Best Known (237, 237+22, s)-Nets in Base 4
(237, 237+22, 3050400)-Net over F4 — Constructive and digital
Digital (237, 259, 3050400)-net over F4, using
- 43 times duplication [i] based on digital (234, 256, 3050400)-net over F4, using
- trace code for nets [i] based on digital (42, 64, 762600)-net over F256, using
- net defined by OOA [i] based on linear OOA(25664, 762600, F256, 22, 22) (dual of [(762600, 22), 16777136, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(25664, 8388600, F256, 22) (dual of [8388600, 8388536, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(25664, large, F256, 22) (dual of [large, large−64, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(25664, large, F256, 22) (dual of [large, large−64, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(25664, 8388600, F256, 22) (dual of [8388600, 8388536, 23]-code), using
- net defined by OOA [i] based on linear OOA(25664, 762600, F256, 22, 22) (dual of [(762600, 22), 16777136, 23]-NRT-code), using
- trace code for nets [i] based on digital (42, 64, 762600)-net over F256, using
(237, 237+22, large)-Net over F4 — Digital
Digital (237, 259, large)-net over F4, using
- 42 times duplication [i] based on digital (235, 257, large)-net over F4, using
- t-expansion [i] based on digital (231, 257, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4257, large, F4, 26) (dual of [large, large−257, 27]-code), using
- 28 times code embedding in larger space [i] based on linear OA(4229, large, F4, 26) (dual of [large, large−229, 27]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [i]
- 28 times code embedding in larger space [i] based on linear OA(4229, large, F4, 26) (dual of [large, large−229, 27]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4257, large, F4, 26) (dual of [large, large−257, 27]-code), using
- t-expansion [i] based on digital (231, 257, large)-net over F4, using
(237, 237+22, large)-Net in Base 4 — Upper bound on s
There is no (237, 259, large)-net in base 4, because
- 20 times m-reduction [i] would yield (237, 239, large)-net in base 4, but