Best Known (187, 212, s)-Nets in Base 3
(187, 212, 132861)-Net over F3 — Constructive and digital
Digital (187, 212, 132861)-net over F3, using
- 32 times duplication [i] based on digital (185, 210, 132861)-net over F3, using
- net defined by OOA [i] based on linear OOA(3210, 132861, F3, 25, 25) (dual of [(132861, 25), 3321315, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3210, 1594333, F3, 25) (dual of [1594333, 1594123, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, 1594337, F3, 25) (dual of [1594337, 1594127, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(3209, 1594323, F3, 25) (dual of [1594323, 1594114, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3196, 1594323, F3, 23) (dual of [1594323, 1594127, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(31, 14, F3, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3210, 1594337, F3, 25) (dual of [1594337, 1594127, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(3210, 1594333, F3, 25) (dual of [1594333, 1594123, 26]-code), using
- net defined by OOA [i] based on linear OOA(3210, 132861, F3, 25, 25) (dual of [(132861, 25), 3321315, 26]-NRT-code), using
(187, 212, 380466)-Net over F3 — Digital
Digital (187, 212, 380466)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3212, 380466, F3, 4, 25) (dual of [(380466, 4), 1521652, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3212, 398584, F3, 4, 25) (dual of [(398584, 4), 1594124, 26]-NRT-code), using
- 32 times duplication [i] based on linear OOA(3210, 398584, F3, 4, 25) (dual of [(398584, 4), 1594126, 26]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3210, 1594336, F3, 25) (dual of [1594336, 1594126, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, 1594337, F3, 25) (dual of [1594337, 1594127, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(3209, 1594323, F3, 25) (dual of [1594323, 1594114, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3196, 1594323, F3, 23) (dual of [1594323, 1594127, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(31, 14, F3, 1) (dual of [14, 13, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3210, 1594337, F3, 25) (dual of [1594337, 1594127, 26]-code), using
- OOA 4-folding [i] based on linear OA(3210, 1594336, F3, 25) (dual of [1594336, 1594126, 26]-code), using
- 32 times duplication [i] based on linear OOA(3210, 398584, F3, 4, 25) (dual of [(398584, 4), 1594126, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3212, 398584, F3, 4, 25) (dual of [(398584, 4), 1594124, 26]-NRT-code), using
(187, 212, large)-Net in Base 3 — Upper bound on s
There is no (187, 212, large)-net in base 3, because
- 23 times m-reduction [i] would yield (187, 189, large)-net in base 3, but