Best Known (45, 45+10, s)-Nets in Base 64
(45, 45+10, 3355440)-Net over F64 — Constructive and digital
Digital (45, 55, 3355440)-net over F64, using
- 641 times duplication [i] based on digital (44, 54, 3355440)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (12, 17, 4194301)-net over F64, using
- net defined by OOA [i] based on linear OOA(6417, 4194301, F64, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(6417, large, F64, 5) (dual of [large, large−17, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(6417, large, F64, 5) (dual of [large, large−17, 6]-code), using
- net defined by OOA [i] based on linear OOA(6417, 4194301, F64, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
- digital (27, 37, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- digital (12, 17, 4194301)-net over F64, using
- (u, u+v)-construction [i] based on
(45, 45+10, large)-Net over F64 — Digital
Digital (45, 55, large)-net over F64, using
- t-expansion [i] based on digital (44, 55, large)-net over F64, using
- 4 times m-reduction [i] based on digital (44, 59, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6459, large, F64, 15) (dual of [large, large−59, 16]-code), using
- 2 times code embedding in larger space [i] 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]
- 2 times code embedding in larger space [i] based on linear OA(6457, large, F64, 15) (dual of [large, large−57, 16]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6459, large, F64, 15) (dual of [large, large−59, 16]-code), using
- 4 times m-reduction [i] based on digital (44, 59, large)-net over F64, using
(45, 45+10, large)-Net in Base 64 — Upper bound on s
There is no (45, 55, large)-net in base 64, because
- 8 times m-reduction [i] would yield (45, 47, large)-net in base 64, but