Best Known (50−6, 50, s)-Nets in Base 3
(50−6, 50, 177155)-Net over F3 — Constructive and digital
Digital (44, 50, 177155)-net over F3, using
- net defined by OOA [i] based on linear OOA(350, 177155, F3, 6, 6) (dual of [(177155, 6), 1062880, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(350, 177155, F3, 5, 6) (dual of [(177155, 5), 885725, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(350, 531465, F3, 6) (dual of [531465, 531415, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(350, 531466, F3, 6) (dual of [531466, 531416, 7]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [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(325, 531441, F3, 4) (dual of [531441, 531416, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(31, 25, F3, 1) (dual of [25, 24, 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(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(350, 531466, F3, 6) (dual of [531466, 531416, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(350, 531465, F3, 6) (dual of [531465, 531415, 7]-code), using
- appending kth column [i] based on linear OOA(350, 177155, F3, 5, 6) (dual of [(177155, 5), 885725, 7]-NRT-code), using
(50−6, 50, 531467)-Net over F3 — Digital
Digital (44, 50, 531467)-net over F3, using
- net defined by OOA [i] based on linear OOA(350, 531467, F3, 6, 6) (dual of [(531467, 6), 3188752, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(350, 531467, F3, 5, 6) (dual of [(531467, 5), 2657285, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(350, 531467, F3, 6) (dual of [531467, 531417, 7]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(3) [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(325, 531441, F3, 4) (dual of [531441, 531416, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(325, 26, F3, 25) (dual of [26, 1, 26]-code or 26-arc in PG(24,3)), using
- dual of repetition code with length 26 [i]
- linear OA(31, 26, F3, 1) (dual of [26, 25, 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(3) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(350, 531467, F3, 6) (dual of [531467, 531417, 7]-code), using
- appending kth column [i] based on linear OOA(350, 531467, F3, 5, 6) (dual of [(531467, 5), 2657285, 7]-NRT-code), using
(50−6, 50, large)-Net in Base 3 — Upper bound on s
There is no (44, 50, large)-net in base 3, because
- 4 times m-reduction [i] would yield (44, 46, large)-net in base 3, but