Best Known (212, 212+32, s)-Nets in Base 3
(212, 212+32, 11075)-Net over F3 — Constructive and digital
Digital (212, 244, 11075)-net over F3, using
- 31 times duplication [i] based on digital (211, 243, 11075)-net over F3, using
- net defined by OOA [i] based on linear OOA(3243, 11075, F3, 32, 32) (dual of [(11075, 32), 354157, 33]-NRT-code), using
- OA 16-folding and stacking [i] based on linear OA(3243, 177200, F3, 32) (dual of [177200, 176957, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3243, 177202, F3, 32) (dual of [177202, 176959, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(25) [i] based on
- linear OA(3232, 177147, F3, 32) (dual of [177147, 176915, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [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(311, 55, F3, 5) (dual of [55, 44, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(31) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3243, 177202, F3, 32) (dual of [177202, 176959, 33]-code), using
- OA 16-folding and stacking [i] based on linear OA(3243, 177200, F3, 32) (dual of [177200, 176957, 33]-code), using
- net defined by OOA [i] based on linear OOA(3243, 11075, F3, 32, 32) (dual of [(11075, 32), 354157, 33]-NRT-code), using
(212, 212+32, 59067)-Net over F3 — Digital
Digital (212, 244, 59067)-net over F3, using
- 31 times duplication [i] based on digital (211, 243, 59067)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3243, 59067, F3, 3, 32) (dual of [(59067, 3), 176958, 33]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3243, 177201, F3, 32) (dual of [177201, 176958, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3243, 177202, F3, 32) (dual of [177202, 176959, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(25) [i] based on
- linear OA(3232, 177147, F3, 32) (dual of [177147, 176915, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [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(311, 55, F3, 5) (dual of [55, 44, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to Ce(31) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3243, 177202, F3, 32) (dual of [177202, 176959, 33]-code), using
- OOA 3-folding [i] based on linear OA(3243, 177201, F3, 32) (dual of [177201, 176958, 33]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3243, 59067, F3, 3, 32) (dual of [(59067, 3), 176958, 33]-NRT-code), using
(212, 212+32, large)-Net in Base 3 — Upper bound on s
There is no (212, 244, large)-net in base 3, because
- 30 times m-reduction [i] would yield (212, 214, large)-net in base 3, but