Best Known (163, 163+23, s)-Nets in Base 3
(163, 163+23, 48315)-Net over F3 — Constructive and digital
Digital (163, 186, 48315)-net over F3, using
- 31 times duplication [i] based on digital (162, 185, 48315)-net over F3, using
- net defined by OOA [i] based on linear OOA(3185, 48315, F3, 23, 23) (dual of [(48315, 23), 1111060, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3185, 531466, F3, 23) (dual of [531466, 531281, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3185, 531469, F3, 23) (dual of [531469, 531284, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- 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(3157, 531441, F3, 20) (dual of [531441, 531284, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(22) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3185, 531469, F3, 23) (dual of [531469, 531284, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3185, 531466, F3, 23) (dual of [531466, 531281, 24]-code), using
- net defined by OOA [i] based on linear OOA(3185, 48315, F3, 23, 23) (dual of [(48315, 23), 1111060, 24]-NRT-code), using
(163, 163+23, 156159)-Net over F3 — Digital
Digital (163, 186, 156159)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3186, 156159, F3, 3, 23) (dual of [(156159, 3), 468291, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3186, 177156, F3, 3, 23) (dual of [(177156, 3), 531282, 24]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3185, 177156, F3, 3, 23) (dual of [(177156, 3), 531283, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3185, 531468, F3, 23) (dual of [531468, 531283, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3185, 531469, F3, 23) (dual of [531469, 531284, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- 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(3157, 531441, F3, 20) (dual of [531441, 531284, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(22) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3185, 531469, F3, 23) (dual of [531469, 531284, 24]-code), using
- OOA 3-folding [i] based on linear OA(3185, 531468, F3, 23) (dual of [531468, 531283, 24]-code), using
- 31 times duplication [i] based on linear OOA(3185, 177156, F3, 3, 23) (dual of [(177156, 3), 531283, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3186, 177156, F3, 3, 23) (dual of [(177156, 3), 531282, 24]-NRT-code), using
(163, 163+23, large)-Net in Base 3 — Upper bound on s
There is no (163, 186, large)-net in base 3, because
- 21 times m-reduction [i] would yield (163, 165, large)-net in base 3, but