Best Known (215−23, 215, s)-Nets in Base 3
(215−23, 215, 434818)-Net over F3 — Constructive and digital
Digital (192, 215, 434818)-net over F3, using
- net defined by OOA [i] based on linear OOA(3215, 434818, F3, 23, 23) (dual of [(434818, 23), 10000599, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3215, 4782999, F3, 23) (dual of [4782999, 4782784, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3215, 4783001, F3, 23) (dual of [4783001, 4782786, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(3211, 4782969, F3, 23) (dual of [4782969, 4782758, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3183, 4782969, F3, 20) (dual of [4782969, 4782786, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(34, 32, F3, 2) (dual of [32, 28, 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(22) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3215, 4783001, F3, 23) (dual of [4783001, 4782786, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3215, 4782999, F3, 23) (dual of [4782999, 4782784, 24]-code), using
(215−23, 215, 1195750)-Net over F3 — Digital
Digital (192, 215, 1195750)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3215, 1195750, F3, 4, 23) (dual of [(1195750, 4), 4782785, 24]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3215, 4783000, F3, 23) (dual of [4783000, 4782785, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3215, 4783001, F3, 23) (dual of [4783001, 4782786, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(3211, 4782969, F3, 23) (dual of [4782969, 4782758, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3183, 4782969, F3, 20) (dual of [4782969, 4782786, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(34, 32, F3, 2) (dual of [32, 28, 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(22) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3215, 4783001, F3, 23) (dual of [4783001, 4782786, 24]-code), using
- OOA 4-folding [i] based on linear OA(3215, 4783000, F3, 23) (dual of [4783000, 4782785, 24]-code), using
(215−23, 215, large)-Net in Base 3 — Upper bound on s
There is no (192, 215, large)-net in base 3, because
- 21 times m-reduction [i] would yield (192, 194, large)-net in base 3, but