Best Known (163−21, 163, s)-Nets in Base 3
(163−21, 163, 17718)-Net over F3 — Constructive and digital
Digital (142, 163, 17718)-net over F3, using
- 32 times duplication [i] based on digital (140, 161, 17718)-net over F3, using
- net defined by OOA [i] based on linear OOA(3161, 17718, F3, 21, 21) (dual of [(17718, 21), 371917, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3161, 177181, F3, 21) (dual of [177181, 177020, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3161, 177186, F3, 21) (dual of [177186, 177025, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(3155, 177147, F3, 22) (dual of [177147, 176992, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3122, 177147, F3, 17) (dual of [177147, 177025, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(36, 39, F3, 3) (dual of [39, 33, 4]-code or 39-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3161, 177186, F3, 21) (dual of [177186, 177025, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3161, 177181, F3, 21) (dual of [177181, 177020, 22]-code), using
- net defined by OOA [i] based on linear OOA(3161, 17718, F3, 21, 21) (dual of [(17718, 21), 371917, 22]-NRT-code), using
(163−21, 163, 69916)-Net over F3 — Digital
Digital (142, 163, 69916)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3163, 69916, F3, 2, 21) (dual of [(69916, 2), 139669, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3163, 88594, F3, 2, 21) (dual of [(88594, 2), 177025, 22]-NRT-code), using
- strength reduction [i] based on linear OOA(3163, 88594, F3, 2, 22) (dual of [(88594, 2), 177025, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3163, 177188, F3, 22) (dual of [177188, 177025, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(3155, 177147, F3, 22) (dual of [177147, 176992, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3122, 177147, F3, 17) (dual of [177147, 177025, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(21) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(3163, 177188, F3, 22) (dual of [177188, 177025, 23]-code), using
- strength reduction [i] based on linear OOA(3163, 88594, F3, 2, 22) (dual of [(88594, 2), 177025, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3163, 88594, F3, 2, 21) (dual of [(88594, 2), 177025, 22]-NRT-code), using
(163−21, 163, large)-Net in Base 3 — Upper bound on s
There is no (142, 163, large)-net in base 3, because
- 19 times m-reduction [i] would yield (142, 144, large)-net in base 3, but