Best Known (37, 37+11, s)-Nets in Base 64
(37, 37+11, 1679736)-Net over F64 — Constructive and digital
Digital (37, 48, 1679736)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 2016)-net over F64, using
- net defined by OOA [i] based on linear OOA(647, 2016, F64, 5, 5) (dual of [(2016, 5), 10073, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(647, 4033, F64, 5) (dual of [4033, 4026, 6]-code), using
- net defined by OOA [i] based on linear OOA(647, 2016, F64, 5, 5) (dual of [(2016, 5), 10073, 6]-NRT-code), using
- digital (30, 41, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6441, large, F64, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6441, 8388601, F64, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(6441, 1677720, F64, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- digital (2, 7, 2016)-net over F64, using
(37, 37+11, large)-Net over F64 — Digital
Digital (37, 48, large)-net over F64, using
- 2 times m-reduction [i] based on digital (37, 50, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6450, large, F64, 13) (dual of [large, large−50, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 1 times code embedding in larger space [i] based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6450, large, F64, 13) (dual of [large, large−50, 14]-code), using
(37, 37+11, large)-Net in Base 64 — Upper bound on s
There is no (37, 48, large)-net in base 64, because
- 9 times m-reduction [i] would yield (37, 39, large)-net in base 64, but