Best Known (109, 109+24, s)-Nets in Base 9
(109, 109+24, 44289)-Net over F9 — Constructive and digital
Digital (109, 133, 44289)-net over F9, using
- 92 times duplication [i] based on digital (107, 131, 44289)-net over F9, using
- net defined by OOA [i] based on linear OOA(9131, 44289, F9, 24, 24) (dual of [(44289, 24), 1062805, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(9131, 531468, F9, 24) (dual of [531468, 531337, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(9131, 531469, F9, 24) (dual of [531469, 531338, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(19) [i] based on
- linear OA(9127, 531441, F9, 24) (dual of [531441, 531314, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(9103, 531441, F9, 20) (dual of [531441, 531338, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(94, 28, F9, 3) (dual of [28, 24, 4]-code or 28-cap in PG(3,9)), using
- construction X applied to Ce(23) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(9131, 531469, F9, 24) (dual of [531469, 531338, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(9131, 531468, F9, 24) (dual of [531468, 531337, 25]-code), using
- net defined by OOA [i] based on linear OOA(9131, 44289, F9, 24, 24) (dual of [(44289, 24), 1062805, 25]-NRT-code), using
(109, 109+24, 531477)-Net over F9 — Digital
Digital (109, 133, 531477)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9133, 531477, F9, 24) (dual of [531477, 531344, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
- linear OA(9127, 531441, F9, 24) (dual of [531441, 531314, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(997, 531441, F9, 19) (dual of [531441, 531344, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(96, 36, F9, 4) (dual of [36, 30, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- 1 times truncation [i] based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
(109, 109+24, large)-Net in Base 9 — Upper bound on s
There is no (109, 133, large)-net in base 9, because
- 22 times m-reduction [i] would yield (109, 111, large)-net in base 9, but