Best Known (85, 85+12, s)-Nets in Base 5
(85, 85+12, 1398106)-Net over F5 — Constructive and digital
Digital (85, 97, 1398106)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 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 (79, 91, 1398100)-net over F5, using
- net defined by OOA [i] based on linear OOA(591, 1398100, F5, 12, 12) (dual of [(1398100, 12), 16777109, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(591, 8388600, F5, 12) (dual of [8388600, 8388509, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(591, large, F5, 12) (dual of [large, large−91, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(591, large, F5, 12) (dual of [large, large−91, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(591, 8388600, F5, 12) (dual of [8388600, 8388509, 13]-code), using
- net defined by OOA [i] based on linear OOA(591, 1398100, F5, 12, 12) (dual of [(1398100, 12), 16777109, 13]-NRT-code), using
- digital (0, 6, 6)-net over F5, using
(85, 85+12, 5808017)-Net over F5 — Digital
Digital (85, 97, 5808017)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(597, 5808017, F5, 12) (dual of [5808017, 5807920, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(597, large, F5, 12) (dual of [large, large−97, 13]-code), using
- 6 times code embedding in larger space [i] based on linear OA(591, large, F5, 12) (dual of [large, large−91, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 9765624 = 510−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 6 times code embedding in larger space [i] based on linear OA(591, large, F5, 12) (dual of [large, large−91, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(597, large, F5, 12) (dual of [large, large−97, 13]-code), using
(85, 85+12, large)-Net in Base 5 — Upper bound on s
There is no (85, 97, large)-net in base 5, because
- 10 times m-reduction [i] would yield (85, 87, large)-net in base 5, but