Best Known (168, 168+20, s)-Nets in Base 3
(168, 168+20, 478300)-Net over F3 — Constructive and digital
Digital (168, 188, 478300)-net over F3, using
- 31 times duplication [i] based on digital (167, 187, 478300)-net over F3, using
- net defined by OOA [i] based on linear OOA(3187, 478300, F3, 20, 20) (dual of [(478300, 20), 9565813, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3187, 4783000, F3, 20) (dual of [4783000, 4782813, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3187, 4783001, F3, 20) (dual of [4783001, 4782814, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- 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(3155, 4782969, F3, 17) (dual of [4782969, 4782814, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(19) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3187, 4783001, F3, 20) (dual of [4783001, 4782814, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3187, 4783000, F3, 20) (dual of [4783000, 4782813, 21]-code), using
- net defined by OOA [i] based on linear OOA(3187, 478300, F3, 20, 20) (dual of [(478300, 20), 9565813, 21]-NRT-code), using
(168, 168+20, 1195750)-Net over F3 — Digital
Digital (168, 188, 1195750)-net over F3, using
- 31 times duplication [i] based on digital (167, 187, 1195750)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3187, 1195750, F3, 4, 20) (dual of [(1195750, 4), 4782813, 21]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3187, 4783000, F3, 20) (dual of [4783000, 4782813, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3187, 4783001, F3, 20) (dual of [4783001, 4782814, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- 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(3155, 4782969, F3, 17) (dual of [4782969, 4782814, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 314−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(19) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3187, 4783001, F3, 20) (dual of [4783001, 4782814, 21]-code), using
- OOA 4-folding [i] based on linear OA(3187, 4783000, F3, 20) (dual of [4783000, 4782813, 21]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3187, 1195750, F3, 4, 20) (dual of [(1195750, 4), 4782813, 21]-NRT-code), using
(168, 168+20, large)-Net in Base 3 — Upper bound on s
There is no (168, 188, large)-net in base 3, because
- 18 times m-reduction [i] would yield (168, 170, large)-net in base 3, but