Best Known (41−8, 41, s)-Nets in Base 7
(41−8, 41, 29420)-Net over F7 — Constructive and digital
Digital (33, 41, 29420)-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 (29, 37, 29412)-net over F7, using
- net defined by OOA [i] based on linear OOA(737, 29412, F7, 8, 8) (dual of [(29412, 8), 235259, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(737, 117648, F7, 8) (dual of [117648, 117611, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(737, 117649, F7, 8) (dual of [117649, 117612, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(737, 117649, F7, 8) (dual of [117649, 117612, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(737, 117648, F7, 8) (dual of [117648, 117611, 9]-code), using
- net defined by OOA [i] based on linear OOA(737, 29412, F7, 8, 8) (dual of [(29412, 8), 235259, 9]-NRT-code), using
- digital (0, 4, 8)-net over F7, using
(41−8, 41, 117671)-Net over F7 — Digital
Digital (33, 41, 117671)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(741, 117671, F7, 8) (dual of [117671, 117630, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
- linear OA(737, 117649, F7, 8) (dual of [117649, 117612, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(719, 117649, F7, 4) (dual of [117649, 117630, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(74, 22, F7, 3) (dual of [22, 18, 4]-code or 22-cap in PG(3,7)), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
(41−8, 41, large)-Net in Base 7 — Upper bound on s
There is no (33, 41, large)-net in base 7, because
- 6 times m-reduction [i] would yield (33, 35, large)-net in base 7, but