Best Known (53−10, 53, s)-Nets in Base 9
(53−10, 53, 106292)-Net over F9 — Constructive and digital
Digital (43, 53, 106292)-net over F9, using
- net defined by OOA [i] based on linear OOA(953, 106292, F9, 10, 10) (dual of [(106292, 10), 1062867, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(953, 531460, F9, 10) (dual of [531460, 531407, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(953, 531463, F9, 10) (dual of [531463, 531410, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(949, 531441, F9, 10) (dual of [531441, 531392, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(931, 531441, F9, 6) (dual of [531441, 531410, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(9) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(953, 531463, F9, 10) (dual of [531463, 531410, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(953, 531460, F9, 10) (dual of [531460, 531407, 11]-code), using
(53−10, 53, 531463)-Net over F9 — Digital
Digital (43, 53, 531463)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(953, 531463, F9, 10) (dual of [531463, 531410, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(949, 531441, F9, 10) (dual of [531441, 531392, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(931, 531441, F9, 6) (dual of [531441, 531410, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(9) ⊂ Ce(5) [i] based on
(53−10, 53, large)-Net in Base 9 — Upper bound on s
There is no (43, 53, large)-net in base 9, because
- 8 times m-reduction [i] would yield (43, 45, large)-net in base 9, but