Best Known (27, 27+8, s)-Nets in Base 7
(27, 27+8, 4209)-Net over F7 — Constructive and digital
Digital (27, 35, 4209)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 8)-net over F7, using
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F7 with g(F) = 0 and N(F) ≥ 8, using
- the rational function field F7(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 7)-sequence over F7, using
- digital (23, 31, 4201)-net over F7, using
- net defined by OOA [i] based on linear OOA(731, 4201, F7, 8, 8) (dual of [(4201, 8), 33577, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(731, 16804, F7, 8) (dual of [16804, 16773, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(731, 16804, F7, 8) (dual of [16804, 16773, 9]-code), using
- net defined by OOA [i] based on linear OOA(731, 4201, F7, 8, 8) (dual of [(4201, 8), 33577, 9]-NRT-code), using
- digital (0, 4, 8)-net over F7, using
(27, 27+8, 16826)-Net over F7 — Digital
Digital (27, 35, 16826)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(735, 16826, F7, 8) (dual of [16826, 16791, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
- linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(716, 16807, F7, 4) (dual of [16807, 16791, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(74, 19, F7, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,7)), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
(27, 27+8, large)-Net in Base 7 — Upper bound on s
There is no (27, 35, large)-net in base 7, because
- 6 times m-reduction [i] would yield (27, 29, large)-net in base 7, but