Best Known (212, 212+29, s)-Nets in Base 3
(212, 212+29, 37964)-Net over F3 — Constructive and digital
Digital (212, 241, 37964)-net over F3, using
- 31 times duplication [i] based on digital (211, 240, 37964)-net over F3, using
- net defined by OOA [i] based on linear OOA(3240, 37964, F3, 29, 29) (dual of [(37964, 29), 1100716, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3240, 531497, F3, 29) (dual of [531497, 531257, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3240, 531500, F3, 29) (dual of [531500, 531260, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(3229, 531441, F3, 29) (dual of [531441, 531212, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(311, 59, F3, 5) (dual of [59, 48, 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(28) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3240, 531500, F3, 29) (dual of [531500, 531260, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3240, 531497, F3, 29) (dual of [531497, 531257, 30]-code), using
- net defined by OOA [i] based on linear OOA(3240, 37964, F3, 29, 29) (dual of [(37964, 29), 1100716, 30]-NRT-code), using
(212, 212+29, 177167)-Net over F3 — Digital
Digital (212, 241, 177167)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3241, 177167, F3, 3, 29) (dual of [(177167, 3), 531260, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3241, 531501, F3, 29) (dual of [531501, 531260, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3240, 531500, F3, 29) (dual of [531500, 531260, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(22) [i] based on
- linear OA(3229, 531441, F3, 29) (dual of [531441, 531212, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- 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(311, 59, F3, 5) (dual of [59, 48, 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(28) ⊂ Ce(22) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3240, 531500, F3, 29) (dual of [531500, 531260, 30]-code), using
- OOA 3-folding [i] based on linear OA(3241, 531501, F3, 29) (dual of [531501, 531260, 30]-code), using
(212, 212+29, large)-Net in Base 3 — Upper bound on s
There is no (212, 241, large)-net in base 3, because
- 27 times m-reduction [i] would yield (212, 214, large)-net in base 3, but