Best Known (205, 205+29, s)-Nets in Base 3
(205, 205+29, 37962)-Net over F3 — Constructive and digital
Digital (205, 234, 37962)-net over F3, using
- 31 times duplication [i] based on digital (204, 233, 37962)-net over F3, using
- net defined by OOA [i] based on linear OOA(3233, 37962, F3, 29, 29) (dual of [(37962, 29), 1100665, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3233, 531469, F3, 29) (dual of [531469, 531236, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [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(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(34, 28, F3, 2) (dual of [28, 24, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- OOA 14-folding and stacking with additional row [i] based on linear OA(3233, 531469, F3, 29) (dual of [531469, 531236, 30]-code), using
- net defined by OOA [i] based on linear OOA(3233, 37962, F3, 29, 29) (dual of [(37962, 29), 1100665, 30]-NRT-code), using
(205, 205+29, 132867)-Net over F3 — Digital
Digital (205, 234, 132867)-net over F3, using
- 31 times duplication [i] based on digital (204, 233, 132867)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3233, 132867, F3, 4, 29) (dual of [(132867, 4), 531235, 30]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3233, 531468, F3, 29) (dual of [531468, 531235, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3233, 531469, F3, 29) (dual of [531469, 531236, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [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(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(34, 28, F3, 2) (dual of [28, 24, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(28) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3233, 531469, F3, 29) (dual of [531469, 531236, 30]-code), using
- OOA 4-folding [i] based on linear OA(3233, 531468, F3, 29) (dual of [531468, 531235, 30]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3233, 132867, F3, 4, 29) (dual of [(132867, 4), 531235, 30]-NRT-code), using
(205, 205+29, large)-Net in Base 3 — Upper bound on s
There is no (205, 234, large)-net in base 3, because
- 27 times m-reduction [i] would yield (205, 207, large)-net in base 3, but