Best Known (212−26, 212, s)-Nets in Base 3
(212−26, 212, 40883)-Net over F3 — Constructive and digital
Digital (186, 212, 40883)-net over F3, using
- 31 times duplication [i] based on digital (185, 211, 40883)-net over F3, using
- net defined by OOA [i] based on linear OOA(3211, 40883, F3, 26, 26) (dual of [(40883, 26), 1062747, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(3211, 531479, F3, 26) (dual of [531479, 531268, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3211, 531483, F3, 26) (dual of [531483, 531272, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(3205, 531441, F3, 26) (dual of [531441, 531236, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3169, 531441, F3, 22) (dual of [531441, 531272, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(25) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3211, 531483, F3, 26) (dual of [531483, 531272, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(3211, 531479, F3, 26) (dual of [531479, 531268, 27]-code), using
- net defined by OOA [i] based on linear OOA(3211, 40883, F3, 26, 26) (dual of [(40883, 26), 1062747, 27]-NRT-code), using
(212−26, 212, 154318)-Net over F3 — Digital
Digital (186, 212, 154318)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3212, 154318, F3, 3, 26) (dual of [(154318, 3), 462742, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3212, 177161, F3, 3, 26) (dual of [(177161, 3), 531271, 27]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3211, 177161, F3, 3, 26) (dual of [(177161, 3), 531272, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3211, 531483, F3, 26) (dual of [531483, 531272, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(3205, 531441, F3, 26) (dual of [531441, 531236, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3169, 531441, F3, 22) (dual of [531441, 531272, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(25) ⊂ Ce(21) [i] based on
- OOA 3-folding [i] based on linear OA(3211, 531483, F3, 26) (dual of [531483, 531272, 27]-code), using
- 31 times duplication [i] based on linear OOA(3211, 177161, F3, 3, 26) (dual of [(177161, 3), 531272, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3212, 177161, F3, 3, 26) (dual of [(177161, 3), 531271, 27]-NRT-code), using
(212−26, 212, large)-Net in Base 3 — Upper bound on s
There is no (186, 212, large)-net in base 3, because
- 24 times m-reduction [i] would yield (186, 188, large)-net in base 3, but