Best Known (232−33, 232, s)-Nets in Base 3
(232−33, 232, 3693)-Net over F3 — Constructive and digital
Digital (199, 232, 3693)-net over F3, using
- 33 times duplication [i] based on digital (196, 229, 3693)-net over F3, using
- net defined by OOA [i] based on linear OOA(3229, 3693, F3, 33, 33) (dual of [(3693, 33), 121640, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(3229, 59089, F3, 33) (dual of [59089, 58860, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3229, 59090, F3, 33) (dual of [59090, 58861, 34]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- linear OA(3221, 59049, F3, 34) (dual of [59049, 58828, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3229, 59090, F3, 33) (dual of [59090, 58861, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(3229, 59089, F3, 33) (dual of [59089, 58860, 34]-code), using
- net defined by OOA [i] based on linear OOA(3229, 3693, F3, 33, 33) (dual of [(3693, 33), 121640, 34]-NRT-code), using
(232−33, 232, 27367)-Net over F3 — Digital
Digital (199, 232, 27367)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3232, 27367, F3, 2, 33) (dual of [(27367, 2), 54502, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3232, 29550, F3, 2, 33) (dual of [(29550, 2), 58868, 34]-NRT-code), using
- strength reduction [i] based on linear OOA(3232, 29550, F3, 2, 34) (dual of [(29550, 2), 58868, 35]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3232, 59100, F3, 34) (dual of [59100, 58868, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- linear OA(3221, 59049, F3, 34) (dual of [59049, 58828, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3181, 59049, F3, 28) (dual of [59049, 58868, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(311, 51, F3, 5) (dual of [51, 40, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(33) ⊂ Ce(27) [i] based on
- OOA 2-folding [i] based on linear OA(3232, 59100, F3, 34) (dual of [59100, 58868, 35]-code), using
- strength reduction [i] based on linear OOA(3232, 29550, F3, 2, 34) (dual of [(29550, 2), 58868, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3232, 29550, F3, 2, 33) (dual of [(29550, 2), 58868, 34]-NRT-code), using
(232−33, 232, large)-Net in Base 3 — Upper bound on s
There is no (199, 232, large)-net in base 3, because
- 31 times m-reduction [i] would yield (199, 201, large)-net in base 3, but