Best Known (232−15, 232, s)-Nets in Base 4
(232−15, 232, 6391313)-Net over F4 — Constructive and digital
Digital (217, 232, 6391313)-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 (156, 171, 3595113)-net over F4, using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F64, using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(6457, 8388598, F64, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(6457, 1198371, F64, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- trace code for nets [i] based on digital (42, 57, 1198371)-net over F64, using
- digital (54, 61, 2796200)-net over F4, using
(232−15, 232, large)-Net over F4 — Digital
Digital (217, 232, large)-net over F4, using
- t-expansion [i] based on digital (213, 232, large)-net over F4, using
- 5 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
- 5 times m-reduction [i] based on digital (213, 237, large)-net over F4, using
(232−15, 232, large)-Net in Base 4 — Upper bound on s
There is no (217, 232, large)-net in base 4, because
- 13 times m-reduction [i] would yield (217, 219, large)-net in base 4, but