Best Known (19, 19+7, s)-Nets in Base 3
(19, 19+7, 245)-Net over F3 — Constructive and digital
Digital (19, 26, 245)-net over F3, using
- net defined by OOA [i] based on linear OOA(326, 245, F3, 7, 7) (dual of [(245, 7), 1689, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(326, 736, F3, 7) (dual of [736, 710, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(325, 729, F3, 7) (dual of [729, 704, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(319, 729, F3, 5) (dual of [729, 710, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(31, 7, F3, 1) (dual of [7, 6, 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(6) ⊂ Ce(4) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(326, 736, F3, 7) (dual of [736, 710, 8]-code), using
(19, 19+7, 368)-Net over F3 — Digital
Digital (19, 26, 368)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(326, 368, F3, 2, 7) (dual of [(368, 2), 710, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(326, 736, F3, 7) (dual of [736, 710, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(325, 729, F3, 7) (dual of [729, 704, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(319, 729, F3, 5) (dual of [729, 710, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(31, 7, F3, 1) (dual of [7, 6, 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(6) ⊂ Ce(4) [i] based on
- OOA 2-folding [i] based on linear OA(326, 736, F3, 7) (dual of [736, 710, 8]-code), using
(19, 19+7, 8594)-Net in Base 3 — Upper bound on s
There is no (19, 26, 8595)-net in base 3, because
- 1 times m-reduction [i] would yield (19, 25, 8595)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 847335 192811 > 325 [i]