Best Known (210−17, 210, s)-Nets in Base 3
(210−17, 210, 1050218)-Net over F3 — Constructive and digital
Digital (193, 210, 1050218)-net over F3, using
- 31 times duplication [i] based on digital (192, 209, 1050218)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (35, 43, 1643)-net over F3, using
- net defined by OOA [i] based on linear OOA(343, 1643, F3, 8, 8) (dual of [(1643, 8), 13101, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(343, 6572, F3, 8) (dual of [6572, 6529, 9]-code), using
- 1 times code embedding in larger space [i] based on linear OA(342, 6571, F3, 8) (dual of [6571, 6529, 9]-code), using
- construction X4 applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(341, 6561, F3, 8) (dual of [6561, 6520, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(333, 6561, F3, 7) (dual of [6561, 6528, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(39, 10, F3, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,3)), using
- dual of repetition code with length 10 [i]
- linear OA(31, 10, F3, 1) (dual of [10, 9, 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(7) ⊂ Ce(6) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(342, 6571, F3, 8) (dual of [6571, 6529, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(343, 6572, F3, 8) (dual of [6572, 6529, 9]-code), using
- net defined by OOA [i] based on linear OOA(343, 1643, F3, 8, 8) (dual of [(1643, 8), 13101, 9]-NRT-code), using
- digital (149, 166, 1048575)-net over F3, using
- net defined by OOA [i] based on linear OOA(3166, 1048575, F3, 17, 17) (dual of [(1048575, 17), 17825609, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3166, 8388601, F3, 17) (dual of [8388601, 8388435, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3166, 8388601, F3, 17) (dual of [8388601, 8388435, 18]-code), using
- net defined by OOA [i] based on linear OOA(3166, 1048575, F3, 17, 17) (dual of [(1048575, 17), 17825609, 18]-NRT-code), using
- digital (35, 43, 1643)-net over F3, using
- (u, u+v)-construction [i] based on
(210−17, 210, large)-Net over F3 — Digital
Digital (193, 210, large)-net over F3, using
- 37 times duplication [i] based on digital (186, 203, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3203, large, F3, 17) (dual of [large, large−203, 18]-code), using
- 37 times code embedding in larger space [i] based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 37 times code embedding in larger space [i] based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3203, large, F3, 17) (dual of [large, large−203, 18]-code), using
(210−17, 210, large)-Net in Base 3 — Upper bound on s
There is no (193, 210, large)-net in base 3, because
- 15 times m-reduction [i] would yield (193, 195, large)-net in base 3, but