Best Known (3, 9, s)-Nets in Base 81
(3, 9, 164)-Net over F81 — Constructive and digital
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
(3, 9, 166)-Net over F81 — Digital
Digital (3, 9, 166)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(819, 166, F81, 6) (dual of [166, 157, 7]-code), using
- construction X applied to C([38,43]) ⊂ C([39,43]) [i] based on
- linear OA(819, 164, F81, 6) (dual of [164, 155, 7]-code), using the BCH-code C(I) with length 164 | 812−1, defining interval I = {38,39,…,43}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(817, 164, F81, 5) (dual of [164, 157, 6]-code), using the BCH-code C(I) with length 164 | 812−1, defining interval I = {39,40,41,42,43}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to C([38,43]) ⊂ C([39,43]) [i] based on
(3, 9, 12070)-Net in Base 81 — Upper bound on s
There is no (3, 9, 12071)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 150127 562993 924241 > 819 [i]