Best Known (215, 215+30, s)-Nets in Base 3
(215, 215+30, 35431)-Net over F3 — Constructive and digital
Digital (215, 245, 35431)-net over F3, using
- t-expansion [i] based on digital (214, 245, 35431)-net over F3, using
- net defined by OOA [i] based on linear OOA(3245, 35431, F3, 31, 31) (dual of [(35431, 31), 1098116, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3245, 531466, F3, 31) (dual of [531466, 531221, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3245, 531469, F3, 31) (dual of [531469, 531224, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(3241, 531441, F3, 31) (dual of [531441, 531200, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3217, 531441, F3, 28) (dual of [531441, 531224, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(30) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3245, 531469, F3, 31) (dual of [531469, 531224, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3245, 531466, F3, 31) (dual of [531466, 531221, 32]-code), using
- net defined by OOA [i] based on linear OOA(3245, 35431, F3, 31, 31) (dual of [(35431, 31), 1098116, 32]-NRT-code), using
(215, 215+30, 145575)-Net over F3 — Digital
Digital (215, 245, 145575)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3245, 145575, F3, 3, 30) (dual of [(145575, 3), 436480, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3245, 177156, F3, 3, 30) (dual of [(177156, 3), 531223, 31]-NRT-code), using
- strength reduction [i] based on linear OOA(3245, 177156, F3, 3, 31) (dual of [(177156, 3), 531223, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3245, 531468, F3, 31) (dual of [531468, 531223, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3245, 531469, F3, 31) (dual of [531469, 531224, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(27) [i] based on
- linear OA(3241, 531441, F3, 31) (dual of [531441, 531200, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3217, 531441, F3, 28) (dual of [531441, 531224, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(30) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3245, 531469, F3, 31) (dual of [531469, 531224, 32]-code), using
- OOA 3-folding [i] based on linear OA(3245, 531468, F3, 31) (dual of [531468, 531223, 32]-code), using
- strength reduction [i] based on linear OOA(3245, 177156, F3, 3, 31) (dual of [(177156, 3), 531223, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3245, 177156, F3, 3, 30) (dual of [(177156, 3), 531223, 31]-NRT-code), using
(215, 215+30, large)-Net in Base 3 — Upper bound on s
There is no (215, 245, large)-net in base 3, because
- 28 times m-reduction [i] would yield (215, 217, large)-net in base 3, but