Best Known (85−10, 85, s)-Nets in Base 5
(85−10, 85, 1677726)-Net over F5 — Constructive and digital
Digital (75, 85, 1677726)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 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 (70, 80, 1677720)-net over F5, using
- net defined by OOA [i] based on linear OOA(580, 1677720, F5, 10, 10) (dual of [(1677720, 10), 16777120, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(580, 8388600, F5, 10) (dual of [8388600, 8388520, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(580, large, F5, 10) (dual of [large, large−80, 11]-code), using
- the primitive narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(580, large, F5, 10) (dual of [large, large−80, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(580, 8388600, F5, 10) (dual of [8388600, 8388520, 11]-code), using
- net defined by OOA [i] based on linear OOA(580, 1677720, F5, 10, 10) (dual of [(1677720, 10), 16777120, 11]-NRT-code), using
- digital (0, 5, 6)-net over F5, using
(85−10, 85, large)-Net over F5 — Digital
Digital (75, 85, large)-net over F5, using
- 54 times duplication [i] based on digital (71, 81, large)-net over F5, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(581, large, F5, 10) (dual of [large, large−81, 11]-code), using
- strength reduction [i] based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 9765626 | 520−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- strength reduction [i] based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(581, large, F5, 10) (dual of [large, large−81, 11]-code), using
(85−10, 85, large)-Net in Base 5 — Upper bound on s
There is no (75, 85, large)-net in base 5, because
- 8 times m-reduction [i] would yield (75, 77, large)-net in base 5, but