Best Known (20, 20+18, s)-Nets in Base 81
(20, 20+18, 730)-Net over F81 — Constructive and digital
Digital (20, 38, 730)-net over F81, using
- net defined by OOA [i] based on linear OOA(8138, 730, F81, 18, 18) (dual of [(730, 18), 13102, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(8138, 6570, F81, 18) (dual of [6570, 6532, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(8138, 6572, F81, 18) (dual of [6572, 6534, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(8127, 6561, F81, 14) (dual of [6561, 6534, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(813, 11, F81, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,81) or 11-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(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(8138, 6572, F81, 18) (dual of [6572, 6534, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(8138, 6570, F81, 18) (dual of [6570, 6532, 19]-code), using
(20, 20+18, 3049)-Net over F81 — Digital
Digital (20, 38, 3049)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8138, 3049, F81, 2, 18) (dual of [(3049, 2), 6060, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8138, 3286, F81, 2, 18) (dual of [(3286, 2), 6534, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8138, 6572, F81, 18) (dual of [6572, 6534, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(8127, 6561, F81, 14) (dual of [6561, 6534, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(813, 11, F81, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,81) or 11-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(17) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(8138, 6572, F81, 18) (dual of [6572, 6534, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(8138, 3286, F81, 2, 18) (dual of [(3286, 2), 6534, 19]-NRT-code), using
(20, 20+18, 5925281)-Net in Base 81 — Upper bound on s
There is no (20, 38, 5925282)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 3 329896 865040 272067 017187 976466 207660 448997 302724 654142 553479 774806 926241 > 8138 [i]