Best Known (206−28, 206, s)-Nets in Base 3
(206−28, 206, 12655)-Net over F3 — Constructive and digital
Digital (178, 206, 12655)-net over F3, using
- 33 times duplication [i] based on digital (175, 203, 12655)-net over F3, using
- net defined by OOA [i] based on linear OOA(3203, 12655, F3, 28, 28) (dual of [(12655, 28), 354137, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(3203, 177170, F3, 28) (dual of [177170, 176967, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3203, 177173, F3, 28) (dual of [177173, 176970, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(24) [i] based on
- 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(3177, 177147, F3, 25) (dual of [177147, 176970, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(34, 26, F3, 2) (dual of [26, 22, 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(27) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(3203, 177173, F3, 28) (dual of [177173, 176970, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(3203, 177170, F3, 28) (dual of [177170, 176967, 29]-code), using
- net defined by OOA [i] based on linear OOA(3203, 12655, F3, 28, 28) (dual of [(12655, 28), 354137, 29]-NRT-code), using
(206−28, 206, 53186)-Net over F3 — Digital
Digital (178, 206, 53186)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3206, 53186, F3, 3, 28) (dual of [(53186, 3), 159352, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3206, 59058, F3, 3, 28) (dual of [(59058, 3), 176968, 29]-NRT-code), using
- 32 times duplication [i] based on linear OOA(3204, 59058, F3, 3, 28) (dual of [(59058, 3), 176970, 29]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3204, 177174, F3, 28) (dual of [177174, 176970, 29]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3203, 177173, F3, 28) (dual of [177173, 176970, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(24) [i] based on
- 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(3177, 177147, F3, 25) (dual of [177147, 176970, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(34, 26, F3, 2) (dual of [26, 22, 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(27) ⊂ Ce(24) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3203, 177173, F3, 28) (dual of [177173, 176970, 29]-code), using
- OOA 3-folding [i] based on linear OA(3204, 177174, F3, 28) (dual of [177174, 176970, 29]-code), using
- 32 times duplication [i] based on linear OOA(3204, 59058, F3, 3, 28) (dual of [(59058, 3), 176970, 29]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3206, 59058, F3, 3, 28) (dual of [(59058, 3), 176968, 29]-NRT-code), using
(206−28, 206, large)-Net in Base 3 — Upper bound on s
There is no (178, 206, large)-net in base 3, because
- 26 times m-reduction [i] would yield (178, 180, large)-net in base 3, but