Best Known (55−12, 55, s)-Nets in Base 9
(55−12, 55, 9844)-Net over F9 — Constructive and digital
Digital (43, 55, 9844)-net over F9, using
- net defined by OOA [i] based on linear OOA(955, 9844, F9, 12, 12) (dual of [(9844, 12), 118073, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(955, 59064, F9, 12) (dual of [59064, 59009, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(955, 59068, F9, 12) (dual of [59068, 59013, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(951, 59049, F9, 12) (dual of [59049, 58998, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(936, 59049, F9, 8) (dual of [59049, 59013, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(94, 19, F9, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,9)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(955, 59068, F9, 12) (dual of [59068, 59013, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(955, 59064, F9, 12) (dual of [59064, 59009, 13]-code), using
(55−12, 55, 59068)-Net over F9 — Digital
Digital (43, 55, 59068)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(955, 59068, F9, 12) (dual of [59068, 59013, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(951, 59049, F9, 12) (dual of [59049, 58998, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(936, 59049, F9, 8) (dual of [59049, 59013, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(94, 19, F9, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,9)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
(55−12, 55, large)-Net in Base 9 — Upper bound on s
There is no (43, 55, large)-net in base 9, because
- 10 times m-reduction [i] would yield (43, 45, large)-net in base 9, but