Best Known (166, 166+23, s)-Nets in Base 3
(166, 166+23, 48316)-Net over F3 — Constructive and digital
Digital (166, 189, 48316)-net over F3, using
- 32 times duplication [i] based on digital (164, 187, 48316)-net over F3, using
- net defined by OOA [i] based on linear OOA(3187, 48316, F3, 23, 23) (dual of [(48316, 23), 1111081, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3187, 531477, F3, 23) (dual of [531477, 531290, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3187, 531483, F3, 23) (dual of [531483, 531296, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(3181, 531441, F3, 23) (dual of [531441, 531260, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3145, 531441, F3, 19) (dual of [531441, 531296, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(36, 42, F3, 3) (dual of [42, 36, 4]-code or 42-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(22) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3187, 531483, F3, 23) (dual of [531483, 531296, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3187, 531477, F3, 23) (dual of [531477, 531290, 24]-code), using
- net defined by OOA [i] based on linear OOA(3187, 48316, F3, 23, 23) (dual of [(48316, 23), 1111081, 24]-NRT-code), using
(166, 166+23, 177161)-Net over F3 — Digital
Digital (166, 189, 177161)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3189, 177161, F3, 3, 23) (dual of [(177161, 3), 531294, 24]-NRT-code), using
- 32 times duplication [i] based on linear OOA(3187, 177161, F3, 3, 23) (dual of [(177161, 3), 531296, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3187, 531483, F3, 23) (dual of [531483, 531296, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(18) [i] based on
- linear OA(3181, 531441, F3, 23) (dual of [531441, 531260, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3145, 531441, F3, 19) (dual of [531441, 531296, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(36, 42, F3, 3) (dual of [42, 36, 4]-code or 42-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(22) ⊂ Ce(18) [i] based on
- OOA 3-folding [i] based on linear OA(3187, 531483, F3, 23) (dual of [531483, 531296, 24]-code), using
- 32 times duplication [i] based on linear OOA(3187, 177161, F3, 3, 23) (dual of [(177161, 3), 531296, 24]-NRT-code), using
(166, 166+23, large)-Net in Base 3 — Upper bound on s
There is no (166, 189, large)-net in base 3, because
- 21 times m-reduction [i] would yield (166, 168, large)-net in base 3, but