Best Known (190, 190+31, s)-Nets in Base 3
(190, 190+31, 11809)-Net over F3 — Constructive and digital
Digital (190, 221, 11809)-net over F3, using
- net defined by OOA [i] based on linear OOA(3221, 11809, F3, 31, 31) (dual of [(11809, 31), 365858, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3221, 177136, F3, 31) (dual of [177136, 176915, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3221, 177147, F3, 31) (dual of [177147, 176926, 32]-code), using
- an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- discarding factors / shortening the dual code based on linear OA(3221, 177147, F3, 31) (dual of [177147, 176926, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3221, 177136, F3, 31) (dual of [177136, 176915, 32]-code), using
(190, 190+31, 44287)-Net over F3 — Digital
Digital (190, 221, 44287)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3221, 44287, F3, 4, 31) (dual of [(44287, 4), 176927, 32]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3221, 177148, F3, 31) (dual of [177148, 176927, 32]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- OOA 4-folding [i] based on linear OA(3221, 177148, F3, 31) (dual of [177148, 176927, 32]-code), using
(190, 190+31, large)-Net in Base 3 — Upper bound on s
There is no (190, 221, large)-net in base 3, because
- 29 times m-reduction [i] would yield (190, 192, large)-net in base 3, but