Best Known (21−4, 21, s)-Nets in Base 8
(21−4, 21, 262147)-Net over F8 — Constructive and digital
Digital (17, 21, 262147)-net over F8, using
- 81 times duplication [i] based on digital (16, 20, 262147)-net over F8, using
- net defined by OOA [i] based on linear OOA(820, 262147, F8, 4, 4) (dual of [(262147, 4), 1048568, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(820, 262147, F8, 3, 4) (dual of [(262147, 3), 786421, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(820, 524294, F8, 4) (dual of [524294, 524274, 5]-code), using
- trace code [i] based on linear OA(6410, 262147, F64, 4) (dual of [262147, 262137, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(6410, 262144, F64, 4) (dual of [262144, 262134, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(647, 262144, F64, 3) (dual of [262144, 262137, 4]-code or 262144-cap in PG(6,64)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- trace code [i] based on linear OA(6410, 262147, F64, 4) (dual of [262147, 262137, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(820, 524294, F8, 4) (dual of [524294, 524274, 5]-code), using
- appending kth column [i] based on linear OOA(820, 262147, F8, 3, 4) (dual of [(262147, 3), 786421, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(820, 262147, F8, 4, 4) (dual of [(262147, 4), 1048568, 5]-NRT-code), using
(21−4, 21, 544398)-Net over F8 — Digital
Digital (17, 21, 544398)-net over F8, using
- net defined by OOA [i] based on linear OOA(821, 544398, F8, 4, 4) (dual of [(544398, 4), 2177571, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(821, 544398, F8, 3, 4) (dual of [(544398, 3), 1633173, 5]-NRT-code), using
(21−4, 21, large)-Net in Base 8 — Upper bound on s
There is no (17, 21, large)-net in base 8, because
- 2 times m-reduction [i] would yield (17, 19, large)-net in base 8, but