Best Known (219, 219+20, s)-Nets in Base 3
(219, 219+20, 839297)-Net over F3 — Constructive and digital
Digital (219, 239, 839297)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (33, 43, 437)-net over F3, using
- net defined by OOA [i] based on linear OOA(343, 437, F3, 10, 10) (dual of [(437, 10), 4327, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(343, 2185, F3, 10) (dual of [2185, 2142, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using
- an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- discarding factors / shortening the dual code based on linear OA(343, 2187, F3, 10) (dual of [2187, 2144, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(343, 2185, F3, 10) (dual of [2185, 2142, 11]-code), using
- net defined by OOA [i] based on linear OOA(343, 437, F3, 10, 10) (dual of [(437, 10), 4327, 11]-NRT-code), using
- digital (176, 196, 838860)-net over F3, using
- net defined by OOA [i] based on linear OOA(3196, 838860, F3, 20, 20) (dual of [(838860, 20), 16777004, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3196, 8388600, F3, 20) (dual of [8388600, 8388404, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3196, large, F3, 20) (dual of [large, large−196, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(3196, large, F3, 20) (dual of [large, large−196, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3196, 8388600, F3, 20) (dual of [8388600, 8388404, 21]-code), using
- net defined by OOA [i] based on linear OOA(3196, 838860, F3, 20, 20) (dual of [(838860, 20), 16777004, 21]-NRT-code), using
- digital (33, 43, 437)-net over F3, using
(219, 219+20, 7686035)-Net over F3 — Digital
Digital (219, 239, 7686035)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3239, 7686035, F3, 20) (dual of [7686035, 7685796, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3239, large, F3, 20) (dual of [large, large−239, 21]-code), using
- 43 times code embedding in larger space [i] based on linear OA(3196, large, F3, 20) (dual of [large, large−196, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 315−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 43 times code embedding in larger space [i] based on linear OA(3196, large, F3, 20) (dual of [large, large−196, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3239, large, F3, 20) (dual of [large, large−239, 21]-code), using
(219, 219+20, large)-Net in Base 3 — Upper bound on s
There is no (219, 239, large)-net in base 3, because
- 18 times m-reduction [i] would yield (219, 221, large)-net in base 3, but