Best Known (6, 6+5, s)-Nets in Base 9
(6, 6+5, 164)-Net over F9 — Constructive and digital
Digital (6, 11, 164)-net over F9, using
- 1 times m-reduction [i] based on digital (6, 12, 164)-net over F9, using
- trace code for nets [i] based on digital (0, 6, 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
- trace code for nets [i] based on digital (0, 6, 82)-net over F81, using
(6, 6+5, 219)-Net over F9 — Digital
Digital (6, 11, 219)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(911, 219, F9, 5) (dual of [219, 208, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(91, 73, F9, 1) (dual of [73, 72, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(93, 73, F9, 2) (dual of [73, 70, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- linear OA(91, 73, F9, 1) (dual of [73, 72, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
(6, 6+5, 351)-Net in Base 9 — Constructive
(6, 11, 351)-net in base 9, using
- net defined by OOA [i] based on OOA(911, 351, S9, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(911, 703, S9, 5), using
- discarding parts of the base [i] based on linear OA(277, 703, F27, 5) (dual of [703, 696, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on OA(911, 703, S9, 5), using
(6, 6+5, 10437)-Net in Base 9 — Upper bound on s
There is no (6, 11, 10438)-net in base 9, because
- 1 times m-reduction [i] would yield (6, 10, 10438)-net in base 9, but
- the generalized Rao bound for nets shows that 9m ≥ 3486 960033 > 910 [i]