Best Known (227−34, 227, s)-Nets in Base 3
(227−34, 227, 3475)-Net over F3 — Constructive and digital
Digital (193, 227, 3475)-net over F3, using
- net defined by OOA [i] based on linear OOA(3227, 3475, F3, 34, 34) (dual of [(3475, 34), 117923, 35]-NRT-code), using
- OA 17-folding and stacking [i] based on linear OA(3227, 59075, F3, 34) (dual of [59075, 58848, 35]-code), using
- 2 times code embedding in larger space [i] based on linear OA(3225, 59073, F3, 34) (dual of [59073, 58848, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(30) [i] based on
- linear OA(3221, 59049, F3, 34) (dual of [59049, 58828, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3201, 59049, F3, 31) (dual of [59049, 58848, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 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(33) ⊂ Ce(30) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(3225, 59073, F3, 34) (dual of [59073, 58848, 35]-code), using
- OA 17-folding and stacking [i] based on linear OA(3227, 59075, F3, 34) (dual of [59075, 58848, 35]-code), using
(227−34, 227, 19691)-Net over F3 — Digital
Digital (193, 227, 19691)-net over F3, using
- 32 times duplication [i] based on digital (191, 225, 19691)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3225, 19691, F3, 3, 34) (dual of [(19691, 3), 58848, 35]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3225, 59073, F3, 34) (dual of [59073, 58848, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(30) [i] based on
- linear OA(3221, 59049, F3, 34) (dual of [59049, 58828, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3201, 59049, F3, 31) (dual of [59049, 58848, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 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(33) ⊂ Ce(30) [i] based on
- OOA 3-folding [i] based on linear OA(3225, 59073, F3, 34) (dual of [59073, 58848, 35]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3225, 19691, F3, 3, 34) (dual of [(19691, 3), 58848, 35]-NRT-code), using
(227−34, 227, large)-Net in Base 3 — Upper bound on s
There is no (193, 227, large)-net in base 3, because
- 32 times m-reduction [i] would yield (193, 195, large)-net in base 3, but