Best Known (41−17, 41, s)-Nets in Base 81
(41−17, 41, 902)-Net over F81 — Constructive and digital
Digital (24, 41, 902)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- digital (16, 33, 820)-net over F81, using
- net defined by OOA [i] based on linear OOA(8133, 820, F81, 17, 17) (dual of [(820, 17), 13907, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(8133, 6561, F81, 17) (dual of [6561, 6528, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(8133, 6561, F81, 17) (dual of [6561, 6528, 18]-code), using
- net defined by OOA [i] based on linear OOA(8133, 820, F81, 17, 17) (dual of [(820, 17), 13907, 18]-NRT-code), using
- digital (0, 8, 82)-net over F81, using
(41−17, 41, 6645)-Net over F81 — Digital
Digital (24, 41, 6645)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8141, 6645, F81, 17) (dual of [6645, 6604, 18]-code), using
- (u, u+v)-construction [i] based on
- linear OA(818, 82, F81, 8) (dual of [82, 74, 9]-code or 82-arc in PG(7,81)), using
- extended Reed–Solomon code RSe(74,81) [i]
- linear OA(8133, 6563, F81, 17) (dual of [6563, 6530, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(8133, 6561, F81, 17) (dual of [6561, 6528, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(8131, 6561, F81, 16) (dual of [6561, 6530, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(818, 82, F81, 8) (dual of [82, 74, 9]-code or 82-arc in PG(7,81)), using
- (u, u+v)-construction [i] based on
(41−17, 41, large)-Net in Base 81 — Upper bound on s
There is no (24, 41, large)-net in base 81, because
- 15 times m-reduction [i] would yield (24, 26, large)-net in base 81, but