Best Known (73−9, 73, s)-Nets in Base 3
(73−9, 73, 132863)-Net over F3 — Constructive and digital
Digital (64, 73, 132863)-net over F3, using
- net defined by OOA [i] based on linear OOA(373, 132863, F3, 9, 9) (dual of [(132863, 9), 1195694, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(373, 531453, F3, 9) (dual of [531453, 531380, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(373, 531454, F3, 9) (dual of [531454, 531381, 10]-code), using
- 1 times truncation [i] based on linear OA(374, 531455, F3, 10) (dual of [531455, 531381, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(373, 531441, F3, 10) (dual of [531441, 531368, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(361, 531441, F3, 8) (dual of [531441, 531380, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(9) ⊂ Ce(7) [i] based on
- 1 times truncation [i] based on linear OA(374, 531455, F3, 10) (dual of [531455, 531381, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(373, 531454, F3, 9) (dual of [531454, 531381, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(373, 531453, F3, 9) (dual of [531453, 531380, 10]-code), using
(73−9, 73, 265727)-Net over F3 — Digital
Digital (64, 73, 265727)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(373, 265727, F3, 2, 9) (dual of [(265727, 2), 531381, 10]-NRT-code), using
- OOA 2-folding [i] based on linear OA(373, 531454, F3, 9) (dual of [531454, 531381, 10]-code), using
- 1 times truncation [i] based on linear OA(374, 531455, F3, 10) (dual of [531455, 531381, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- linear OA(373, 531441, F3, 10) (dual of [531441, 531368, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(361, 531441, F3, 8) (dual of [531441, 531380, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(9) ⊂ Ce(7) [i] based on
- 1 times truncation [i] based on linear OA(374, 531455, F3, 10) (dual of [531455, 531381, 11]-code), using
- OOA 2-folding [i] based on linear OA(373, 531454, F3, 9) (dual of [531454, 531381, 10]-code), using
(73−9, 73, large)-Net in Base 3 — Upper bound on s
There is no (64, 73, large)-net in base 3, because
- 7 times m-reduction [i] would yield (64, 66, large)-net in base 3, but