Best Known (239−18, 239, s)-Nets in Base 4
(239−18, 239, 3728524)-Net over F4 — Constructive and digital
Digital (221, 239, 3728524)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (22, 31, 256)-net over F4, using
- net defined by OOA [i] based on linear OOA(431, 256, F4, 9, 9) (dual of [(256, 9), 2273, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(431, 1025, F4, 9) (dual of [1025, 994, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1025 | 410−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(431, 1025, F4, 9) (dual of [1025, 994, 10]-code), using
- net defined by OOA [i] based on linear OOA(431, 256, F4, 9, 9) (dual of [(256, 9), 2273, 10]-NRT-code), 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 (22, 31, 256)-net over F4, using
(239−18, 239, large)-Net over F4 — Digital
Digital (221, 239, large)-net over F4, using
- 42 times duplication [i] based on digital (219, 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
(239−18, 239, large)-Net in Base 4 — Upper bound on s
There is no (221, 239, large)-net in base 4, because
- 16 times m-reduction [i] would yield (221, 223, large)-net in base 4, but