Best Known (151−18, 151, s)-Nets in Base 3
(151−18, 151, 59053)-Net over F3 — Constructive and digital
Digital (133, 151, 59053)-net over F3, using
- net defined by OOA [i] based on linear OOA(3151, 59053, F3, 18, 18) (dual of [(59053, 18), 1062803, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(3151, 531477, F3, 18) (dual of [531477, 531326, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3151, 531483, F3, 18) (dual of [531483, 531332, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- 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(3109, 531441, F3, 14) (dual of [531441, 531332, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3151, 531483, F3, 18) (dual of [531483, 531332, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(3151, 531477, F3, 18) (dual of [531477, 531326, 19]-code), using
(151−18, 151, 177161)-Net over F3 — Digital
Digital (133, 151, 177161)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3151, 177161, F3, 3, 18) (dual of [(177161, 3), 531332, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3151, 531483, F3, 18) (dual of [531483, 531332, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- 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(3109, 531441, F3, 14) (dual of [531441, 531332, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(18) ⊂ Ce(13) [i] based on
- OOA 3-folding [i] based on linear OA(3151, 531483, F3, 18) (dual of [531483, 531332, 19]-code), using
(151−18, 151, large)-Net in Base 3 — Upper bound on s
There is no (133, 151, large)-net in base 3, because
- 16 times m-reduction [i] would yield (133, 135, large)-net in base 3, but