Best Known (219, 219+22, s)-Nets in Base 4
(219, 219+22, 763632)-Net over F4 — Constructive and digital
Digital (219, 241, 763632)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (37, 48, 1032)-net over F4, using
- trace code for nets [i] based on digital (1, 12, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 12, 258)-net over F256, using
- digital (171, 193, 762600)-net over F4, using
- net defined by OOA [i] based on linear OOA(4193, 762600, F4, 22, 22) (dual of [(762600, 22), 16777007, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4193, 8388600, F4, 22) (dual of [8388600, 8388407, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4193, large, F4, 22) (dual of [large, large−193, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(4193, large, F4, 22) (dual of [large, large−193, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(4193, 8388600, F4, 22) (dual of [8388600, 8388407, 23]-code), using
- net defined by OOA [i] based on linear OOA(4193, 762600, F4, 22, 22) (dual of [(762600, 22), 16777007, 23]-NRT-code), using
- digital (37, 48, 1032)-net over F4, using
(219, 219+22, large)-Net over F4 — Digital
Digital (219, 241, large)-net over F4, using
- 44 times duplication [i] based on digital (215, 237, large)-net over F4, using
- t-expansion [i] based on digital (213, 237, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4237, large, F4, 24) (dual of [large, large−237, 25]-code), using
- 21 times code embedding in larger space [i] based on linear OA(4216, large, F4, 24) (dual of [large, large−216, 25]-code), using
- the primitive narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- 21 times code embedding in larger space [i] based on linear OA(4216, large, F4, 24) (dual of [large, large−216, 25]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4237, large, F4, 24) (dual of [large, large−237, 25]-code), using
- t-expansion [i] based on digital (213, 237, large)-net over F4, using
(219, 219+22, large)-Net in Base 4 — Upper bound on s
There is no (219, 241, large)-net in base 4, because
- 20 times m-reduction [i] would yield (219, 221, large)-net in base 4, but