Best Known (86, 86+14, s)-Nets in Base 7
(86, 86+14, 823546)-Net over F7 — Constructive and digital
Digital (86, 100, 823546)-net over F7, using
- net defined by OOA [i] based on linear OOA(7100, 823546, F7, 14, 14) (dual of [(823546, 14), 11529544, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(7100, 5764822, F7, 14) (dual of [5764822, 5764722, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(7100, 5764828, F7, 14) (dual of [5764828, 5764728, 15]-code), using
- 1 times truncation [i] based on linear OA(7101, 5764829, F7, 15) (dual of [5764829, 5764728, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- linear OA(797, 5764801, F7, 15) (dual of [5764801, 5764704, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(74, 28, F7, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,7)), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- 1 times truncation [i] based on linear OA(7101, 5764829, F7, 15) (dual of [5764829, 5764728, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(7100, 5764828, F7, 14) (dual of [5764828, 5764728, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(7100, 5764822, F7, 14) (dual of [5764822, 5764722, 15]-code), using
(86, 86+14, 5764828)-Net over F7 — Digital
Digital (86, 100, 5764828)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7100, 5764828, F7, 14) (dual of [5764828, 5764728, 15]-code), using
- 1 times truncation [i] based on linear OA(7101, 5764829, F7, 15) (dual of [5764829, 5764728, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- linear OA(797, 5764801, F7, 15) (dual of [5764801, 5764704, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(74, 28, F7, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,7)), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- 1 times truncation [i] based on linear OA(7101, 5764829, F7, 15) (dual of [5764829, 5764728, 16]-code), using
(86, 86+14, large)-Net in Base 7 — Upper bound on s
There is no (86, 100, large)-net in base 7, because
- 12 times m-reduction [i] would yield (86, 88, large)-net in base 7, but