Best Known (87−12, 87, s)-Nets in Base 7
(87−12, 87, 960809)-Net over F7 — Constructive and digital
Digital (75, 87, 960809)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (0, 6, 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 (69, 81, 960801)-net over F7, using
- net defined by OOA [i] based on linear OOA(781, 960801, F7, 12, 12) (dual of [(960801, 12), 11529531, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(781, 5764806, F7, 12) (dual of [5764806, 5764725, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(781, 5764809, F7, 12) (dual of [5764809, 5764728, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(781, 5764801, F7, 12) (dual of [5764801, 5764720, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(773, 5764801, F7, 11) (dual of [5764801, 5764728, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(70, 8, F7, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(781, 5764809, F7, 12) (dual of [5764809, 5764728, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(781, 5764806, F7, 12) (dual of [5764806, 5764725, 13]-code), using
- net defined by OOA [i] based on linear OOA(781, 960801, F7, 12, 12) (dual of [(960801, 12), 11529531, 13]-NRT-code), using
- digital (0, 6, 8)-net over F7, using
(87−12, 87, 5764839)-Net over F7 — Digital
Digital (75, 87, 5764839)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(787, 5764839, F7, 12) (dual of [5764839, 5764752, 13]-code), using
- 2 times code embedding in larger space [i] based on linear OA(785, 5764837, F7, 12) (dual of [5764837, 5764752, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(781, 5764801, F7, 12) (dual of [5764801, 5764720, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(749, 5764801, F7, 8) (dual of [5764801, 5764752, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(74, 36, F7, 3) (dual of [36, 32, 4]-code or 36-cap in PG(3,7)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(785, 5764837, F7, 12) (dual of [5764837, 5764752, 13]-code), using
(87−12, 87, large)-Net in Base 7 — Upper bound on s
There is no (75, 87, large)-net in base 7, because
- 10 times m-reduction [i] would yield (75, 77, large)-net in base 7, but