Best Known (198, 198+30, s)-Nets in Base 3
(198, 198+30, 11812)-Net over F3 — Constructive and digital
Digital (198, 228, 11812)-net over F3, using
- 1 times m-reduction [i] based on digital (198, 229, 11812)-net over F3, using
- net defined by OOA [i] based on linear OOA(3229, 11812, F3, 31, 31) (dual of [(11812, 31), 365943, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3229, 177181, F3, 31) (dual of [177181, 176952, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3229, 177188, F3, 31) (dual of [177188, 176959, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] 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]
- linear OA(3188, 177147, F3, 26) (dual of [177147, 176959, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(30) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3229, 177188, F3, 31) (dual of [177188, 176959, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3229, 177181, F3, 31) (dual of [177181, 176952, 32]-code), using
- net defined by OOA [i] based on linear OOA(3229, 11812, F3, 31, 31) (dual of [(11812, 31), 365943, 32]-NRT-code), using
(198, 198+30, 59062)-Net over F3 — Digital
Digital (198, 228, 59062)-net over F3, using
- 31 times duplication [i] based on digital (197, 227, 59062)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3227, 59062, F3, 3, 30) (dual of [(59062, 3), 176959, 31]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3227, 177186, F3, 30) (dual of [177186, 176959, 31]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] 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]
- linear OA(3188, 177147, F3, 26) (dual of [177147, 176959, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(30) ⊂ Ce(25) [i] based on
- OOA 3-folding [i] based on linear OA(3227, 177186, F3, 30) (dual of [177186, 176959, 31]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3227, 59062, F3, 3, 30) (dual of [(59062, 3), 176959, 31]-NRT-code), using
(198, 198+30, large)-Net in Base 3 — Upper bound on s
There is no (198, 228, large)-net in base 3, because
- 28 times m-reduction [i] would yield (198, 200, large)-net in base 3, but