Best Known (60, 60+17, s)-Nets in Base 81
(60, 60+17, 1048739)-Net over F81 — Constructive and digital
Digital (60, 77, 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 (48, 65, 1048575)-net over F81, using
- net defined by OOA [i] based on linear OOA(8165, 1048575, F81, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(8165, 8388601, F81, 17) (dual of [8388601, 8388536, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(8165, large, F81, 17) (dual of [large, large−65, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8165, large, F81, 17) (dual of [large, large−65, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(8165, 8388601, F81, 17) (dual of [8388601, 8388536, 18]-code), using
- net defined by OOA [i] based on linear OOA(8165, 1048575, F81, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- digital (4, 12, 164)-net over F81, using
(60, 60+17, large)-Net over F81 — Digital
Digital (60, 77, large)-net over F81, using
- 4 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
(60, 60+17, large)-Net in Base 81 — Upper bound on s
There is no (60, 77, large)-net in base 81, because
- 15 times m-reduction [i] would yield (60, 62, large)-net in base 81, but