Best Known (172−20, 172, s)-Nets in Base 3
(172−20, 172, 159433)-Net over F3 — Constructive and digital
Digital (152, 172, 159433)-net over F3, using
- 32 times duplication [i] based on digital (150, 170, 159433)-net over F3, using
- net defined by OOA [i] based on linear OOA(3170, 159433, F3, 20, 20) (dual of [(159433, 20), 3188490, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3170, 1594330, F3, 20) (dual of [1594330, 1594160, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3170, 1594336, F3, 20) (dual of [1594336, 1594166, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3170, 1594323, F3, 20) (dual of [1594323, 1594153, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(3170, 1594336, F3, 20) (dual of [1594336, 1594166, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3170, 1594330, F3, 20) (dual of [1594330, 1594160, 21]-code), using
- net defined by OOA [i] based on linear OOA(3170, 159433, F3, 20, 20) (dual of [(159433, 20), 3188490, 21]-NRT-code), using
(172−20, 172, 398584)-Net over F3 — Digital
Digital (152, 172, 398584)-net over F3, using
- 32 times duplication [i] based on digital (150, 170, 398584)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3170, 398584, F3, 4, 20) (dual of [(398584, 4), 1594166, 21]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3170, 1594336, F3, 20) (dual of [1594336, 1594166, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3170, 1594323, F3, 20) (dual of [1594323, 1594153, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3157, 1594323, F3, 19) (dual of [1594323, 1594166, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- OOA 4-folding [i] based on linear OA(3170, 1594336, F3, 20) (dual of [1594336, 1594166, 21]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3170, 398584, F3, 4, 20) (dual of [(398584, 4), 1594166, 21]-NRT-code), using
(172−20, 172, large)-Net in Base 3 — Upper bound on s
There is no (152, 172, large)-net in base 3, because
- 18 times m-reduction [i] would yield (152, 154, large)-net in base 3, but