Best Known (47−8, 47, s)-Nets in Base 7
(47−8, 47, 205893)-Net over F7 — Constructive and digital
Digital (39, 47, 205893)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- digital (35, 43, 205885)-net over F7, using
- net defined by OOA [i] based on linear OOA(743, 205885, F7, 8, 8) (dual of [(205885, 8), 1647037, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(743, 823540, F7, 8) (dual of [823540, 823497, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(743, 823540, F7, 8) (dual of [823540, 823497, 9]-code), using
- net defined by OOA [i] based on linear OOA(743, 205885, F7, 8, 8) (dual of [(205885, 8), 1647037, 9]-NRT-code), using
- digital (0, 4, 8)-net over F7, using
(47−8, 47, 823568)-Net over F7 — Digital
Digital (39, 47, 823568)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(747, 823568, F7, 8) (dual of [823568, 823521, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
- linear OA(743, 823543, F7, 8) (dual of [823543, 823500, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(722, 823543, F7, 4) (dual of [823543, 823521, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(74, 25, F7, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,7)), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
(47−8, 47, large)-Net in Base 7 — Upper bound on s
There is no (39, 47, large)-net in base 7, because
- 6 times m-reduction [i] would yield (39, 41, large)-net in base 7, but