Best Known (57, 57+16, s)-Nets in Base 81
(57, 57+16, 1048739)-Net over F81 — Constructive and digital
Digital (57, 73, 1048739)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (4, 12, 164)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 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 (0, 8, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81 (see above)
- digital (0, 4, 82)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (45, 61, 1048575)-net over F81, using
- net defined by OOA [i] based on linear OOA(8161, 1048575, F81, 16, 16) (dual of [(1048575, 16), 16777139, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(8161, 8388600, F81, 16) (dual of [8388600, 8388539, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(8161, large, F81, 16) (dual of [large, large−61, 17]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(8161, large, F81, 16) (dual of [large, large−61, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(8161, 8388600, F81, 16) (dual of [8388600, 8388539, 17]-code), using
- net defined by OOA [i] based on linear OOA(8161, 1048575, F81, 16, 16) (dual of [(1048575, 16), 16777139, 17]-NRT-code), using
- digital (4, 12, 164)-net over F81, using
(57, 57+16, large)-Net over F81 — Digital
Digital (57, 73, large)-net over F81, using
- 4 times m-reduction [i] based on digital (57, 77, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8177, large, F81, 20) (dual of [large, large−77, 21]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8177, large, F81, 20) (dual of [large, large−77, 21]-code), using
(57, 57+16, large)-Net in Base 81 — Upper bound on s
There is no (57, 73, large)-net in base 81, because
- 14 times m-reduction [i] would yield (57, 59, large)-net in base 81, but