Best Known (65−32, 65, s)-Nets in Base 81
(65−32, 65, 470)-Net over F81 — Constructive and digital
Digital (33, 65, 470)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 17, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- digital (16, 48, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- digital (1, 17, 100)-net over F81, using
(65−32, 65, 2189)-Net over F81 — Digital
Digital (33, 65, 2189)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8165, 2189, F81, 3, 32) (dual of [(2189, 3), 6502, 33]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8165, 6567, F81, 32) (dual of [6567, 6502, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(8165, 6569, F81, 32) (dual of [6569, 6504, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(28) [i] based on
- linear OA(8163, 6561, F81, 32) (dual of [6561, 6498, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(8157, 6561, F81, 29) (dual of [6561, 6504, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(812, 8, F81, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,81)), using
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- Reed–Solomon code RS(79,81) [i]
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- construction X applied to Ce(31) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(8165, 6569, F81, 32) (dual of [6569, 6504, 33]-code), using
- OOA 3-folding [i] based on linear OA(8165, 6567, F81, 32) (dual of [6567, 6502, 33]-code), using
(65−32, 65, 4815799)-Net in Base 81 — Upper bound on s
There is no (33, 65, 4815800)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 11259 688778 571123 371865 711881 607712 705079 699061 821633 810638 534481 598400 582864 884049 059961 371416 772213 288418 943343 627207 424001 > 8165 [i]