Best Known (56, 56+16, s)-Nets in Base 81
(56, 56+16, 1048691)-Net over F81 — Constructive and digital
Digital (56, 72, 1048691)-net over F81, using
- 811 times duplication [i] based on digital (55, 71, 1048691)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (2, 10, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- 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 (2, 10, 116)-net over F81, using
- (u, u+v)-construction [i] based on
(56, 56+16, large)-Net over F81 — Digital
Digital (56, 72, large)-net over F81, using
- t-expansion [i] based on digital (54, 72, large)-net over F81, using
- 1 times m-reduction [i] based on digital (54, 73, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8173, large, F81, 19) (dual of [large, large−73, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8173, large, F81, 19) (dual of [large, large−73, 20]-code), using
- 1 times m-reduction [i] based on digital (54, 73, large)-net over F81, using
(56, 56+16, large)-Net in Base 81 — Upper bound on s
There is no (56, 72, large)-net in base 81, because
- 14 times m-reduction [i] would yield (56, 58, large)-net in base 81, but