Best Known (44, 44+7, s)-Nets in Base 3
(44, 44+7, 177151)-Net over F3 — Constructive and digital
Digital (44, 51, 177151)-net over F3, using
- 31 times duplication [i] based on digital (43, 50, 177151)-net over F3, using
- net defined by OOA [i] based on linear OOA(350, 177151, F3, 7, 7) (dual of [(177151, 7), 1240007, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(350, 531454, F3, 7) (dual of [531454, 531404, 8]-code), using
- construction X 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(31, 13, F3, 1) (dual of [13, 12, 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(350, 531454, F3, 7) (dual of [531454, 531404, 8]-code), using
- net defined by OOA [i] based on linear OOA(350, 177151, F3, 7, 7) (dual of [(177151, 7), 1240007, 8]-NRT-code), using
(44, 44+7, 265728)-Net over F3 — Digital
Digital (44, 51, 265728)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(351, 265728, F3, 2, 7) (dual of [(265728, 2), 531405, 8]-NRT-code), using
- OOA 2-folding [i] based on linear OA(351, 531456, F3, 7) (dual of [531456, 531405, 8]-code), using
- 1 times code embedding in larger space [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 code embedding in larger space [i] based on linear OA(350, 531455, F3, 7) (dual of [531455, 531405, 8]-code), using
- OOA 2-folding [i] based on linear OA(351, 531456, F3, 7) (dual of [531456, 531405, 8]-code), using
(44, 44+7, large)-Net in Base 3 — Upper bound on s
There is no (44, 51, large)-net in base 3, because
- 5 times m-reduction [i] would yield (44, 46, large)-net in base 3, but