Best Known (55, 55+15, s)-Nets in Base 64
(55, 55+15, 1199736)-Net over F64 — Constructive and digital
Digital (55, 70, 1199736)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (6, 13, 1365)-net over F64, using
- net defined by OOA [i] based on linear OOA(6413, 1365, F64, 7, 7) (dual of [(1365, 7), 9542, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OOA 3-folding and stacking with additional row [i] based on linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using
- net defined by OOA [i] based on linear OOA(6413, 1365, F64, 7, 7) (dual of [(1365, 7), 9542, 8]-NRT-code), using
- 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
- digital (6, 13, 1365)-net over F64, using
(55, 55+15, large)-Net over F64 — Digital
Digital (55, 70, large)-net over F64, using
- t-expansion [i] based on digital (53, 70, large)-net over F64, using
- 1 times m-reduction [i] based on digital (53, 71, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6471, large, F64, 18) (dual of [large, large−71, 19]-code), using
- 2 times code embedding in larger space [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 2 times code embedding in larger space [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6471, large, F64, 18) (dual of [large, large−71, 19]-code), using
- 1 times m-reduction [i] based on digital (53, 71, large)-net over F64, using
(55, 55+15, large)-Net in Base 64 — Upper bound on s
There is no (55, 70, large)-net in base 64, because
- 13 times m-reduction [i] would yield (55, 57, large)-net in base 64, but