Best Known (221−14, 221, s)-Nets in Base 4
(221−14, 221, 7589684)-Net over F4 — Constructive and digital
Digital (207, 221, 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 (146, 160, 4793484)-net over F4, using
- trace code for nets [i] based on digital (26, 40, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(25640, large, F256, 14) (dual of [large, large−40, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(25640, 8388597, F256, 14) (dual of [8388597, 8388557, 15]-code), using
- net defined by OOA [i] based on linear OOA(25640, 1198371, F256, 14, 14) (dual of [(1198371, 14), 16777154, 15]-NRT-code), using
- trace code for nets [i] based on digital (26, 40, 1198371)-net over F256, using
- digital (54, 61, 2796200)-net over F4, using
(221−14, 221, large)-Net over F4 — Digital
Digital (207, 221, large)-net over F4, using
- t-expansion [i] based on digital (204, 221, large)-net over F4, using
- 6 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
- 6 times m-reduction [i] based on digital (204, 227, large)-net over F4, using
(221−14, 221, large)-Net in Base 4 — Upper bound on s
There is no (207, 221, large)-net in base 4, because
- 12 times m-reduction [i] would yield (207, 209, large)-net in base 4, but