Best Known (20, 20+4, s)-Nets in Base 7
(20, 20+4, 411783)-Net over F7 — Constructive and digital
Digital (20, 24, 411783)-net over F7, using
- net defined by OOA [i] based on linear OOA(724, 411783, F7, 4, 4) (dual of [(411783, 4), 1647108, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(724, 411783, F7, 3, 4) (dual of [(411783, 3), 1235325, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(72, 8, F7, 3, 2) (dual of [(8, 3), 22, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;22,7) [i]
- linear OOA(722, 411775, F7, 3, 4) (dual of [(411775, 3), 1235303, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(722, 823550, F7, 4) (dual of [823550, 823528, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(722, 823543, F7, 4) (dual of [823543, 823521, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(715, 823543, F7, 3) (dual of [823543, 823528, 4]-code or 823543-cap in PG(14,7)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(70, 7, F7, 0) (dual of [7, 7, 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(3) ⊂ Ce(2) [i] based on
- OA 2-folding and stacking [i] based on linear OA(722, 823550, F7, 4) (dual of [823550, 823528, 5]-code), using
- linear OOA(72, 8, F7, 3, 2) (dual of [(8, 3), 22, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(724, 411783, F7, 3, 4) (dual of [(411783, 3), 1235325, 5]-NRT-code), using
(20, 20+4, 1745891)-Net over F7 — Digital
Digital (20, 24, 1745891)-net over F7, using
- net defined by OOA [i] based on linear OOA(724, 1745891, F7, 4, 4) (dual of [(1745891, 4), 6983540, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(724, 1745891, F7, 3, 4) (dual of [(1745891, 3), 5237649, 5]-NRT-code), using
(20, 20+4, large)-Net in Base 7 — Upper bound on s
There is no (20, 24, large)-net in base 7, because
- 2 times m-reduction [i] would yield (20, 22, large)-net in base 7, but