Best Known (223−18, 223, s)-Nets in Base 4
(223−18, 223, 3728286)-Net over F4 — Constructive and digital
Digital (205, 223, 3728286)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (6, 15, 18)-net over F4, using
- 3 times m-reduction [i] based on digital (6, 18, 18)-net over F4, using
- digital (190, 208, 3728268)-net over F4, using
- trace code for nets [i] based on digital (34, 52, 932067)-net over F256, using
- net defined by OOA [i] based on linear OOA(25652, 932067, F256, 18, 18) (dual of [(932067, 18), 16777154, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(25652, large, F256, 18) (dual of [large, large−52, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(25652, large, F256, 18) (dual of [large, large−52, 19]-code), using
- net defined by OOA [i] based on linear OOA(25652, 932067, F256, 18, 18) (dual of [(932067, 18), 16777154, 19]-NRT-code), using
- trace code for nets [i] based on digital (34, 52, 932067)-net over F256, using
- digital (6, 15, 18)-net over F4, using
(223−18, 223, large)-Net over F4 — Digital
Digital (205, 223, large)-net over F4, using
- t-expansion [i] based on digital (204, 223, large)-net over F4, using
- 4 times m-reduction [i] based on digital (204, 227, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4227, large, F4, 23) (dual of [large, large−227, 24]-code), using
- 22 times code embedding in larger space [i] based on linear OA(4205, large, F4, 23) (dual of [large, large−205, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- 22 times code embedding in larger space [i] based on linear OA(4205, large, F4, 23) (dual of [large, large−205, 24]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4227, large, F4, 23) (dual of [large, large−227, 24]-code), using
- 4 times m-reduction [i] based on digital (204, 227, large)-net over F4, using
(223−18, 223, large)-Net in Base 4 — Upper bound on s
There is no (205, 223, large)-net in base 4, because
- 16 times m-reduction [i] would yield (205, 207, large)-net in base 4, but