Best Known (234−15, 234, s)-Nets in Base 4
(234−15, 234, 7589684)-Net over F4 — Constructive and digital
Digital (219, 234, 7589684)-net over F4, using
- 41 times duplication [i] based on digital (218, 233, 7589684)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (54, 61, 2796200)-net over F4, using
- net defined by OOA [i] based on linear OOA(461, 2796200, F4, 7, 7) (dual of [(2796200, 7), 19573339, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(461, 8388601, F4, 7) (dual of [8388601, 8388540, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(461, large, F4, 7) (dual of [large, large−61, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(461, large, F4, 7) (dual of [large, large−61, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(461, 8388601, F4, 7) (dual of [8388601, 8388540, 8]-code), using
- net defined by OOA [i] based on linear OOA(461, 2796200, F4, 7, 7) (dual of [(2796200, 7), 19573339, 8]-NRT-code), using
- digital (157, 172, 4793484)-net over F4, using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- digital (54, 61, 2796200)-net over F4, using
- (u, u+v)-construction [i] based on
(234−15, 234, large)-Net over F4 — Digital
Digital (219, 234, large)-net over F4, using
- t-expansion [i] based on digital (213, 234, large)-net over F4, using
- 3 times m-reduction [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
- 3 times m-reduction [i] based on digital (213, 237, large)-net over F4, using
(234−15, 234, large)-Net in Base 4 — Upper bound on s
There is no (219, 234, large)-net in base 4, because
- 13 times m-reduction [i] would yield (219, 221, large)-net in base 4, but