Best Known (49−6, 49, s)-Nets in Base 3
(49−6, 49, 177151)-Net over F3 — Constructive and digital
Digital (43, 49, 177151)-net over F3, using
- net defined by OOA [i] based on linear OOA(349, 177151, F3, 6, 6) (dual of [(177151, 6), 1062857, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(349, 177151, F3, 5, 6) (dual of [(177151, 5), 885706, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(349, 531453, F3, 6) (dual of [531453, 531404, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(349, 531454, F3, 6) (dual of [531454, 531405, 7]-code), using
- 1 times truncation [i] based on linear OA(350, 531455, F3, 7) (dual of [531455, 531405, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(349, 531441, F3, 7) (dual of [531441, 531392, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(337, 531441, F3, 5) (dual of [531441, 531404, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(313, 14, F3, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,3)), using
- dual of repetition code with length 14 [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 X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- 1 times truncation [i] based on linear OA(350, 531455, F3, 7) (dual of [531455, 531405, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(349, 531454, F3, 6) (dual of [531454, 531405, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(349, 531453, F3, 6) (dual of [531453, 531404, 7]-code), using
- appending kth column [i] based on linear OOA(349, 177151, F3, 5, 6) (dual of [(177151, 5), 885706, 7]-NRT-code), using
(49−6, 49, 531454)-Net over F3 — Digital
Digital (43, 49, 531454)-net over F3, using
- net defined by OOA [i] based on linear OOA(349, 531454, F3, 6, 6) (dual of [(531454, 6), 3188675, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(349, 531454, F3, 5, 6) (dual of [(531454, 5), 2657221, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(349, 531454, F3, 6) (dual of [531454, 531405, 7]-code), using
- 1 times truncation [i] based on linear OA(350, 531455, F3, 7) (dual of [531455, 531405, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(349, 531441, F3, 7) (dual of [531441, 531392, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(337, 531441, F3, 5) (dual of [531441, 531404, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(313, 14, F3, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,3)), using
- dual of repetition code with length 14 [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 X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- 1 times truncation [i] based on linear OA(350, 531455, F3, 7) (dual of [531455, 531405, 8]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(349, 531454, F3, 6) (dual of [531454, 531405, 7]-code), using
- appending kth column [i] based on linear OOA(349, 531454, F3, 5, 6) (dual of [(531454, 5), 2657221, 7]-NRT-code), using
(49−6, 49, large)-Net in Base 3 — Upper bound on s
There is no (43, 49, large)-net in base 3, because
- 4 times m-reduction [i] would yield (43, 45, large)-net in base 3, but