Best Known (64, 64+16, s)-Nets in Base 81
(64, 64+16, 1050297)-Net over F81 — Constructive and digital
Digital (64, 80, 1050297)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (11, 19, 1722)-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 (7, 15, 1640)-net over F81, using
- net defined by OOA [i] based on linear OOA(8115, 1640, F81, 8, 8) (dual of [(1640, 8), 13105, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(8115, 6560, F81, 8) (dual of [6560, 6545, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(8115, 6560, F81, 8) (dual of [6560, 6545, 9]-code), using
- net defined by OOA [i] based on linear OOA(8115, 1640, F81, 8, 8) (dual of [(1640, 8), 13105, 9]-NRT-code), using
- 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 (11, 19, 1722)-net over F81, using
(64, 64+16, large)-Net over F81 — Digital
Digital (64, 80, large)-net over F81, using
- t-expansion [i] based on digital (60, 80, large)-net over F81, using
- 1 times m-reduction [i] based on digital (60, 81, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8181, large, F81, 21) (dual of [large, large−81, 22]-code), using
- 1 times m-reduction [i] based on digital (60, 81, large)-net over F81, using
(64, 64+16, large)-Net in Base 81 — Upper bound on s
There is no (64, 80, large)-net in base 81, because
- 14 times m-reduction [i] would yield (64, 66, large)-net in base 81, but