Best Known (75−9, 75, s)-Nets in Base 5
(75−9, 75, 2097156)-Net over F5 — Constructive and digital
Digital (66, 75, 2097156)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (62, 71, 2097150)-net over F5, using
- net defined by OOA [i] based on linear OOA(571, 2097150, F5, 9, 9) (dual of [(2097150, 9), 18874279, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(571, 8388601, F5, 9) (dual of [8388601, 8388530, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(571, large, F5, 9) (dual of [large, large−71, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(571, large, F5, 9) (dual of [large, large−71, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(571, 8388601, F5, 9) (dual of [8388601, 8388530, 10]-code), using
- net defined by OOA [i] based on linear OOA(571, 2097150, F5, 9, 9) (dual of [(2097150, 9), 18874279, 10]-NRT-code), using
- digital (0, 4, 6)-net over F5, using
(75−9, 75, large)-Net over F5 — Digital
Digital (66, 75, large)-net over F5, using
- 53 times duplication [i] based on digital (63, 72, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(572, large, F5, 9) (dual of [large, large−72, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(571, large, F5, 9) (dual of [large, large−71, 10]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,8], and designed minimum distance d ≥ |I|+1 = 10 [i]
- 1 times code embedding in larger space [i] based on linear OA(571, large, F5, 9) (dual of [large, large−71, 10]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(572, large, F5, 9) (dual of [large, large−72, 10]-code), using
(75−9, 75, large)-Net in Base 5 — Upper bound on s
There is no (66, 75, large)-net in base 5, because
- 7 times m-reduction [i] would yield (66, 68, large)-net in base 5, but