Best Known (54−12, 54, s)-Nets in Base 81
(54−12, 54, 1398264)-Net over F81 — Constructive and digital
Digital (42, 54, 1398264)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (3, 9, 164)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 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, 6, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81 (see above)
- digital (0, 3, 82)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (33, 45, 1398100)-net over F81, using
- net defined by OOA [i] based on linear OOA(8145, 1398100, F81, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(8145, 8388600, F81, 12) (dual of [8388600, 8388555, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(8145, large, F81, 12) (dual of [large, large−45, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(8145, 8388600, F81, 12) (dual of [8388600, 8388555, 13]-code), using
- net defined by OOA [i] based on linear OOA(8145, 1398100, F81, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- digital (3, 9, 164)-net over F81, using
(54−12, 54, large)-Net over F81 — Digital
Digital (42, 54, large)-net over F81, using
- 3 times m-reduction [i] based on digital (42, 57, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8157, large, F81, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8157, large, F81, 15) (dual of [large, large−57, 16]-code), using
(54−12, 54, large)-Net in Base 81 — Upper bound on s
There is no (42, 54, large)-net in base 81, because
- 10 times m-reduction [i] would yield (42, 44, large)-net in base 81, but