Best Known (75, 75+14, s)-Nets in Base 7
(75, 75+14, 117653)-Net over F7 — Constructive and digital
Digital (75, 89, 117653)-net over F7, using
- net defined by OOA [i] based on linear OOA(789, 117653, F7, 14, 14) (dual of [(117653, 14), 1647053, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(789, 823571, F7, 14) (dual of [823571, 823482, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(789, 823575, F7, 14) (dual of [823575, 823486, 15]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- linear OA(785, 823543, F7, 15) (dual of [823543, 823458, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(74, 32, F7, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,7)), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(789, 823575, F7, 14) (dual of [823575, 823486, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(789, 823571, F7, 14) (dual of [823571, 823482, 15]-code), using
(75, 75+14, 823575)-Net over F7 — Digital
Digital (75, 89, 823575)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(789, 823575, F7, 14) (dual of [823575, 823486, 15]-code), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
- linear OA(785, 823543, F7, 15) (dual of [823543, 823458, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(757, 823543, F7, 10) (dual of [823543, 823486, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(74, 32, F7, 3) (dual of [32, 28, 4]-code or 32-cap in PG(3,7)), using
- construction X applied to Ce(14) ⊂ Ce(9) [i] based on
(75, 75+14, large)-Net in Base 7 — Upper bound on s
There is no (75, 89, large)-net in base 7, because
- 12 times m-reduction [i] would yield (75, 77, large)-net in base 7, but