Best Known (197−17, 197, s)-Nets in Base 3
(197−17, 197, 1048758)-Net over F3 — Constructive and digital
Digital (180, 197, 1048758)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (23, 31, 183)-net over F3, using
- net defined by OOA [i] based on linear OOA(331, 183, F3, 8, 8) (dual of [(183, 8), 1433, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(331, 732, F3, 8) (dual of [732, 701, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(331, 735, F3, 8) (dual of [735, 704, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(331, 729, F3, 8) (dual of [729, 698, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(325, 729, F3, 7) (dual of [729, 704, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(331, 735, F3, 8) (dual of [735, 704, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(331, 732, F3, 8) (dual of [732, 701, 9]-code), using
- net defined by OOA [i] based on linear OOA(331, 183, F3, 8, 8) (dual of [(183, 8), 1433, 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 (23, 31, 183)-net over F3, using
(197−17, 197, 5509598)-Net over F3 — Digital
Digital (180, 197, 5509598)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3197, 5509598, F3, 17) (dual of [5509598, 5509401, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3197, large, F3, 17) (dual of [large, large−197, 18]-code), using
- 31 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]
- 31 times code embedding in larger space [i] based on linear OA(3166, large, F3, 17) (dual of [large, large−166, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3197, large, F3, 17) (dual of [large, large−197, 18]-code), using
(197−17, 197, large)-Net in Base 3 — Upper bound on s
There is no (180, 197, large)-net in base 3, because
- 15 times m-reduction [i] would yield (180, 182, large)-net in base 3, but