Best Known (41, 41+40, s)-Nets in Base 81
(41, 41+40, 730)-Net over F81 — Constructive and digital
Digital (41, 81, 730)-net over F81, using
- t-expansion [i] based on digital (36, 81, 730)-net over F81, using
- net from sequence [i] based on digital (36, 729)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 36 and N(F) ≥ 730, using
- the Hermitian function field over F81 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 36 and N(F) ≥ 730, using
- net from sequence [i] based on digital (36, 729)-sequence over F81, using
(41, 41+40, 2189)-Net over F81 — Digital
Digital (41, 81, 2189)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8181, 2189, F81, 3, 40) (dual of [(2189, 3), 6486, 41]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8181, 6567, F81, 40) (dual of [6567, 6486, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(8181, 6569, F81, 40) (dual of [6569, 6488, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(36) [i] based on
- linear OA(8179, 6561, F81, 40) (dual of [6561, 6482, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(8173, 6561, F81, 37) (dual of [6561, 6488, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(39) ⊂ Ce(36) [i] based on
- discarding factors / shortening the dual code based on linear OA(8181, 6569, F81, 40) (dual of [6569, 6488, 41]-code), using
- OOA 3-folding [i] based on linear OA(8181, 6567, F81, 40) (dual of [6567, 6486, 41]-code), using
(41, 41+40, 5566469)-Net in Base 81 — Upper bound on s
There is no (41, 81, 5566470)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 38662 309503 680585 086997 120137 531408 605852 261557 302845 263112 283249 598437 469342 817297 733725 943644 029300 529881 154547 393712 105827 748519 540716 174405 725899 312001 > 8181 [i]