Best Known (40−9, 40, s)-Nets in Base 8
(40−9, 40, 8200)-Net over F8 — Constructive and digital
Digital (31, 40, 8200)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 9)-net over F8, using
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 0 and N(F) ≥ 9, using
- the rational function field F8(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- digital (27, 36, 8191)-net over F8, using
- net defined by OOA [i] based on linear OOA(836, 8191, F8, 9, 9) (dual of [(8191, 9), 73683, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(836, 32765, F8, 9) (dual of [32765, 32729, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(836, 32765, F8, 9) (dual of [32765, 32729, 10]-code), using
- net defined by OOA [i] based on linear OOA(836, 8191, F8, 9, 9) (dual of [(8191, 9), 73683, 10]-NRT-code), using
- digital (0, 4, 9)-net over F8, using
(40−9, 40, 32787)-Net over F8 — Digital
Digital (31, 40, 32787)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(840, 32787, F8, 9) (dual of [32787, 32747, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
- linear OA(836, 32768, F8, 9) (dual of [32768, 32732, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(821, 32768, F8, 5) (dual of [32768, 32747, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(84, 19, F8, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,8)), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
(40−9, 40, large)-Net in Base 8 — Upper bound on s
There is no (31, 40, large)-net in base 8, because
- 7 times m-reduction [i] would yield (31, 33, large)-net in base 8, but