Best Known (212−29, 212, s)-Nets in Base 3
(212−29, 212, 12654)-Net over F3 — Constructive and digital
Digital (183, 212, 12654)-net over F3, using
- 32 times duplication [i] based on digital (181, 210, 12654)-net over F3, using
- net defined by OOA [i] based on linear OOA(3210, 12654, F3, 29, 29) (dual of [(12654, 29), 366756, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3210, 177157, F3, 29) (dual of [177157, 176947, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3210, 177158, F3, 29) (dual of [177158, 176948, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3210, 177147, F3, 29) (dual of [177147, 176937, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 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(28) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3210, 177158, F3, 29) (dual of [177158, 176948, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(3210, 177157, F3, 29) (dual of [177157, 176947, 30]-code), using
- net defined by OOA [i] based on linear OOA(3210, 12654, F3, 29, 29) (dual of [(12654, 29), 366756, 30]-NRT-code), using
(212−29, 212, 49559)-Net over F3 — Digital
Digital (183, 212, 49559)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3212, 49559, F3, 3, 29) (dual of [(49559, 3), 148465, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3212, 59053, F3, 3, 29) (dual of [(59053, 3), 176947, 30]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3211, 59053, F3, 3, 29) (dual of [(59053, 3), 176948, 30]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3211, 177159, F3, 29) (dual of [177159, 176948, 30]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3210, 177158, F3, 29) (dual of [177158, 176948, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3210, 177147, F3, 29) (dual of [177147, 176937, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 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(28) ⊂ Ce(27) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3210, 177158, F3, 29) (dual of [177158, 176948, 30]-code), using
- OOA 3-folding [i] based on linear OA(3211, 177159, F3, 29) (dual of [177159, 176948, 30]-code), using
- 31 times duplication [i] based on linear OOA(3211, 59053, F3, 3, 29) (dual of [(59053, 3), 176948, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3212, 59053, F3, 3, 29) (dual of [(59053, 3), 176947, 30]-NRT-code), using
(212−29, 212, large)-Net in Base 3 — Upper bound on s
There is no (183, 212, large)-net in base 3, because
- 27 times m-reduction [i] would yield (183, 185, large)-net in base 3, but