Best Known (53, 53+12, s)-Nets in Base 9
(53, 53+12, 88577)-Net over F9 — Constructive and digital
Digital (53, 65, 88577)-net over F9, using
- net defined by OOA [i] based on linear OOA(965, 88577, F9, 12, 12) (dual of [(88577, 12), 1062859, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(965, 531462, F9, 12) (dual of [531462, 531397, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(965, 531463, F9, 12) (dual of [531463, 531398, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(961, 531441, F9, 12) (dual of [531441, 531380, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(943, 531441, F9, 8) (dual of [531441, 531398, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(94, 22, F9, 3) (dual of [22, 18, 4]-code or 22-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(965, 531463, F9, 12) (dual of [531463, 531398, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(965, 531462, F9, 12) (dual of [531462, 531397, 13]-code), using
(53, 53+12, 531463)-Net over F9 — Digital
Digital (53, 65, 531463)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(965, 531463, F9, 12) (dual of [531463, 531398, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(961, 531441, F9, 12) (dual of [531441, 531380, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(943, 531441, F9, 8) (dual of [531441, 531398, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(94, 22, F9, 3) (dual of [22, 18, 4]-code or 22-cap in PG(3,9)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
(53, 53+12, large)-Net in Base 9 — Upper bound on s
There is no (53, 65, large)-net in base 9, because
- 10 times m-reduction [i] would yield (53, 55, large)-net in base 9, but