F81, 8) (dual of [531441, 531419, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(11) ⊂ Ce(7) [i] based on
- OA 6-folding and stacking [i] based on linear OA(8137, 531456, F81, 12) (dual of [531456, 531419, 13]-code), using
(37−12, 37, 420173)-Net over F81 — Digital
Digital (25, 37, 420173)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8137, 420173, F81, 12) (dual of [420173, 420136, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8137, 531456, F81, 12) (dual of [531456, 531419, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [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(8122, 531441, F81, 8) (dual of [531441, 531419, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(11) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(8137, 531456, F81, 12) (dual of [531456, 531419, 13]-code), using
(37−12, 37, large)-Net in Base 81 — Upper bound on s
There is no (25, 37, large)-net in base 81, because
- 10 times m-reduction [i] would yield (25, 27, large)-net in base 81, but