Best Known (135−24, 135, s)-Nets in Base 9
(135−24, 135, 44289)-Net over F9 — Constructive and digital
Digital (111, 135, 44289)-net over F9, using
- 94 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
(135−24, 135, 531481)-Net over F9 — Digital
Digital (111, 135, 531481)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9135, 531481, F9, 24) (dual of [531481, 531346, 25]-code), using
- construction X with 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
- linear OA(9133, 531479, F9, 23) (dual of [531479, 531346, 24]-code), using Gilbert–Varšamov bound and bm = 9133 > Vbs−1(k−1) = 59938 108098 658100 137643 095770 698979 389616 418654 127537 342492 219678 225290 068123 272247 715983 404350 493689 492021 383801 393899 885169 [i]
- linear OA(90, 2, F9, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(9133, 531477, F9, 24) (dual of [531477, 531344, 25]-code), using
- construction X with Varšamov bound [i] based on
(135−24, 135, large)-Net in Base 9 — Upper bound on s
There is no (111, 135, large)-net in base 9, because
- 22 times m-reduction [i] would yield (111, 113, large)-net in base 9, but