Best Known (75, 75+11, s)-Nets in Base 5
(75, 75+11, 1677726)-Net over F5 — Constructive and digital
Digital (75, 86, 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, 81, 1677720)-net over F5, using
- net defined by OOA [i] based on linear OOA(581, 1677720, F5, 11, 11) (dual of [(1677720, 11), 18454839, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(581, 8388601, F5, 11) (dual of [8388601, 8388520, 12]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(581, 8388601, F5, 11) (dual of [8388601, 8388520, 12]-code), using
- net defined by OOA [i] based on linear OOA(581, 1677720, F5, 11, 11) (dual of [(1677720, 11), 18454839, 12]-NRT-code), using
- digital (0, 5, 6)-net over F5, using
(75, 75+11, 4194307)-Net over F5 — Digital
Digital (75, 86, 4194307)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(586, 4194307, F5, 2, 11) (dual of [(4194307, 2), 8388528, 12]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(55, 6, F5, 2, 5) (dual of [(6, 2), 7, 6]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(2;7,5) [i]
- linear OOA(581, 4194301, F5, 2, 11) (dual of [(4194301, 2), 8388521, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(581, 8388602, F5, 11) (dual of [8388602, 8388521, 12]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(581, large, F5, 11) (dual of [large, large−81, 12]-code), using
- OOA 2-folding [i] based on linear OA(581, 8388602, F5, 11) (dual of [8388602, 8388521, 12]-code), using
- linear OOA(55, 6, F5, 2, 5) (dual of [(6, 2), 7, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
(75, 75+11, large)-Net in Base 5 — Upper bound on s
There is no (75, 86, large)-net in base 5, because
- 9 times m-reduction [i] would yield (75, 77, large)-net in base 5, but