Best Known (52, 52+18, s)-Nets in Base 9
(52, 52+18, 1458)-Net over F9 — Constructive and digital
Digital (52, 70, 1458)-net over F9, using
- net defined by OOA [i] based on linear OOA(970, 1458, F9, 18, 18) (dual of [(1458, 18), 26174, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(970, 13122, F9, 18) (dual of [13122, 13052, 19]-code), using
- trace code [i] based on linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- trace code [i] based on linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(970, 13122, F9, 18) (dual of [13122, 13052, 19]-code), using
(52, 52+18, 11073)-Net over F9 — Digital
Digital (52, 70, 11073)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(970, 11073, F9, 18) (dual of [11073, 11003, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(970, 13122, F9, 18) (dual of [13122, 13052, 19]-code), using
- trace code [i] based on linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- trace code [i] based on linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(970, 13122, F9, 18) (dual of [13122, 13052, 19]-code), using
(52, 52+18, large)-Net in Base 9 — Upper bound on s
There is no (52, 70, large)-net in base 9, because
- 16 times m-reduction [i] would yield (52, 54, large)-net in base 9, but